From 9bedfd5aaa69224041064d5d4f315bbf87c8c31c Mon Sep 17 00:00:00 2001 From: Michael Zhang Date: Thu, 23 May 2024 21:24:38 -0500 Subject: [PATCH] dissertation source --- .../VanDoornDissertation/amsalphaurl.bst | 1726 +++++ .../VanDoornDissertation/dissertation.bbl | 395 + .../VanDoornDissertation/dissertation.pdf | Bin 0 -> 1328373 bytes .../VanDoornDissertation/dissertation.tex | 6598 +++++++++++++++++ resources/VanDoornDissertation/lstlean.tex | 293 + resources/VanDoornDissertation/macros.tex | 783 ++ 6 files changed, 9795 insertions(+) create mode 100644 resources/VanDoornDissertation/amsalphaurl.bst create mode 100644 resources/VanDoornDissertation/dissertation.bbl create mode 100644 resources/VanDoornDissertation/dissertation.pdf create mode 100644 resources/VanDoornDissertation/dissertation.tex create mode 100644 resources/VanDoornDissertation/lstlean.tex create mode 100644 resources/VanDoornDissertation/macros.tex diff --git a/resources/VanDoornDissertation/amsalphaurl.bst b/resources/VanDoornDissertation/amsalphaurl.bst new file mode 100644 index 0000000..1722838 --- /dev/null +++ b/resources/VanDoornDissertation/amsalphaurl.bst @@ -0,0 +1,1726 @@ +%%% Modification of BibTeX style file /usr/local/texlive/2008/texmf-dist/bibtex/bst/ams/amsalpha.bst +%%% ... by urlbst, version 0.6 (marked with "% urlbst") +%%% See +%%% Added webpage entry type, and url and lastchecked fields. +%%% Added eprint support. +%%% Added DOI support. +%%% Added hyperref support. +%%% Original headers follow... + +%%% ==================================================================== +%%% @BibTeX-style-file{ +%%% filename = "amsalpha.bst", +%%% version = "2.0", +%%% date = "2000/03/27", +%%% time = "13:49:36 EST", +%%% checksum = "00166 1404 4124 29978", +%%% author = "American Mathematical Society", +%%% address = "American Mathematical Society, +%%% Technical Support, +%%% Electronic Products and Services, +%%% P. O. Box 6248, +%%% Providence, RI 02940, +%%% USA", +%%% telephone = "401-455-4080 or (in the USA and Canada) +%%% 800-321-4AMS (321-4267)", +%%% FAX = "401-331-3842", +%%% email = "tech-support@ams.org (Internet)", +%%% copyright = "Copyright 1995 American Mathematical Society, +%%% all rights reserved. Copying of this file is +%%% authorized only if either: +%%% (1) you make absolutely no changes to your copy, +%%% including name; OR +%%% (2) if you do make changes, you first rename it +%%% to some other name.", +%%% codetable = "ISO/ASCII", +%%% keywords = "bibtex, bibliography, amslatex, ams-latex", +%%% supported = "yes", +%%% abstract = "BibTeX bibliography style `amsalpha' for BibTeX +%%% versions 0.99a or later and LaTeX version 2e. +%%% Produces alphabetic-label bibliography items in +%%% a form typical for American Mathematical Society +%%% publications.", +%%% docstring = "The checksum field above contains a CRC-16 +%%% checksum as the first value, followed by the +%%% equivalent of the standard UNIX wc (word +%%% count) utility output of lines, words, and +%%% characters. This is produced by Robert +%%% Solovay's checksum utility.", +%%% } +%%% ==================================================================== + +% See the file btxbst.doc for extra documentation other than +% what is included here. And see btxhak.tex for a description +% of the BibTeX language and how to use it. + +% This defines the types of fields that can occur in a database entry +% for this particular bibliography style. Except for `language', +% this is the standard list from alpha.bst. + +%% Types of entries currently allowed in a BibTeX file: +%% +%% ARTICLE -- An article from a journal or magazine. +%% +%% BOOK -- A book with an explicit publisher. +%% +%% BOOKLET -- A work that is printed and bound, +%% but without a named publisher or sponsoring institution. +%% +%% CONFERENCE -- The same as INPROCEEDINGS, +%% included for Scribe compatibility. +%% +%% INBOOK -- A part of a book, +%% which may be a chapter (or section or whatever) and/or a range of pages. +%% +%% INCOLLECTION -- A part of a book having its own title. +%% +%% INPROCEEDINGS -- An article in a conference proceedings. +%% +%% MANUAL -- Technical documentation. +%% +%% MASTERSTHESIS -- A Master's thesis. +%% +%% MISC -- Use this type when nothing else fits. +%% +%% PHDTHESIS -- A PhD thesis. +%% +%% PROCEEDINGS -- The proceedings of a conference. +%% +%% TECHREPORT -- A report published by a school or other institution, +%% usually numbered within a series. +%% +%% UNPUBLISHED -- A document having an author and title, but not formally +%% published. + +ENTRY + { address + author + booktitle + chapter + edition + editor + howpublished + institution + journal + key + language + month + mrnumber + note + number + organization + pages + publisher + school + series + title + type + volume + year + eprint % urlbst + doi % urlbst + url % urlbst + lastchecked % urlbst + } + {} + { label extra.label sort.label bysame } + +% Removed after.sentence, after.block---not needed. + +INTEGERS { output.state before.all mid.sentence } + +STRINGS { urlintro eprinturl eprintprefix doiprefix doiurl openinlinelink closeinlinelink } % urlbst... +INTEGERS { hrefform inlinelinks makeinlinelink addeprints adddoiresolver } +% Following constants may be adjusted by hand, if desired +FUNCTION {init.urlbst.variables} +{ + "" 'urlintro := % prefix before URL + "http://arxiv.org/abs/" 'eprinturl := % prefix to make URL from eprint ref + "arXiv:" 'eprintprefix := % text prefix printed before eprint ref + "http://dx.doi.org/" 'doiurl := % prefix to make URL from DOI + "doi:" 'doiprefix := % text prefix printed before DOI ref + #1 'addeprints := % 0=no eprints; 1=include eprints + #1 'adddoiresolver := % 0=no DOI resolver; 1=include it + #2 'hrefform := % 0=no crossrefs; 1=hypertex xrefs; 2=hyperref refs + #0 'inlinelinks := % 0=URLs explicit; 1=URLs attached to titles + % the following are internal state variables, not config constants + #0 'makeinlinelink := % state variable managed by setup.inlinelink + "" 'openinlinelink := % ditto + "" 'closeinlinelink := % ditto +} +INTEGERS { + bracket.state + outside.brackets + open.brackets + within.brackets + close.brackets +} +% ...urlbst to here +FUNCTION {init.state.consts} +{ #0 'outside.brackets := % urlbst + #1 'open.brackets := + #2 'within.brackets := + #3 'close.brackets := + + #0 'before.all := + #1 'mid.sentence := +} + +% Scratch variables: + +STRINGS { s t } + +% Utility functions + +FUNCTION {shows} +{ duplicate$ ":::: `" swap$ * "'" * top$ +} + +FUNCTION {showstack} +{"STACK=====================================================================" +top$ +stack$ +"ENDSTACK==================================================================" +top$ +} + +FUNCTION {not} +{ { #0 } + { #1 } + if$ +} + +FUNCTION {and} +{ 'skip$ + { pop$ #0 } + if$ +} + +FUNCTION {or} +{ { pop$ #1 } + 'skip$ + if$ +} + +FUNCTION {field.or.null} +{ duplicate$ empty$ + { pop$ "" } + 'skip$ + if$ +} + +FUNCTION {emphasize} +{ duplicate$ empty$ + { pop$ "" } + { "\emph{" swap$ * "}" * } + if$ +} + +% n.dashify is used to make sure page ranges get the TeX code +% (two hyphens) for en-dashes. + +FUNCTION {n.dashify} +{ 't := + "" + { t empty$ not } + { t #1 #1 substring$ "-" = + { t #1 #2 substring$ "--" = not + { "--" * + t #2 global.max$ substring$ 't := + } + { { t #1 #1 substring$ "-" = } + { "-" * + t #2 global.max$ substring$ 't := + } + while$ + } + if$ + } + { t #1 #1 substring$ * + t #2 global.max$ substring$ 't := + } + if$ + } + while$ +} + +% tie.or.space.connect connects two items with a ~ if the +% second item is less than 3 letters long, otherwise it just puts an +% ordinary space. + +FUNCTION {tie.or.space.connect} +{ duplicate$ text.length$ #3 < + { "~" } + { " " } + if$ + swap$ * * +} + +FUNCTION {add.space.if.necessary} +{ duplicate$ "" = + 'skip$ + { " " * } + if$ +} + +% either.or.check gives a warning if two mutually exclusive fields +% were used in the database. + +FUNCTION {either.or.check} +{ empty$ + 'pop$ + { "can't use both " swap$ * " fields in " * cite$ * warning$ } + if$ +} + +% output.nonnull is called by output. + +% urlbst +FUNCTION {output.nonnull.original} +% remove the top item from the stack because it's in the way. +{ 's := + output.state mid.sentence = +% If we're in mid-sentence, add a comma to the new top item and write it + { ", " * write$ } +% Otherwise, if we're at the beginning of a bibitem, + { output.state before.all = +% just write out the top item from the stack; + 'write$ +% and the last alternative is that we're at the end of the current +% bibitem, so we add a period to the top stack item and write it out. + { add.period$ " " * write$ } + if$ + mid.sentence 'output.state := + } + if$ +% Put the top item back on the stack that we removed earlier. + s +} + +% urlbst... +% The following three functions are for handling inlinelink. They wrap +% a block of text which is potentially output with write$ by multiple +% other functions, so we don't know the content a priori. +% They communicate between each other using the variables makeinlinelink +% (which is true if a link should be made), and closeinlinelink (which holds +% the string which should close any current link. They can be called +% at any time, but start.inlinelink will be a no-op unless something has +% previously set makeinlinelink true, and the two ...end.inlinelink functions +% will only do their stuff if start.inlinelink has previously set +% closeinlinelink to be non-empty. +FUNCTION {setup.inlinelink} +{ makeinlinelink + { hrefform #1 = % hypertex + { "\special {html: }{" * 'openinlinelink := + "\special {html:}" 'closeinlinelink := + } + { hrefform #2 = % hyperref + { "\href{" url * "}{" * 'openinlinelink := + "}" 'closeinlinelink := + } + 'skip$ + if$ % hrefform #2 = + } + if$ % hrefform #1 = + #0 'makeinlinelink := + } + 'skip$ + if$ % makeinlinelink +} +FUNCTION {add.inlinelink} +{ openinlinelink empty$ + 'skip$ + { openinlinelink swap$ * closeinlinelink * + "" 'openinlinelink := + } + if$ +} +FUNCTION {output.nonnull} +{ % Save the thing we've been asked to output + 's := + % If the bracket-state is close.brackets, then add a close-bracket to + % what is currently at the top of the stack, and set bracket.state + % to outside.brackets + bracket.state close.brackets = + { "]" * + outside.brackets 'bracket.state := + } + 'skip$ + if$ + bracket.state outside.brackets = + { % We're outside all brackets -- this is the normal situation. + % Write out what's currently at the top of the stack, using the + % original output.nonnull function. + s + add.inlinelink + output.nonnull.original % invoke the original output.nonnull + } + { % Still in brackets. Add open-bracket or (continuation) comma, add the + % new text (in s) to the top of the stack, and move to the close-brackets + % state, ready for next time (unless inbrackets resets it). If we come + % into this branch, then output.state is carefully undisturbed. + bracket.state open.brackets = + { " [" * } + { ", " * } % bracket.state will be within.brackets + if$ + s * + close.brackets 'bracket.state := + } + if$ +} + +% Call this function just before adding something which should be presented in +% brackets. bracket.state is handled specially within output.nonnull. +FUNCTION {inbrackets} +{ bracket.state close.brackets = + { within.brackets 'bracket.state := } % reset the state: not open nor closed + { open.brackets 'bracket.state := } + if$ +} + +FUNCTION {format.lastchecked} +{ lastchecked empty$ + { "" } + { inbrackets "cited " lastchecked * } + if$ +} +% ...urlbst to here + +% Output checks to see if the stack top is empty; if not, it +% calls output.nonnull to write it out. + +FUNCTION {output} +{ duplicate$ empty$ + 'pop$ + 'output.nonnull + if$ +} + +% Standard warning message for a missing or empty field. For the user +% we call any such field `missing' without respect to the distinction +% made by BibTeX between missing and empty. + +FUNCTION {missing.warning} +{ "missing " swap$ * " in " * cite$ * warning$ } + +% Output.check is like output except that it gives a warning on-screen +% if the given field in the database entry is empty. t is the field +% name. + +FUNCTION {output.check} +{ 't := + duplicate$ empty$ + { pop$ t missing.warning } + 'output.nonnull + if$ +} + +FUNCTION {output.bibitem.original} +{ newline$ + "\bibitem[" write$ + label write$ + "]{" write$ + cite$ write$ + "}" write$ + newline$ +% This empty string is the first thing that will be written +% the next time write$ is called. Done this way because each +% item is saved on the stack until we find out what punctuation +% should be added after it. Therefore we need an empty first item. + "" + before.all 'output.state := +} + +FUNCTION {output.nonempty.mrnumber} +{ duplicate$ missing$ + { pop$ "" } + 'skip$ + if$ + duplicate$ empty$ + 'pop$ + { " \MR{" swap$ * "}" * write$ } + if$ +} + +FUNCTION {fin.entry.original} +{ add.period$ + write$ + mrnumber output.nonempty.mrnumber + newline$ +} + +% Removed new.block, new.block.checka, new.block.checkb, new.sentence, +% new.sentence.checka, and new.sentence.checkb functions here, since they +% don't seem to be needed in the AMS style. Also moved some real +% basic functions like `and' and 'or' earlier in the file. + +INTEGERS { nameptr namesleft numnames } + +% The extra section to write out a language field was added +% for AMSPLAIN.BST. Not present in plain.bst. + +FUNCTION {format.language} +{ language empty$ + { "" } + { " (" language * ")" * } + if$ +} + +% This version of format.names puts names in the format +% +% First von Last, Jr. +% +% (i.e., first name first, no abbreviating to initials). + +FUNCTION {format.names} +{ 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't := + nameptr #1 > + { namesleft #1 > + { ", " * t * } + { numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " and " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {format.authors} +{ author empty$ + { "" } + { bysame "\bysame" = + { bysame } + { author format.names } + if$ + } + if$ +} + +FUNCTION {format.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { " (eds.)" * } + { " (ed.)" * } + if$ + } + if$ +} + +FUNCTION {format.nonauthor.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { ", eds." * } + { ", ed." * } + if$ + } + if$ +} + +FUNCTION {format.title} +{ title empty$ + { "" } + { title "t" change.case$ emphasize } + if$ +} + +FUNCTION {format.journal.vol.year} +{ journal empty$ + { "" "journal name" missing.warning } + { journal + volume empty$ + 'skip$ + { " \textbf{" * volume * "}" * } + if$ + year empty$ + { "year" missing.warning } + { " (" * year * ")" * } + if$ + } + if$ +} + +% For formatting the issue number for a journal article. + +FUNCTION {format.number} +{ number empty$ + { "" } + { "no.~" number * } + if$ +} + +% For formatting miscellaneous dates + +FUNCTION {format.date} +{ year empty$ + { month empty$ + { "" } + { "there's a month but no year in " cite$ * warning$ + month + } + if$ + } + { month empty$ + 'year + { month " " * year * } + if$ + } + if$ +} + +%% The volume, series and number information is sort of tricky. +%% This code handles it as follows: +%% If the series is present, and the volume, but not the number, +%% then we do "\emph{Book title}, Series Name, vol. 000" +%% If the series is present, and the number, but not the volume, +%% then we do "\emph{Book title}, Series Name, no. 000" +%% If the series is present, and both number and volume, +%% then we do "\emph{Book title}, vol. XX, Series Name, no. 000" +%% Finally, if the series is absent, +%% then we do "\emph{Book title}, vol. XX" +%% or "\emph{Book title}, no. 000" +%% and if both volume and number are present, give a warning message. + +FUNCTION {format.bookvolume.series.number} +{ volume empty$ + { "" % Push the empty string as a placeholder in case everything else + % is empty too. + series empty$ + 'skip$ + { pop$ series } % if series is not empty put in stack + if$ + number empty$ + 'skip$ + { duplicate$ empty$ % if no preceding material, + 'skip$ % do nothing, otherwise + { ", " * } % add a comma and space to separate. + if$ + "no." number tie.or.space.connect * % add the number information + } + if$ + } +%% If the volume is NOT EMPTY: + { "vol." volume tie.or.space.connect % vol. XX + number empty$ + { series empty$ + 'skip$ + { series ", " * swap$ *} % Series Name, vol. XX + if$ + } + { series empty$ + { "can't use both volume and number if series info is missing" + warning$ + "in BibTeX entry type `" type$ * "'" * top$ + } + { ", " * series * ", no." * number tie.or.space.connect } + if$ + } + if$ + } + if$ + +} % end of format.bookvolume.series.number + +%% format.inproc.title.where.editors is used by inproceedings entry types + +%% No case changing or emphasizing for the title. We want initial +%% caps, roman. +%% We add parentheses around the address (place where conference +%% was held). +%% Likewise we add parentheses around the editors' names. + +FUNCTION {format.inproc.title.address.editors} +{ booktitle empty$ + { "" } + { booktitle + address empty$ + 'skip$ + { add.space.if.necessary "(" * address * ")" * } + if$ + editor empty$ + 'skip$ + { add.space.if.necessary "(" * format.nonauthor.editors * ")" * } + if$ + } + if$ +} + +%% format.incoll.title.editors is similar to format.inproc... but +%% omits the address. For collections that are not proceedings volumes. + +FUNCTION {format.incoll.title.editors} +{ booktitle empty$ + { "" } + { editor empty$ + { booktitle } + { booktitle + add.space.if.necessary "(" * format.nonauthor.editors * ")" * + } + if$ + } + if$ +} + +FUNCTION {format.edition} +{ edition empty$ + { "" } + { output.state mid.sentence = + { edition "l" change.case$ " ed." * } + { edition "t" change.case$ " ed." * } + if$ + } + if$ +} + +INTEGERS { multiresult } + +FUNCTION {multi.page.check} +{ 't := + #0 'multiresult := + { multiresult not + t empty$ not + and + } + { t #1 #1 substring$ + duplicate$ "-" = + swap$ duplicate$ "," = + swap$ "+" = + or or + { #1 'multiresult := } + { t #2 global.max$ substring$ 't := } + if$ + } + while$ + multiresult +} + +FUNCTION {format.pages} +{ pages empty$ + { "" } + { pages n.dashify } + if$ +} + +FUNCTION {format.book.pages} +{ pages empty$ + { "" } + { pages multi.page.check + { "pp.~" pages n.dashify * } + { "p.~" pages * } + if$ + } + if$ +} + +FUNCTION {format.chapter.pages} +{ chapter empty$ + 'format.book.pages + { type empty$ + { "ch.~" } + { type "l" change.case$ " " * } + if$ + chapter * + pages empty$ + 'skip$ + { ", " * format.book.pages * } + if$ + } + if$ +} + +FUNCTION {empty.misc.check} +{ author empty$ title empty$ howpublished empty$ + month empty$ year empty$ note empty$ + and and and and and + key empty$ not and + { "all relevant fields are empty in " cite$ * warning$ } + 'skip$ + if$ +} + +FUNCTION {format.thesis.type} +{ type empty$ + 'skip$ + { pop$ + type "t" change.case$ + } + if$ +} + +FUNCTION {format.tr.number} +{ type empty$ + { "Tech. Report" } + 'type + if$ + number empty$ + { "t" change.case$ } + { number tie.or.space.connect } + if$ +} + +% The format.crossref functions haven't been paid much attention +% at the present time (June 1990) and could probably use some +% work. MJD + +FUNCTION {format.article.crossref} +{ key empty$ + { journal empty$ + { "need key or journal for " cite$ * " to crossref " * crossref * + warning$ + "" + } + { "in " journal * } + if$ + } + { "in " key * } + if$ + " \cite{" * crossref * "}" * +} + +FUNCTION {format.crossref.editor} +{ editor #1 "{vv~}{ll}" format.name$ + editor num.names$ duplicate$ + #2 > + { pop$ " et~al." * } + { #2 < + 'skip$ + { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { " et~al." * } + { " and " * editor #2 "{vv~}{ll}" format.name$ * } + if$ + } + if$ + } + if$ +} + +FUNCTION {format.book.crossref} +{ volume empty$ + { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ + "in " + } + { "vol." volume tie.or.space.connect + " of " * + } + if$ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { series empty$ + { "need editor, key, or series for " cite$ * " to crossref " * + crossref * warning$ + "" * + } + { series * } + if$ + } + { key * } + if$ + } + { format.crossref.editor * } + if$ + " \cite{" * crossref * "}" * +} + +FUNCTION {format.incoll.inproc.crossref} +{ editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { booktitle empty$ + { "need editor, key, or booktitle for " cite$ * " to crossref " * + crossref * warning$ + "" + } + { "in \emph{" booktitle * "}" * } + if$ + } + { "in " key * } + if$ + } + { "in " format.crossref.editor * } + if$ + " \cite{" * crossref * "}" * +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% The main functions for each entry type. + +% journal, vol and year are formatted together because they are +% not separated by commas. + +% urlbst... +FUNCTION {new.block} % dummy new.block function +{ + % empty +} + +% Functions for making hypertext links. +% In all cases, the stack has (link-text href-url) +% +% make 'null' specials +FUNCTION {make.href.null} +{ + pop$ +} +% make hypertex specials +FUNCTION {make.href.hypertex} +{ + "\special {html: }" * swap$ * + "\special {html:}" * +} +% make hyperref specials +FUNCTION {make.href.hyperref} +{ + "\href {" swap$ * "} {\path{" * swap$ * "}}" * +} +FUNCTION {make.href} +{ hrefform #2 = + 'make.href.hyperref % hrefform = 2 + { hrefform #1 = + 'make.href.hypertex % hrefform = 1 + 'make.href.null % hrefform = 0 (or anything else) + if$ + } + if$ +} + +% If inlinelinks is true, then format.url should be a no-op, since it's +% (a) redundant, and (b) could end up as a link-within-a-link. +FUNCTION {format.url} +{ inlinelinks #1 = url empty$ or + { "" } + { hrefform #1 = + { % special case -- add HyperTeX specials + urlintro "\url{" url * "}" * url make.href.hypertex * } + { urlintro "\url{" * url * "}" * } + if$ + } + if$ +} + +FUNCTION {format.eprint} +{ eprint empty$ + { "" } + { eprintprefix eprint * eprinturl eprint * make.href } + if$ +} + +FUNCTION {format.doi} +{ doi empty$ + { "" } + { doiprefix doi * doiurl doi * make.href } + if$ +} + +% Output a URL. We can't use the more normal idiom (something like +% `format.url output'), because the `inbrackets' within +% format.lastchecked applies to everything between calls to `output', +% so that `format.url format.lastchecked * output' ends up with both +% the URL and the lastchecked in brackets. +FUNCTION {output.url} +{ url empty$ + 'skip$ + { new.block + format.url output + format.lastchecked output + } + if$ +} + +FUNCTION {output.web.refs} +{ + new.block + addeprints eprint empty$ not and + { format.eprint output.nonnull } + 'skip$ + if$ + adddoiresolver doi empty$ not and + { + url empty$ + { format.doi output.nonnull } + { + doiurl doi * url = + { format.doi output.nonnull } + { format.doi output.nonnull output.url } + if$ + } + if$ + } + { output.url } + if$ +} + +% Wrapper for output.bibitem.original. +% If the URL field is not empty, set makeinlinelink to be true, +% so that an inline link will be started at the next opportunity +FUNCTION {output.bibitem} +{ outside.brackets 'bracket.state := + output.bibitem.original + inlinelinks url empty$ not and + { #1 'makeinlinelink := } + { #0 'makeinlinelink := } + if$ +} + +% Wrapper for fin.entry.original +FUNCTION {fin.entry} +{ output.web.refs % urlbst + makeinlinelink % ooops, it appears we didn't have a title for inlinelink + { setup.inlinelink % add some artificial link text here, as a fallback + "[link]" output.nonnull } + 'skip$ + if$ + bracket.state close.brackets = % urlbst + { "]" * } + 'skip$ + if$ + fin.entry.original +} + +% Webpage entry type. +% Title and url fields required; +% author, note, year, month, and lastchecked fields optional +% See references +% ISO 690-2 http://www.nlc-bnc.ca/iso/tc46sc9/standard/690-2e.htm +% http://www.classroom.net/classroom/CitingNetResources.html +% http://neal.ctstateu.edu/history/cite.html +% http://www.cas.usf.edu/english/walker/mla.html +% for citation formats for web pages. +FUNCTION {webpage} +{ output.bibitem + author empty$ + { editor empty$ + 'skip$ % author and editor both optional + { format.editors output.nonnull } + if$ + } + { editor empty$ + { format.authors output.nonnull } + { "can't use both author and editor fields in " cite$ * warning$ } + if$ + } + if$ + new.block + title empty$ 'skip$ 'setup.inlinelink if$ + format.title "title" output.check + inbrackets "online" output + new.block + year empty$ + 'skip$ + { format.date "year" output.check } + if$ + % We don't need to output the URL details ('lastchecked' and 'url'), + % because fin.entry does that for us, using output.web.refs. The only + % reason we would want to put them here is if we were to decide that + % they should go in front of the rather miscellaneous information in 'note'. + new.block + note output + fin.entry +} +% ...urlbst to here + + +FUNCTION {article} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + crossref missing$ + { format.journal.vol.year "journal, volume, and year" output.check + format.number output + format.pages "pages" output.check + } + { format.article.crossref output.nonnull + format.pages "pages" output.check + } + if$ + format.language * + note output + fin.entry +} + +FUNCTION {book} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + format.edition output + crossref missing$ + { format.bookvolume.series.number output + publisher "publisher" output.check + address output + } + { format.book.crossref output.nonnull + } + if$ + format.date "year" output.check + format.language * + note output + fin.entry +} + +FUNCTION {booklet} +{ output.bibitem + format.authors output + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + howpublished output + address output + format.date output + note output + fin.entry +} + +FUNCTION {inbook} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + format.edition output + crossref missing$ + { format.bookvolume.series.number output + format.chapter.pages "chapter and pages" output.check + publisher "publisher" output.check + address output + } + { format.chapter.pages "chapter and pages" output.check + format.book.crossref output.nonnull + } + if$ + format.date "year" output.check + format.language * + note output + fin.entry +} + +FUNCTION {incollection} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + crossref missing$ + { format.incoll.title.editors "booktitle" output.check + format.bookvolume.series.number output + publisher "publisher" output.check + address output + format.edition output + format.date "year" output.check + } + { format.incoll.inproc.crossref output.nonnull + } + if$ + note output + format.book.pages output + format.language * + fin.entry +} + +FUNCTION {inproceedings} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + crossref missing$ + { format.inproc.title.address.editors "booktitle" output.check + format.bookvolume.series.number output + organization output + publisher output + format.date "year" output.check + } + { format.incoll.inproc.crossref output.nonnull + } + if$ + note output + format.book.pages output + format.language * + fin.entry +} + +FUNCTION {conference} { inproceedings } + +FUNCTION {manual} +{ output.bibitem + author empty$ + { organization empty$ + 'skip$ + { organization output.nonnull + address output + } + if$ + } + { format.authors output.nonnull } + if$ + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + author empty$ + { organization empty$ + { address output } + 'skip$ + if$ + } + { organization output + address output + } + if$ + format.edition output + format.date output + note output + fin.entry +} + +FUNCTION {mastersthesis} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + "Master's thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + note output + format.book.pages output + fin.entry +} + +FUNCTION {misc} +{ output.bibitem + format.authors output + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title output + howpublished output + format.date output + note output + format.book.pages output + fin.entry + empty.misc.check +} + +FUNCTION {phdthesis} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + "Ph.D. thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + note output + format.book.pages output + fin.entry +} + +FUNCTION {proceedings} +{ output.bibitem + editor empty$ + { organization output } + { format.editors output.nonnull } + if$ + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + format.bookvolume.series.number output + address empty$ + { editor empty$ + 'skip$ + { organization output } + if$ + publisher output + format.date "year" output.check + } + { address output.nonnull + editor empty$ + 'skip$ + { organization output } + if$ + publisher output + format.date "year" output.check + } + if$ + note output + fin.entry +} + +FUNCTION {techreport} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + format.tr.number output.nonnull + institution "institution" output.check + address output + format.date "year" output.check + note output + fin.entry +} + +FUNCTION {unpublished} +{ output.bibitem + format.authors "author" output.check + title empty$ 'skip$ 'setup.inlinelink if$ % urlbst + format.title "title" output.check + note "note" output.check + format.date output + fin.entry +} + +FUNCTION {default.type} { misc } + +MACRO {jan} {"January"} + +MACRO {feb} {"February"} + +MACRO {mar} {"March"} + +MACRO {apr} {"April"} + +MACRO {may} {"May"} + +MACRO {jun} {"June"} + +MACRO {jul} {"July"} + +MACRO {aug} {"August"} + +MACRO {sep} {"September"} + +MACRO {oct} {"October"} + +MACRO {nov} {"November"} + +MACRO {dec} {"December"} + +READ + +FUNCTION {sortify} +{ purify$ + "l" change.case$ +} + +INTEGERS { len } + +FUNCTION {chop.word} +{ 's := + 'len := + s #1 len substring$ = + { s len #1 + global.max$ substring$ } + 's + if$ +} + +INTEGERS { et.al.char.used } + +FUNCTION {initialize.et.al.char.used} +{ #0 'et.al.char.used := +} + +EXECUTE {initialize.et.al.char.used} + +FUNCTION {format.lab.names} +{ 's := + s num.names$ 'numnames := + numnames #1 > + { numnames #4 > + { #3 'namesleft := } + { numnames 'namesleft := } + if$ + #1 'nameptr := + "" + { namesleft #0 > } + { nameptr numnames = + { s nameptr "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { "{\etalchar{+}}" * + #1 'et.al.char.used := + } + { s nameptr "{v{}}{l{}}" format.name$ * } + if$ + } + { s nameptr "{v{}}{l{}}" format.name$ * } + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ + numnames #4 > + { "{\etalchar{+}}" * + #1 'et.al.char.used := + } + 'skip$ + if$ + } + { s #1 "{v{}}{l{}}" format.name$ + duplicate$ text.length$ #2 < + { pop$ s #1 "{ll}" format.name$ #3 text.prefix$ } + 'skip$ + if$ + } + if$ +} + +FUNCTION {author.key.label} +{ author empty$ + { key empty$ + { cite$ #1 #3 substring$ } + { key #3 text.prefix$ } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.editor.key.label} +{ author empty$ + { editor empty$ + { key empty$ + { cite$ #1 #3 substring$ } + { key #3 text.prefix$ } + if$ + } + { editor format.lab.names } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.key.organization.label} +{ author empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + { key #3 text.prefix$ } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {editor.key.organization.label} +{ editor empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + { key #3 text.prefix$ } + if$ + } + { editor format.lab.names } + if$ +} + +FUNCTION {calc.label} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.key.label + { type$ "proceedings" = + 'editor.key.organization.label + { type$ "manual" = + 'author.key.organization.label + 'author.key.label + if$ + } + if$ + } + if$ + duplicate$ + year field.or.null purify$ #-1 #2 substring$ + * + 'label := + year field.or.null purify$ #-1 #4 substring$ + * + sortify 'sort.label := +} + +FUNCTION {sort.format.names} +{ 's := + #1 'nameptr := + "" + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { nameptr #1 > + { " " * } + 'skip$ + if$ + s nameptr "{vv{ } }{ll{ }}{ ff{ }}{ jj{ }}" format.name$ 't := + nameptr numnames = t "others" = and + { "et al" * } + { t sortify * } + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {sort.format.title} +{ 't := + "A " #2 + "An " #3 + "The " #4 t chop.word + chop.word + chop.word + sortify + #1 global.max$ substring$ +} + +FUNCTION {author.sort} +{ author empty$ + { key empty$ + { "to sort, need author or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.editor.sort} +{ author empty$ + { editor empty$ + { key empty$ + { "to sort, need author, editor, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { editor sort.format.names } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.organization.sort} +{ author empty$ + { organization empty$ + { key empty$ + { "to sort, need author, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {editor.organization.sort} +{ editor empty$ + { organization empty$ + { key empty$ + { "to sort, need editor, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { editor sort.format.names } + if$ +} + +FUNCTION {presort} +{ calc.label + sort.label + " " + * + type$ "book" = + type$ "inbook" = + or + 'author.editor.sort + { type$ "proceedings" = + 'editor.organization.sort + { type$ "manual" = + 'author.organization.sort + 'author.sort + if$ + } + if$ + } + if$ + * + " " + * + year field.or.null sortify + * + " " + * + title field.or.null + sort.format.title + * + #1 entry.max$ substring$ + 'sort.key$ := +} + +ITERATE {presort} + +SORT + +STRINGS { + longest.label last.sort.label next.extra prev.author this.author +} + +INTEGERS { longest.label.width last.extra.num } + +FUNCTION {initialize.longest.label} +{ "" 'longest.label := + #0 int.to.chr$ 'last.sort.label := + "" 'next.extra := + #0 'longest.label.width := + #0 'last.extra.num := + "abcxyz" 'prev.author := + "" 'this.author := +} + +FUNCTION {forward.pass} +{ last.sort.label sort.label = + { last.extra.num #1 + 'last.extra.num := + last.extra.num int.to.chr$ 'extra.label := + } + { "a" chr.to.int$ 'last.extra.num := + "" 'extra.label := + sort.label 'last.sort.label := + } + if$ + author empty$ { editor empty$ { "" } 'editor if$ } 'author if$ + 'this.author := + this.author prev.author = + { "\bysame" 'bysame := } + { "" 'bysame := + this.author "" = + { "abcxyz" } + 'this.author + if$ + 'prev.author := + } + if$ +} + +FUNCTION {reverse.pass} +{ next.extra "b" = + { "a" 'extra.label := } + 'skip$ + if$ + label extra.label * 'label := + label width$ longest.label.width > + { label 'longest.label := + label width$ 'longest.label.width := + } + 'skip$ + if$ + extra.label 'next.extra := +} + +EXECUTE {initialize.longest.label} + +ITERATE {forward.pass} + +REVERSE {reverse.pass} + +FUNCTION {begin.bib} +{ et.al.char.used + { "\newcommand{\etalchar}[1]{$^{#1}$}" write$ newline$ } + 'skip$ + if$ + preamble$ empty$ + 'skip$ + { preamble$ write$ newline$ } + if$ + "\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}" + write$ newline$ + "\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }" + write$ newline$ + "% \MRhref is called by the amsart/book/proc definition of \MR." + write$ newline$ + "\providecommand{\MRhref}[2]{%" + write$ newline$ + " \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}" + write$ newline$ + "}" + write$ newline$ + "\providecommand{\href}[2]{#2}" + write$ newline$ + "\begin{thebibliography}{" longest.label * "}" * + write$ newline$ +} + +EXECUTE {begin.bib} + +EXECUTE {init.urlbst.variables} +EXECUTE {init.state.consts} + +ITERATE {call.type$} + +FUNCTION {end.bib} +{ newline$ + "\end{thebibliography}" write$ newline$ +} + +EXECUTE {end.bib} +%% \CharacterTable +%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z +%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z +%% Digits \0\1\2\3\4\5\6\7\8\9 +%% Exclamation \! Double quote \" Hash (number) \# +%% Dollar \$ Percent \% Ampersand \& +%% Acute accent \' Left paren \( Right paren \) +%% Asterisk \* Plus \+ Comma \, +%% Minus \- Point \. Solidus \/ +%% Colon \: Semicolon \; Less than \< +%% Equals \= Greater than \> Question mark \? +%% Commercial at \@ Left bracket \[ Backslash \\ +%% Right bracket \] Circumflex \^ Underscore \_ +%% Grave accent \` Left brace \{ Vertical bar \| +%% Right brace \} Tilde \~} diff --git a/resources/VanDoornDissertation/dissertation.bbl b/resources/VanDoornDissertation/dissertation.bbl new file mode 100644 index 0000000..d6832ca --- /dev/null +++ b/resources/VanDoornDissertation/dissertation.bbl @@ -0,0 +1,395 @@ +\newcommand{\etalchar}[1]{$^{#1}$} +\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} +\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } +% \MRhref is called by the amsart/book/proc definition of \MR. +\providecommand{\MRhref}[2]{% + \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} +} +\providecommand{\href}[2]{#2} +\begin{thebibliography}{dMKA{\etalchar{+}}15} + +\bibitem[AB04]{awodey2004Propositions} +Steve Awodey and Andrej Bauer, \emph{Propositions as [{T}ypes]}, Journal of + Logic and Computation \textbf{14} (2004), no.~4, 447--471. + +\bibitem[AC10]{asperti2010itp} +Andrea Asperti and Claudio~Sacerdoti Coen, \emph{Some considerations on the + usability of interactive provers}, International Conference on Intelligent + Computer Mathematics, Springer, 2010, pp.~147--156. + +\bibitem[ACD{\etalchar{+}}16]{altenkirch2016qiits} +Thorsten Altenkirch, Paolo Capriotti, Gabe Dijkstra, Nicolai Kraus, and Fredrik + Nordvall~Forsberg, \emph{{Quotient inductive-inductive types}}, ArXiv + e-prints (2016), \href {http://arxiv.org/abs/1612.02346} + {\path{arXiv:1612.02346}}. + +\bibitem[AH61]{atiyah1961spectral} +Michael~F Atiyah and Friedrich Hirzebruch, \emph{Vector bundles and homogeneous + spaces}, Differential geometry, Proceedings of Symposia in Pure Mathematics, + no.~3, 1961, pp.~7--38. + +\bibitem[AHW17]{angiuli2017computational} +Carlo Angiuli, Robert Harper, and Todd Wilson, \emph{Computational + higher-dimensional type theory}, POPL '17: Proceedings of the 44th Annual ACM + SIGPLAN-SIGACT Symposium on Principles of Programming Languages, ACM, 2017, + \href {http://dx.doi.org/10.1145/3009837.3009861} + {\path{doi:10.1145/3009837.3009861}}. + +\bibitem[AKL15]{avigad2015limits} +Jeremy Avigad, Chris Kapulkin, and Peter~LeFanu Lumsdaine, \emph{Homotopy + limits in type theory}, Mathematical Structures in Computer Science + \textbf{25} (2015), no.~05, 1040--1070. + +\bibitem[AW09]{awodey2009homotopy} +Steve Awodey and Michael~A. Warren, \emph{Homotopy theoretic models of identity + types}, Math. Proc. Camb. Phil. Soc., vol. 146, Cambridge Univ Press, 2009, + pp.~45--55. + +\bibitem[BCH14]{bezem2014cubicalsets} +Marc Bezem, Thierry Coquand, and Simon Huber, \emph{A model of type theory in + cubical sets}, 19th International Conference on Types for Proofs and Programs + (TYPES 2013), vol.~26, 2014, pp.~107--128. + +\bibitem[BGLL{\etalchar{+}}16]{bauer2016coqhott} +Andrej Bauer, Jason Gross, Peter~LeFanu LeFanu~Lumsdaine, Michael Shulman, + Matthieu Sozeau, and Bas Spitters, \emph{{The HoTT Library: A formalization + of homotopy type theory in Coq}}, ArXiv e-prints (2016), \href + {http://arxiv.org/abs/1610.04591} {\path{arXiv:1610.04591}}. + +\bibitem[BH18]{buchholtz2018cellular} +Ulrik Buchholtz and Kuen-Bang {Hou (Favonia)}, \emph{{Cellular Cohomology in + Homotopy Type Theory}}, ArXiv e-prints (2018), \href + {http://arxiv.org/abs/1802.02191} {\path{arXiv:1802.02191}}. + +\bibitem[BHC{\etalchar{+}}]{hottagda} +Guillaume Brunerie, Kuen-Bang {Hou (Favonia)}, Evan Cavallo, Eric Finster, + Jesper Cockx, Christian Sattler, Chris Jeris, Michael Shulman, et~al., + \emph{Homotopy type theory in {A}gda}, + \url{https://github.com/HoTT/HoTT-Agda}. + +\bibitem[Bla79]{blass1979injectivity} +Andreas Blass, \emph{Injectivity, projectivity, and the axiom of choice}, + Transactions of the American Mathematical Society \textbf{255} (1979), + 31--59. + +\bibitem[BR16]{buchholtz2016cayleydickson} +Ulrik Buchholtz and Egbert Rijke, \emph{The {C}ayley-{D}ickson construction in + {H}omotopy {T}ype {T}heory}, ArXiv e-prints (2016), \href + {http://arxiv.org/abs/1610.01134} {\path{arXiv:1610.01134}}. + +\bibitem[Bru16]{brunerie2016spheres} +Guillaume Brunerie, \emph{On the homotopy groups of spheres in homotopy type + theory}, Ph.D. thesis, University of Nice Sophia Antipolis, 2016, + \url{https://arxiv.org/abs/1606.05916}. + +\bibitem[BvDR18]{buchholtz2018groups} +Ulrik Buchholtz, Floris van Doorn, and Egbert Rijke, \emph{{Higher Groups in + Homotopy Type Theory}}, ArXiv e-prints (2018), \href + {http://arxiv.org/abs/1802.04315} {\path{arXiv:1802.04315}}. + +\bibitem[Car18]{carneiro2018leantheory} +Mario Carneiro, \emph{The type theory of {L}ean}, 2018, online, + \url{https://github.com/digama0/lean-type-theory/releases}. + +\bibitem[Cav15]{cavallo2015cohomology} +Evan Cavallo, \emph{Synthetic cohomology in homotopy type theory}, Master's + thesis, Carnegie Mellon University, 2015, + \url{http://www.cs.cmu.edu/~ecavallo/works/thesis.pdf}. + +\bibitem[CCHM]{cubicaltt} +Cyril Cohen, Thierry Coquand, Simon Huber, and Anders M{\"o}rtberg, + \emph{Cubical type theory}, code library, + \url{https://github.com/mortberg/cubicaltt}. + +\bibitem[CCHM16]{cohen2016cubical} +\bysame, \emph{Cubical type theory: a constructive interpretation of the + univalence axiom}, November 2016, \href {http://arxiv.org/abs/1611.02108} + {\path{arXiv:1611.02108}}. + +\bibitem[CF58]{curry1958combinatorylogic} +Haskell~B. 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Licata, and Peter~LeFanu + Lumsdaine, \emph{A mechanization of the {Blakers-Massey} connectivity theorem + in {Homotopy Type Theory}}, Proceedings of the 31st Annual ACM/IEEE Symposium + on Logic in Computer Science, ACM, 2016, pp.~565--574. + +\bibitem[{Hou}17]{favonia2017thesis} +Kuen-Bang {Hou (Favonia)}, \emph{Higher-dimensional types in the mechanization + of homotopy theory}, Ph.D. thesis, Carnegie Mellon University, 2017. + +\bibitem[How80]{howard1980formulae} +William~A. Howard, \emph{The formulae-as-types notion of construction}, To H.B. + Curry: essays on combinatory logic, lambda calculus and formalism \textbf{44} + (1980), 479--490. + +\bibitem[HP13]{holmbergperoux2014models} +Maximilien Holmberg-Péroux, \emph{The serre spectral sequence}, preprint + (2013), \url{http://homepages.math.uic.edu/~mholmb2/serre.pdf}. + +\bibitem[HS98]{hofmann1998groupoid} +Martin Hofmann and Thomas Streicher, \emph{The groupoid interpretation of type + theory}, Twenty-five years of constructive type theory ({V}enice, 1995), + Oxford Logic Guides, vol.~36, Oxford Univ. Press, New York, 1998, + pp.~83--111. + +\bibitem[HS16]{favonia2016seifert} +Kuen-Bang {Hou (Favonia)} and Michael Shulman, \emph{The {S}eifert-van {K}ampen + theorem in homotopy type theory}, 25th EACSL Annual Conference on Computer + Science Logic (CSL 2016), Leibniz International Proceedings in Informatics + (LIPIcs), vol.~62, 2016, pp.~22:1--22:16, \href + {http://dx.doi.org/10.4230/LIPIcs.CSL.2016.22} + {\path{doi:10.4230/LIPIcs.CSL.2016.22}}. + +\bibitem[KECA14]{kraus2014anonymousexistence} +Nicolai Kraus, Mart{\'\i}n Escard{\'o}, Thierry Coquand, and Thorsten + Altenkirch, \emph{Notions of anonymous existence in {Martin-L{\"o}f} type + theory}, Submitted to the special issue of TLCA'13 (2014). + +\bibitem[KL12]{kapulkin2012simplicialnew} +Chris Kapulkin and Peter~LeFanu Lumsdaine, \emph{{The Simplicial Model of + Univalent Foundations (after Voevodsky)}}, ArXiv e-prints (2012), \href + {http://arxiv.org/abs/1211.2851} {\path{arXiv:1211.2851}}. + +\bibitem[Kra15]{kraus2014universalproperty} +Nicolai Kraus, \emph{The general universal property of the propositional + truncation}, 20th International Conference on Types for Proofs and Programs + (TYPES 2014), Leibniz International Proceedings in Informatics (LIPIcs), + vol.~39, 2015, pp.~111--145, \href + {http://dx.doi.org/10.4230/LIPIcs.TYPES.2014.111} + {\path{doi:10.4230/LIPIcs.TYPES.2014.111}}. + +\bibitem[Kra16]{kraus2016hits} +\bysame, \emph{Constructions with non-recursive higher inductive types}, + Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer + Science, ACM, 2016, pp.~595--604. + +\bibitem[LF14]{licata2014em} +Daniel~R. Licata and Eric Finster, \emph{{E}ilenberg-{M}ac{L}ane spaces in + homotopy type theory}, Proceedings of the Joint Meeting of the Twenty-Third + EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth + Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), ACM, 2014, + p.~66. + +\bibitem[Lic11]{licata2011trick} +Daniel~R. Licata, \emph{Running circles around (in) your proof assistant; or, + quotients that compute}, blog post, April 2011, + \url{http://homotopytypetheory.org/2011/04/23/running-circles-around-in-your-proof-assistant/}. + +\bibitem[LS13]{licatashulman2013} +Daniel~R. Licata and Michael Shulman, \emph{Calculating the fundamental group + of the circle in homotopy type theory}, 2013 28th {A}nnual {ACM}/{IEEE} + {S}ymposium on {L}ogic in {C}omputer {S}cience ({LICS} 2013), IEEE Computer + Soc., Los Alamitos, CA, 2013, pp.~223--232. + +\bibitem[LS17]{lumsdaine2017HITsemantics} +Peter~LeFanu Lumsdaine and Michael Shulman, \emph{{Semantics of higher + inductive types}}, ArXiv e-prints (2017), \href + {http://arxiv.org/abs/1705.07088} {\path{arXiv:1705.07088}}. + +\bibitem[Lum11]{lumsdaine2011hits} +Peter~LeFanu Lumsdaine, \emph{Higher inductive types: a tour of the menagerie}, + blog post, April 2011, + \url{https://homotopytypetheory.org/2011/04/24/higher-inductive-types-a-tour-of-the-menagerie/}. + +\bibitem[Luo12]{luo2012universes} +Zhaohui Luo, \emph{Notes on universes in type theory}, preprint, 2012, + \url{http://www.cs.rhul.ac.uk/home/zhaohui/universes.pdf}. + +\bibitem[Mas52]{massey1952exactcouple} +William~S Massey, \emph{Exact couples in 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{http://arxiv.org/abs/1701.07538} {\path{arXiv:1701.07538}}. + +\bibitem[RSS17]{rijke2017modalities} +Egbert Rijke, Michael Shulman, and Bas Spitters, \emph{Modalities in homotopy + type theory}, ArXiv e-prints (2017), \href {http://arxiv.org/abs/1706.07526} + {\path{arXiv:1706.07526}}. + +\bibitem[S{\etalchar{+}}11]{shulman2011spectrification} +Michael Shulman et~al., \emph{{higher inductive type}}, 2011, nLab article, + \url{https://ncatlab.org/nlab/revision/higher+inductive+type/31}. + +\bibitem[Ser51]{serre1951homology} +Jean-Pierre Serre, \emph{Homologie singuli{\`e}re des espaces fibr{\'e}s}, + Annals of Mathematics (1951), 425--505. + +\bibitem[Shu11a]{shulman2011pi1S1} +Michael Shulman, \emph{A formal proof that $\pi_1(s^1)=\mathbb{Z}$}, blog post, + April 2011, + \url{https://homotopytypetheory.org/2011/04/29/a-formal-proof-that-pi1s1-is-z/}. + +\bibitem[Shu11b]{shulman2011HoTThits} +\bysame, \emph{Homotopy type theory, vi}, forum post, April 2011, + \url{https://golem.ph.utexas.edu/category/2011/04/homotopy_type_theory_vi.html}. + +\bibitem[Shu11c]{shulman2011intervalimpliesfunext} +\bysame, \emph{An interval type implies function extensionality}, blog post, + April 2011, + \url{https://homotopytypetheory.org/2011/04/04/an-interval-type-implies-function-extensionality/}. + +\bibitem[Shu13]{shulman2013spectral} +\bysame, \emph{{Spectral sequences in HoTT}}, blog posts, August 2013, + \url{https://ncatlab.org/homotopytypetheory/revision/spectral+sequences/5}. + +\bibitem[Shu17]{shulman2017topos} +\bysame, \emph{Elementary $(\infty,1)$-topoi}, blog post, April 2017, + \url{https://golem.ph.utexas.edu/category/2017/04/elementary_1topoi.html}. + +\bibitem[Sna81]{snaith1981ktheory} +Victor Snaith, \emph{Localized stable homotopy of some classifying spaces}, + Mathematical Proceedings of the Cambridge Philosophical Society, vol.~89, + Cambridge University Press, 1981, pp.~325--330. + +\bibitem[Str14]{streicher2014simplicial} +Thomas Streicher, \emph{A model of type theory in simplicial sets: A brief + introduction to {V}oevodsky's homotopy type theory}, Journal of Applied Logic + \textbf{12} (2014), no.~1, 45 -- 49, Logic Categories Semantics, \href + {http://dx.doi.org/https://doi.org/10.1016/j.jal.2013.04.001} + {\path{doi:https://doi.org/10.1016/j.jal.2013.04.001}}. + +\bibitem[{The}18]{redprl} +{The RedPRL Development Team}, \emph{{RedPRL} -- the {P}eople's {R}efinement + {L}ogic}, 2018, \url{http://www.redprl.org/}. + +\bibitem[{Uni}13]{hottbook} +The {Univalent Foundations Program}, \emph{Homotopy type theory: Univalent + foundations of mathematics}, \url{http://homotopytypetheory.org/book}, + Institute for Advanced Study, 2013. + +\bibitem[VAG{\etalchar{+}}]{unimath} +Vladimir Voevodsky, Benedikt Ahrens, Daniel Grayson, et~al., \emph{{U}ni{M}ath + --- {U}nivalent {M}athematics}, code library, + \url{https://github.com/UniMath}. + +\bibitem[vD16]{vandoorn2016proptrunc} +Floris van Doorn, \emph{Constructing the propositional truncation using + non-recursive hits}, Proceedings of the 5th ACM SIGPLAN Conference on + Certified Programs and Proofs, ACM, 2016, pp.~122--129. + +\bibitem[vDvRB17]{vandoorn2017leanhott} +Floris van Doorn, Jakob von Raumer, and Ulrik Buchholtz, \emph{Homotopy type + theory in lean}, pp.~479--495, Springer, 2017, \href + {http://dx.doi.org/10.1007/978-3-319-66107-0_30} + {\path{doi:10.1007/978-3-319-66107-0_30}}. + +\bibitem[vG12]{berg2010models} +Benno {van den Berg} and Richard Garner, \emph{{Topological and simplicial + models of identity types}}, ACM transactions on computational logic (TOCL) + \textbf{13} (2012), no.~1, 3. + +\bibitem[Voe06]{voevodsky2006} +Vladimir Voevodsky, \emph{A very short note on the homotopy + {$\lambda$}-calculus}, online, 2006, + \url{http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf}. + +\bibitem[Voe09]{voevodsky2009typesystems} +\bysame, \emph{Notes on type systems}, online, 2009, + \url{http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/expressions_current_1.pdf}. + +\bibitem[Voe14]{voevodsky2014univalence} +\bysame, \emph{{The equivalence axiom and univalent models of type theory. + (Talk at CMU on February 4, 2010)}}, \href {http://arxiv.org/abs/1402.5556} + {\path{arXiv:1402.5556}}. + +\bibitem[Voe15]{voevodsky2015lecture} +\bysame, \emph{Oxford lectures on {U}ni{M}ath}, filmed by Kohei Kishida, + available at \url{https://www.math.ias.edu/vladimir/Lectures}, 2015. + +\bibitem[vR16]{raumer2016doublegroupoids} +Jakob von Raumer, \emph{Formalizing double groupoids and cross modules in the + lean theorem prover}, Mathematical Software -- ICMS 2016, Springer + International Publishing, 2016, pp.~28--33. + +\bibitem[Zha17]{zhan2017auto2} +Bohua Zhan, \emph{Formalization of the fundamental group in untyped set theory + using auto2}, International Conference on Interactive Theorem Proving, + Springer, Springer, 2017, pp.~514--530, \href + {http://dx.doi.org/10.1007/978-3-319-66107-0_32} + {\path{doi:10.1007/978-3-319-66107-0_32}}. + +\end{thebibliography} diff --git a/resources/VanDoornDissertation/dissertation.pdf b/resources/VanDoornDissertation/dissertation.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8b072c386552f435548e58c21b6e553f91dfd6a7 GIT binary patch literal 1328373 zcmb?jcOcdK`?sel;B|(S7)4wJ*8)Fh(@h*AB47QLJ)I1h#wLx>u{>QteNM{ zD{U-NoS)&rEv2UoFW6RQD#`o1tS!BXj;?>6y+C!_DuUnpqSR)oA(GM20V|WVU%)Fo zVXyigH5IOxm~`peSu&}M*#b=wdG*!Z=Uoy70t#6AYQI`ll}zu-80l!X zb)?a$EpDkscqm-EXUabj$ge9G$d|BRYSC{%Syfq8nFn#lQZ1vm?Y@lU_-I<9R=#5lGZeVvuOj;s!iYVz81C+m)Z z;eOrR;8WjDlri0;Li%HIZalbyrja@KMgtIxF`|$$2_zmvZ8Aw zPbb<*DM>9G1`C`kdt*kbbfBp(QY~bkMV~vJjFJwI!my5)FFCodv}E?63`DnUs=;^i z3cZ{gjW2W37iJT!kH?JH-qCD4C-R@OaZlMK=y?*a>AQOCQppvWmpbV~ew1;7a-m6Tp6V-7H>bT?DBd;1AP>K$x=7W{dcQUeMtcKGSu|+cu_yOq?3!wT7sTB{eJSW7*7q zVI*5rWDcea>7AGR81^#lp->Y4z4%{R~HCglhr9j 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+\newcommand{\sequence}[3][]{(#2,#3)} +\newcommand{\msm}[2]{\Sigma(#1,#2)} +\newenvironment{constr}{% + \begin{proof}[Construction]% +}{\end{proof}} +\newcommand{\tfcolim}{\mathsf{colim}} +\newcommand{\kshiftequiv}{\mathsf{kshift\underline{~}equiv}} + + +\usepackage{tocloft} +\setlength{\cftbeforetoctitleskip}{0pt} + +%% center environment which ignores margins +\newenvironment{xcenter} + {\par\vspace*{3mm}\setbox0=\hbox\bgroup\ignorespaces} + {\unskip\egroup\noindent\makebox[\textwidth]{\box0}\par\vspace*{3mm}} + +%watermark +% \usepackage{draftwatermark} +% \SetWatermarkText{DRAFT} +% \SetWatermarkScale{1} + +\begin{document} +% \frontmatter +\pdfbookmark[chapter]{Front Matter}{front} +\pdfbookmark[section]{Title Page}{front} +\title{On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory} +\author{Floris van Doorn} +\pagenumbering{roman} +\begin{titlepage} + \begin{center} + \vspace*{20mm} + \LARGE + %Dissertation\\[5mm] + \textbf{On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory} + \par\vspace*{15mm}\par + \Large + Floris van Doorn + \par\vspace*{15mm}\par + May 2018\\[15mm] %\today\nn + \normalsize + % Advisor: Jeremy Avigad\nn + Dissertation Committee:\\ + Jeremy Avigad\\ + Steve Awodey\\ + Ulrik Buchholtz\\ + Mike Shulman\\[30mm] +Submitted in partial fulfillment of the requirements for the degree of\\ +Doctor of Philosophy in Pure and Applied Logic\\ +Department of Philosophy\\ +Carnegie Mellon University +\end{center} +\end{titlepage} +% \pdfbookmark[section]{Abstract}{abstract} +% \abstract{Abstract goes here} +% \setcounter{page}{2} +% \thispagestyle{plain} +% \clearpage +%\SetWatermarkText{} +\setcounter{page}{2} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents + + +\chapter{Introduction}\label{cha:introduction} +\pagenumbering{arabic} + +The goal of this dissertation is to present synthetic homotopy theory in the setting +of \emph{homotopy type theory}. We will present various results in this +framework, most notably the construction of the Atiyah-Hirzebruch and Serre +spectral sequences for cohomology, which have been fully formalized in the Lean +proof assistant. + +Homotopy type theory, often abbreviated HoTT, is a version of type theory. Type +theory is a language for formal mathematics, in which every object has a +computational interpretation, so that it can also function as a programming +language. It can be used as a foundation of mathematics as an alternative to set +theory. + +A key feature of HoTT is that the equality in a space corresponds to the path +spaces; a path between two points $a$ and $b$ is a proof that $a=b$. Two paths +that are not homotopic give different (unequal) proofs of this equality. The +fact that we identify proofs of an equality with a path means that every +construction in HoTT respects paths. + +Many different researchers contributed to the homotopical interpretation of type theory. +Steve Awodey and Michael Warren gave a model of type theory in abstract homotopy theory~\cite{awodey2009homotopy}. +Benno van den Berg and Richard Garner published a paper addressing the coherence issue~\cite{berg2010models}. +Independently, Vladimir Voevodsky gave a model of type theory without identity types in simplicial sets +and formulated the \emph{univalence axiom}, which he proved consistent~\cite{voevodsky2006,voevodsky2009typesystems}. +The univalence axiom states that +homotopy equivalences between two types (spaces) corresponds to equality between +them~\cite{voevodsky2014univalence}. This means that every construction done in +HoTT automatically respects homotopy equivalence, which is a very convenient +property. Also, Voevodsky proved that a consequence of the univalence axiom is +\emph{function extensionality}. This states that two functions are equal when +they are homotopic. + +The fact that all constructions are homotopy invariant also leads to some +challenges. It is not always clear whether we can define a concept of homotopy theory in +homotopy type theory. For example, \emph{singular homology} is a homotopy +invariant notion, but in the construction we use the set of all simplices in a +space, which is not a homotopy invariant notion. In this case, we can define +homology in a different way (see \autoref{sec:spectral-sequence-homology}). +However, for other definitions, such as the Grassmannian manifolds, it is an +open problem whether they can be constructed in homotopy type theory. + +A new concept in homotopy type theory is the concept of \emph{higher inductive +types}. These are types that generalize both cell complexes in homotopy theory, +and inductively generated types (like $\N$) in type theory. Higher inductive +types can be used to construct many spaces and operations on spaces often +encountered in homotopy theory. + +Type theory is a convenient language for computer proof assistants. These are +programs that allow you to write formal proofs in a specified language, and +then the computer checks whether the proof is correct and complete. There are +many major results formalized in proof assistants, such as the four colour +theorem~\cite{gonthier2005fourcolour}, Feit-Thompson theorem~\cite{gonthier2013oddorder} +and the Kepler conjecture (Hales' Theorem)~\cite{hales2017kepler}. +HoTT is a type theory, and it has been implemented in various proof assistants, +such as Coq~\cite{bauer2016coqhott}, Agda~\cite{hottagda}, cubicaltt~\cite{cubicaltt}, Lean~\cite{vandoorn2017leanhott} +and various experimental proof assistants. One disadvantage of formally verifying +proofs in a proof assistant is that it takes a lot of work spelling out all +details. For example, doing very basic homotopy theory (not using homotopy type +theory) already takes a lot of effort~\cite{zhan2017auto2}. In HoTT this effect +is mitigated, because many homotopical concepts are close to the foundations of +the type theory, making formal proofs only a little more work than a paper proof. + +Various results have been proven and formalized in HoTT, such as the the +Seifert--van Kampen theorem~\cite{favonia2016seifert}, the Blakers--Massey +theorem~\cite{favonia2016blakersmassey} and a development of cellular +cohomology~\cite{buchholtz2018cellular}. Another main result (which has not been +formalized) is the computation of $\pi_4(\S^3)$~\cite{brunerie2016spheres}, which +relies on conjectured properties of the smash product, which we will discuss in +\autoref{sec:smash-product}. + +HoTT gives novel proof methods and new insights to homotopy theory. A basic +property of HoTT is \emph{path induction}, which states that when proving +something for a path with one free endpoint, one may assume that the path is the +constant path. This corresponds to the fact that the path space with one fixed +endpoint is contractible. Another technique is the encode-decode method, for +calculating the path space of certain spaces~\cite{licatashulman2013}. Moreover, +the proof of the Blakers--Massey theorem has been translated back to homotopy +theory, resulting in a new proof with novel ideas~\cite{rezk2014blakersmassey}. + +Homotopy type theory has models in most model categories~\cite{awodey2009homotopy,berg2010models}, +which are categorical models for homotopy theory. These models were +inspired by the groupoid model~\cite{hofmann1998groupoid}. Other models for HoTT +include the simplicial set model~\cite{voevodsky2009typesystems,kapulkin2012simplicialnew,streicher2014simplicial} and the cubical +set model~\cite{bezem2014cubicalsets,cohen2016cubical}. More generally, all Grothendieck +$(\infty,1)$-toposes model HoTT~\cite{cisinski2014models}.\footnote{General Grothendieck $(\infty,1)$-toposes model HoTT with universes \'a la Tarski. +This notion is weaker than universes \'a la Russell, which are usually considered in HoTT. We explain Russell universes in \autoref{sec:universes}.} Moreover, it is +conjectured that all elementary $(\infty,1)$-toposes form models of HoTT~\cite{shulman2017topos}. + +\subsubsection*{Type Theory} +Homotopy type theory is based on Martin-L\"of type theory (also called intuitionistic type theory +or constructive type theory)~\cite{martinlof1975typetheory,martinlof1984typetheory}. In this type theory there are types, like the +integers $\Z$, vectors $\R^n$; and complex functions $\C\to\C$. There are also +terms, which have a unique type.\footnote{To be more precise: in many type theories there are terms with multiple types, for example due to universe cumulativity, but we will ignore these issues. Moreover, the type theory of Lean has unique typing~\cite{carneiro2018leantheory}.} +For example the number $-2$ has type $\Z$ +(written as $-2:\Z$), the vector $(1,2,3,\ldots,n)$ has type $\R^n$ and we have the +exponential function $\exp:\C\to\C$. One can think of types as sets of objects +(and indeed, there is a model of type theory where the types are exactly sets), +but there are different interpretations, such as the types-as-spaces +interpretation that homotopy type theory provides. The fact that terms have a unique type means that the +$2:\Z$ and the $2:\R$ are different objects. It might be helpful to think of data types in a programming language, in which the \texttt{int} $2$ is stored differently in memory than the \texttt{float} $2$. Of course, the canonical inclusion +$i:\Z\hookrightarrow\R$ does satisfy $i(2)=2$. Type theory has a primitive +notion of computation, so that for example $2+3$ computes to $5$. Every function +that is explicitly defined in type theory therefore describes an algorithm +that can be executed. This means that type theory can be used as a programming +language, and many programming languages make use of a type system. +The congruence closure of this notion of computation is called +\emph{definitional equality} or \emph{judgmental equality}, +and if two terms are judgmentally equal, one can replace one for the other in any term. + +There are several methods to construct new types out of existing ones. For +example we can form the function type $A\to B$ for types $A$ and $B$, the +cartesian product type $A\times B$ and the coproduct or sum $A+B$. Propositions +can also be interpreted as types by the \emph{Curry-Howard isomorphism}~\cite{curry1958combinatorylogic,howard1980formulae}, and +under this interpretation $A\times B$ is the conjunction of $A$ and $B$, the sum +$A+B$ is the disjunction and $A\to B$ is the implication. Furthermore, there are +dependent function types $\prd{x:A}P(x)$ and dependent sum types $\sm{x:A}P(x)$, +which correspond to the universal quantification $\forall(x:A), P(x)$ and +existential quantification $\exists(x:A), P(x)$, respectively. So for example +the transitivity of $\le$ on $\N$ can be expressed as $\prd{k,m,n:\N}k \le m \to +m \le n \to k \le n$, and a term of this type is a proof that $\le$ is +transitive. The $P$ in $\prd{x:A}P(x)$ and $\sm{x:A}P(x)$ is called a +\emph{dependent type}, since it is a type depending on a term $x:A$. It has type +$P:A\to\type$, where $\type$ is the universe of (small) types. The dependent +function type $\prd{x:A}P(x)$ consists of functions $f$ that send terms $a:A$ +to a term $f(a):P(a)$. Note that the type of $f(a)$ depends on the input $a$. +The dependent sum type $\sm{x:A}P(x)$ consists of dependent pairs $(a,x)$ with +$a:A$ and $x:P(a)$, where the type of $x$ depends on $a$. + +Given two terms $a, b : A$, we can form the \emph{identity type} which we write as $a =_A b$ or $a=b$. As a proposition we view $a=_A b$ as the statement that $a$ and $b$ are +equal. In homotopy type theory these identity types correspond to the path space +of the type $A$. + +\subsubsection*{Homotopy Type Theory} + +There are various versions of dependent type theory with different rules for the +identity type. Some type theories have a \emph{reflection rule}, which states that +if we have a proof $p:a=b$, then $a$ and $b$ are judgmentally equal. Type theories with this rule are often called \emph{extensional}. +This is a convenient rule, but these type theories have meta-theoretic properties that are often seen as undesirable. +For example, checking whether a term $t$ has type $A$ is not decidable anymore. +Since this operation can be viewed as ``checking the correctness of a proof,'' one often wants to work in a type theory with decidable type-checking. + +In \emph{intensional} type theory, without the reflection rule, multiple approaches can be taken for the identity type. +In some versions, there is a rule that any two proofs of the same +equality are themselves equal. This rule, often called \emph{uniqueness of +identity proofs} or \emph{axiom K} states that if $p,q : a = b$, then there is a +proof of $p = q$. In homotopy type theory, this rule is rejected. In the +types-as-spaces interpretation of homotopy type theory, terms of the identity +type $a =_A b$ are interpreted as paths in $A$ from $a$ to $b$. +We have familiar operations on paths: given two paths $p:a=_Ab$ and $q:b=_Ac$, +we write $p\cdot q:a=c$ for the concatenation of $p$ and $q$. +Furthermore, we have the inverse path $p\sy:b=a$ and the constant path $\refl_a:a=a$. +We also have higher paths, the identity type $p =_{a=_Ab} q$ consists of homotopies from path $p$ to $q$. We can form +higher path types between two homotopies, and there are also operations on these higher paths. +In this way every type comes equipped with the structure of a higher groupoid. + +In 2011, higher inductive types were introduced in homotopy type +theory~\cite{shulman2011intervalimpliesfunext,lumsdaine2011hits,shulman2011HoTThits,shulman2011pi1S1}. With +ordinary inductive types we specify constructors that generate the type, for +example the natural numbers are generated by zero $0:\N$ and the successor +function $\mathsf{succ}:\N\to\N$. Higher inductive types are generated not only +by these ``point constructors'' but also by ``path constructors,'' which specify +the inhabitants of paths or higher paths in the type. For example, the circle +$\S^1$ is generated by a point $\star:\S^1$ and a loop $\ell:\star=\star$. The +rest of the structure of $\S^1$ is built from these constructors. Using higher +inductive types we can construct many other spaces in homotopy theory, +such as Eilenberg-MacLane spaces and homotopy pushouts. + +As mentioned before in this introduction, we can use HoTT to do homotopy theory. +We think of types as spaces and we think of maps between types as continuous +maps between those spaces. Then we can define usual notions in homotopy theory, +as long as they are homotopy invariant: homotopy equivalences, suspensions, +spheres, etcetera. This is a \emph{synthetic} way to do homotopy theory: many +concepts, such as spaces and paths are uninterpreted constants of the type +theory. This is opposed to \emph{analytic} homotopy theory, where one studies +topological spaces up to homotopy equivalence. This distinction is similar to +the distinction for elementary geometry, which we can do synthetically (points +and lines are undefined concepts) or analytically (we are working in $\R^2$). +Synthetic geometry limits the things one can state or prove, but these proofs +are applicable in every model of the axioms. The same is true for synthetic +homotopy theory: the proofs performed synthetically are true in all models of +HoTT. + +In this dissertation I will not be very precise about the exact rules of the type theory we are using. We will +present the constructions and proofs in such a way that they can be performed in the ``HoTT +book''~\cite{hottbook}. Most of the results in this dissertation have been +formalized in the Lean proof assistant~\cite{moura2015lean}. The HoTT mode we +used in Lean has very similar rules to the HoTT book, and the differences are +not relevant for the constructions in this dissertation. A concept closely related to +homotopy type theory is \emph{univalent mathematics}, a term coined by Vladimir +Voevodsky for the development of mathematics where one takes homotopy types as +primitive objects, and reasons about them using type-theoretic reasoning and the +univalence axiom. This is pursued in the proof assistant UniMath~\cite{unimath}. There are also +radically different type theories which are studied in homotopy type +theory. These are called ``cubical type theories'' because they all have a +primitive notion of cubes. Examples include the cubical type theory described in~\cite{cohen2016cubical}, +which was implemented in the proof assistant cubicaltt~\cite{cubicaltt}, +and computational higher-dimensional type theory~\cite{angiuli2017computational}, +on which the proof assistant RedPRL is based~\cite{redprl}. +These type theories are extensions of the type theory presented in +the HoTT book, which we will call book-HoTT. In book-HoTT the univalence axiom +is an axiom: an uninterpreted constant of a certain type. This breaks the +computational behavior of the type theory. For example not every closed term of +type $\N$ computes to either $0$ or the successor of another number. These +cubical type theories add primitive concepts to the theory to make the +univalence axiom provable, and therefore all terms in these system do compute. + +We will often want to compare homotopy type theory with ordinary homotopy +theory. We will use the adverb ``classically'' to refer to the concepts and +theorems in homotopy theory that do not involve HoTT.\footnote{This use of classically has nothing to do with the word classical in ``classical logic,'' involving the law of excluded middle or the axiom of choice. In homotopy type theory one can consistently assume the law of excluded middle or the axiom of choice, formulated in a precise way so that it corresponds to what it usually means. However, doing so removes the computational content of all notions defined using it.} +Conversely, we will say that something is provable in HoTT if we can prove it in book-HoTT. + +\subsubsection*{Contents} + +In \autoref{cha:preliminaries} we review the basic concepts in homotopy type +theory. For a more detailed and thorough exposition, we refer to~\cite{hottbook}. +Alternative introductions can be found in~\cite{favonia2017thesis} and~\cite{brunerie2016spheres}. In +\autoref{sec:martin-lof-type} we introduce the basic concepts of type theory: +functions, pairs, universes, and inductive types such as the identity type. In +\autoref{sec:homotopy-type-theory} we will introduce the basics of homotopy type +theory. In particular we will formally state the univalence axiom and present +higher inductive types. In \autoref{sec:lean} we will discuss the Lean proof +assistant in more detail. + +In \autoref{cha:high-induct-types} we will study higher inductive types +internally in HoTT. The main problem we will focus on is the interdefinability +of higher inductive types. In particular, we try to construct various higher +inductive types from the homotopy pushout. We will define the propositional +truncation in \autoref{sec:prop-trunc}, nonrecursive higher inductive types with +2-path constructors in \autoref{sec:non-recursive-2} and work towards defining +certain \emph{localizations} in \autoref{sec:colimits}. + +In \autoref{cha:homotopy-theory} we present some synthetic homotopy theory in +HoTT. In \autoref{sec:computing-pi3s2} we will describe the formalization of the +long exact sequence of homotopy groups and its application to compute +$\pi_3(\S^2)$. Although this construction has been described before in HoTT +in~\cite[Section 8.4]{hottbook} and~\cite[Section 2.5.1]{brunerie2016spheres}, +no formally verified proof has been given before. In +\autoref{sec:eilenb-macl-spac} we will study Eilenberg-MacLane spaces, which are +spaces with only one nontrivial homotopy group. Eilenberg-MacLane spaces have +been defined in HoTT before~\cite{licata2014em}. Here we prove the (classically +known) results that Eilenberg-MacLane spaces are unique, and give an equivalence +of categories between the category of (abelian) groups and an appropriate class of pointed +types. In \autoref{sec:smash-product} we will discuss the smash product. The +ultimate goal is to prove that the smash product forms a 1-coherent symmetric monoidal +product on pointed types, and we will give one approach towards proving this +using a Yoneda-style argument. + +In \autoref{cha:serre-spectr-sequ} we develop the theory of spectral sequences +in HoTT. We give the construction of a spectral sequence from an exact couple +(in \autoref{sec:exact-couples}) and show how to construct an exact couple from +a tower of spectra (in \autoref{sec:spectra}). We construct the +classically-known Atiyah-Hirzebruch and Serre spectral sequences for cohomology +(in \autoref{sec:spectral-sequence-cohomology}), and give some ideas towards +doing the same for their counterparts in homology (in +\autoref{sec:spectral-sequence-homology}). +%Finally, we give some applications in \autoref{sec:applications-spectral-sequences}. + +\chapter{Preliminaries}\label{cha:preliminaries} + +In this chapter we will give a brief overview of type theory and homotopy type +theory. We cannot cover all the subtleties, so readers new to (homotopy) type +theory should consult the homotopy type theory book~\cite{hottbook}. + +In \autoref{sec:lean} we will discuss the proof assistant \emph{Lean}. All main +results in this dissertation have been formalized in Lean. + +\section{Martin-L\"of Type Theory}\label{sec:martin-lof-type} + +As mentioned in the introduction, homotopy type theory is based on a system called \emph{Martin-L\"of type theory} or \emph{intuitionistic type theory}. +There are types and there are terms, which have a unique type. There is a notion of computation. Two terms $t$ and $s$ are considered \emph{judgmentally equal} or \emph{definitionally equal}, +denoted $t \equiv s$ if $t$ and $s$ compute to the same term. + +We are working in dependent type theory, which means that types can depend on terms. +For example, there is a type of vectors of length $n:\N$ in type $A$, denoted $\Vector_A(n)$. +In this case $\Vector_A$ is a dependent type over $\N$. +An example term in this type family is $(5,6,7,8) : \Vector_\N(4)$. +When we say that a term has a unique type, we mean that it has a unique type up to definitional equality. +In our example, we also have that $(5,6,7,8) : \Vector_\N(2+2)$, because $2+2\equiv 4$. +More generally, if we have two definitionally equal types $A\equiv B$ and if $t : A$, then $t : B$. +Logically (under the types-as-propositions interpretation) dependent types are predicates. +We will explain the topological interpretation of dependent types at the end of \autoref{sec:pair-types}. + +In the remainder of this section we will discuss the type formers of Martin-L\"of type theory more closely. + +\subsection{Function Types}\label{sec:function-types} +Given a type $A$ and a family of types $B$ depending on $A$, we can form the \emph{dependent function type} (also called \emph{product type} or \emph{pi type}) $$(x : A) \to B(x)\qquad\text{or}\qquad\prd{x:A}B(x).$$ +We will use the former notation in this document. A term $f:(x : A)\to B(x)$ is a function that sends each element $a : A$ to an element\footnote{Formally, $B(a)$ is the term $B(x)$ where we substitute $a$ for $x$. In \autoref{sec:universes} we will see that we can treat $B$ as a function into a universe, and that alternatively we can view $B(x)$ and $B(a)$ as function applications.} $f(a):B(a)$. We also use the notation $fa$ or $f\ a$ for $f(a)$. Note that the type of $f(x)$ depends on $x$. We can form functions using \emph{lambda-abstraction}. Given a term $t(x) : B(x)$, we can form the term $\lam{x}t(x): (x : A) \to B(x)$, which is the function $x\mapsto t(x)$, i.e. the function that sends $x$ to $t(x)$. We get the computation rule +$$(\lam{x}t(x))a\equiv t(a)$$ +for $a : A$, which is called the \emph{beta-rule} or \emph{beta-reduction}. We also have an \emph{eta-rule}, which states that every function is a lambda abstraction. This means that for $f : (x : A) \to B(x)$ we have +$$f\equiv \lam{x}f(x).$$ +We will often define functions by writing $f(x)\defeq t$ (where $x$ may occur in $t$), which formally means that we define $f$ as $\lam{x}t$. + +An important special case occurs when $B$ does not depend on $A$. In this case the dependent function type $(x : A) \to B$ is written as $A \to B$, which is the type of functions from type $A$ to type $B$. + +Logically, the type $A \to B$ is interpreted as the implication $A \Rightarrow B$ and the type $(x : A) \to B(x)$ is interpreted as the universal quantification $\forall(x : A), B(x)$. Topologically, a function $f : A \to B$ corresponds to a continuous map from $A$ to $B$. The type $A \to B$ is the mapping space from $A$ to $B$. We will explain the topological interpretation of $(x : A) \to B(x)$ at the end of \autoref{sec:pair-types}. + +We can define the identity function +$$\idfunc\equiv\idfunc[A]\defeq \lam{x:A}x:A\to A$$ +and the composition of functions: if $f : A \to B$ and $g : B \to C$, then $g \o f\defeq \lam{x}g(f(x)):A\to C$. Given $b:B$, we also have a constant function $\const_b\defeq \lam{x}b:A \to B$. + +We will often write some arguments of a function implicitly. Such arguments are written with curly braces in the type. For example, given a dependent type $C$ over $\N$, we write +$$g : \{n : \N\} \to C(n) \to C(n+1)$$ +to emphasize that the first argument of $g$ is implicit. In this case, for $c : C(n)$ we will write $g(c)$ for $g$ applied (implicitly) to $n$ and applied to $c$. The curly braces are only to indicate how we write function application for functions with this type, for all other purposes the types $\{x : A\} \to B(x)$ and $(x : A) \to B(x)$ are the same. + +\subsection{Pair Types}\label{sec:pair-types} +Given a type family $B$ depending on a type $A$, we can form the \emph{dependent pair type} (also called \emph{dependent sum type} or \emph{sigma type}) +$$(x : A) \times B(x)\qquad\text{or}\qquad\sm{x:A}B(x).$$ +We will use the former notation in this document. A term of type $(x : A)\times B(x)$ is a pair consisting of an element $a : A$ and an element $b : B(a)$. Given $a : A$ and $b : B(a)$, we can form the term $(a,b):(x : A)\times B(x)$, and we have projections +$$p_1:(x : A)\times B(x)\to A\quad \text{ and }\quad p_2:(z : (x : A)\times B(x))\to B(p_1(z)).$$ +We will sometimes write $x.i$ for $p_i(x)$. There are beta rules $p_1(a,b)\equiv a$ and $p_2(a,b)\equiv b$ and an eta rule stating that for any $z:(x : A)\times B(x)$ we have $z\equiv (p_1z, p_2z)$. In Lean, there is no eta rule for dependent pair types, but instead there is an induction principle, similar to those of inductive types (see \autoref{sec:inductive-types}). %For more details, see \autoref{sec:lean-eta}. + +If $B(x)$ does not depend on $x$, we write $(x : A) \times B$ simply as $A\times B$. In this case we retrieve the usual cartesian product of $A$ and $B$. + +Logically we can think of $A\times B$ as the conjunction of $A$ and $B$, as +described above. Furthermore, we can think of $(x : A)\times B(x)$ as a +proof-relevant version of the existential quantifier $\exists(x : A). B(x)$. It +is proof-relevant in the sense that from a proof of $(x : A)\times B(x)$ we can +extract a witness $a : A$ such that $B(a)$ holds. In \autoref{sec:truncatedness} +we will define an existential quantifier from which the witness cannot be +extracted. + +Topologically, we think of $A\times B$ as the product space of $A$ and $B$. +The map $p_1 : (x : A) \times B(x) \to A$ corresponds to a \emph{fibration}. A fibration is a map that has the homotopy lifting property with respect to any space, which is given by transport, to be defined in \autoref{sec:paths}. Under this interpretation, $(x : A) \times B(x)$ is the total space of the fibration $p_1$, and $B(a)$ is the \emph{fiber} of $p_1$ at point $a$. The type $(x : A)\to B(x)$ is the type of sections of $p_1$. These observations are usually summarized as ``dependent types correspond to fibrations.'' We will often call dependent functions \emph{sections}. + +\subsection{Universes}\label{sec:universes} +In our discussions below we need one or more \emph{universes} in our type theory. There are different styles of universes in type theory~\cite{martinlof1984typetheory}, we will describe the universes \'a la Russell. A universe $\type$ is a type that has types as its terms. That is to say, if $A:\type$, then $A$ is a type. It is closed under all type-forming operations. For example, for pi-types this means that if $A : \type$ and for $a:A$ we have $B(a):\type$, then +$$(a : A) \to B(a):\type.$$ +We can now interpret dependent types in $\type$, such as $B$ above, as functions $B:A\to\type$. + +In the proofs in this document we can often get away with assuming only a single universe. However, it is useful to have the property that all types have a type themselves, and we cannot do that with a single universe $\type$, because positing $\type:\type$ is inconsistent~\cite{girard1972paradox}. Instead, we will assume that we have a tower of universes $$\type_0:\type_1:\type_2:\cdots$$ +such that for every type $A$ there is an $i$ such that $A:\type_i$. In this case every dependent type can be interpreted as a function $A\to\type_i$ for some $i$. As is customary, we usually omit writing universe levels explicitly, and we will perform constructions polymorphic over all universes. For example, if we write $$\idfunc : \{A : \type\} \to A \to A,$$ we really mean that for any universe level $i$ we have $$\idfunc[i] : \{A : \type_i\} \to A \to A.$$ + +One rule that is sometimes assumed is \emph{universe cumulativity}, which states that if $A : \type_i$, then $A : \type_j$ for $j\geq i$. +This can be problematic, and lead to violation of nice properties of the type theory, such as subject reduction or canonicity~\cite{luo2012universes}. +In this document (and in Lean), we do not assume universe cumulativity. +Instead, using inductive types (see \autoref{sec:inductive-types}) we can construct for $A : \type_i$ a new type $\ulift A:\type_j$ for $j\geq i$ such that $A\simeq \ulift A$. + +\subsection{Inductive Types}\label{sec:inductive-types} + +\emph{Inductive types} are types that are inductively generated by some +\emph{constructors}. A simple example is $\N$, which is inductively generated by +$0$ and the successor function $S\defeq\lam{x}x+1$. In this section we will +discuss some inductive types that we will need in this dissertation. We will +talk about the empty type, the unit type, the booleans, the natural numbers and +the sum type. The dependent pair type (\autoref{sec:pair-types}) is also an inductive type. + +\subsubsection*{The empty type} +The empty type $\emptyt:\type_0$ is a type without inhabitants. There are no constructors, and we have as induction principle that if $P:\emptyt\to\U_i$, then $$\ind{\emptyt}:(x : \emptyt) \to P(x).$$ This conveys that $\emptyt$ indeed has no inhabitants, because if we view $P$ as a predicate, we can prove anything about all inhabitants of $\emptyt$. We can define negation $\neg A\defeq A \to \emptyt$. + +\subsubsection*{The unit type} +The unit type $\unit:\type_0$ is a type with exactly one inhabitant $\star:\unit$. The induction principle states that if $P:\unit\to\U_i$, then $$\ind{\unit}:P(\star) \to (x : \unit) \to P(x).$$ This states that $\star$ is the only inhabitant of $\unit$, because if we can prove something for $\star$, then it holds for all inhabitants of $\unit$. There is a computation rule $$\ind{\unit}(p,\star)\equiv p.$$ + +\subsubsection*{The booleans} +The type of booleans $\bool:\type_0$ has exactly two inhabitants $\btrue,\bfalse:\bool$. Its induction principle states that if $P:\bool\to\U_i$, then $$\ind{\bool}:P(\btrue) \to P(\bfalse) \to (x : \bool) \to P(x).$$ +The computation rules are +$$\ind{\bool}(p_\btrue,p_\bfalse,\btrue)\equiv p_\btrue\qquad\text{and}\qquad +\ind{\bool}(p_\btrue,p_\bfalse,\bfalse)\equiv p_\bfalse.$$ + +\subsubsection*{The natural numbers} +A more interesting type is the type of natural numbers $\N:\type_0$. It has a constructor $0:\N$ and a unary constructor $S:\N \to \N$, and it is freely generated by these constructors. This means that if $P:\N\to\U_i$ and if we have $p_0:P(0)$ and $p_S : (k : \N) \to P(k) \to P(S\ k)$, then $$\ind{\N}(p_0,p_S):(n : \N) \to P(n).$$ +If we view $P$ as a predicate, this is the usual induction principle for $\N$: to prove something for all numbers we need to prove it for $0$ and we need to prove it for $k+1$ assuming it holds for $k$, for an arbitrary $k$. However, this induction principle also allows us to define (dependent) functions from $\N$. These functions satisfy the computation rules +$$\ind{\nat}(p_0,p_S,0)\equiv p_0\qquad\text{and}\qquad +\ind{\nat}(p_0,p_S,S\ n)\equiv p_S(n,\ind{\nat}(p_0,p_S,n)).$$ +Often, we will want to give a name $f$ to $\ind{\nat}(p_0,p_S)$, and we will instead denote the recursive definition of $f$ using pattern matching notation: +$$f(0)\defeq p_0\qquad\text{and}\qquad f(S\ n)\defeq p_S(n,f(n)).$$ +For example, we can define addition and multiplication $+,\cdot:\N\to\N\to\N$ recursively (in the second argument) as +\begin{align*} + n+0&\defeq n & n\cdot 0&\defeq 0\\ + n+(S\ m)&\defeq S(n + m) & n \cdot (S\ m)&\defeq n \cdot m + n. +\end{align*} +Note that $n+1\equiv S\ n$, and we will often write $n+1$ instead of $S\ n$ from now on. + +\subsubsection*{The sum type} +Given two types $A$ and $B$, we can form the \emph{sum type} or \emph{coproduct} $A+B$ with constructors $\inl:A\to A+B$ and $\inr : B \to A + B$. +The induction principle states that for $P:A+B\to\type$ with maps $p_{\inl} : (a : A) \to P(\inl a)$ and $p_{\inr} : (b : B) \to P(\inr b)$ we get a section +$$\ind{{+}}(p_{\inl},p_{\inr}):(x : A + B) \to P(x)$$ +with computation rules +$$\ind{{+}}(p_{\inl},p_{\inr},\inl(a))\equiv p_{\inl}(a)\qquad\text{and}\qquad +\ind{{+}}(p_{\inl},p_{\inr},\inr(b))\equiv p_{\inr}(b)$$ + +Logically, the type $A+B$ is the proof-relevant disjunction of $A$ and $B$. It is +proof-relevant in the sense that a proof of $A + B$ is of the form +$\inl a$ or $\inr b$. Therefore, a proof comes with a proof of either $A$ or $B$. In +\autoref{sec:truncatedness} we will see a disjunction that does not have this +property. + +\subsubsection*{General Inductive Types} + +In \autoref{sec:inductive-types} we saw various instances of inductive types. +Also the sigma-types from \autoref{sec:pair-types} (without eta rule) are an instance of an +inductive type. We will now explain inductive types and families of inductive +types in general. For a more detailed description, see~\cite[Section ``Inductive Types'']{carneiro2018leantheory}. + +When defining an inductive type, we have to list its constructors. For example, we could define the sum type as follows. Given $A\ B : \type$, we define + +\begin{inductive} + \texttt{inductive} $A + B : \type \defeqp$ \\ + $\bullet\ \inl : A \to A + B;$\\ + $\bullet\ \inr : B \to A + B.$ +\end{inductive} +This defines the type $A+B$ with constructors $\inl$ and $\inr$ of the specified +type. Each constructor must have as target the inductive type currently being +defined (in this case $A+B$).\footnote{For \emph{higher inductive types} +(\autoref{sec:high-induct-types}) the conclusion can also be a (higher) path in +the type currently being defined.} Constructors can be \emph{recursive}, +meaning that the type being defined can occur in the domain of a constructor. +For example, here is the type of $\omega$-branching trees with leaves labeled by +a type $C$. +\begin{inductive} + \texttt{inductive} $\wtree_C : \type \defeqp$ \\ + $\bullet\ \leaf : C\to\wtree_C;$\\ + $\bullet\ \node : (\N \to \wtree_C)\to\wtree_C.$ +\end{inductive} +A restriction on recursive constructors is that the inductive type being defined can only occur in strictly positive positions, that is as the target of one of the arguments of the constructor. + +Every inductive type has an induction principle. We can algorithmically find the type of the induction principle from the constructors. The first argument of the induction principle (often left implicit) is the \emph{motive}, which is an arbitrary type family over the inductive type being defined, for $\wtree_C$ this has type $P:\wtree_C\to\type$. Then for every constructor $c$ there is an argument that mimics the type of $c$ and has as target $P(c(\cdots))$. +For $\wtree_C$ these arguments have type $p_{\leaf}:(c:C)\to P(\leaf c)$ and +$$p_{\node}:(f:\N\to\wtree_C)\to ((n : \N) \to P(f\ n))\to P(\node f).$$ +Note that for each recursive argument $f$ of the constructor we assume an \emph{induction hypothesis} of type $P(f(\cdots))$. The induction principle then gives a section of $P$. So for example we get +$$\ind{\wtree_C}(p_{\leaf},p_{\node}):(x : \wtree_C) \to P(x).$$ +Finally, the computation rules states that if the induction principle acts on a constructor, then it will reduce to the argument corresponding to that constructor. For $\wtree_C$ this means (abbreviating $s\defeq \ind{\wtree_C}(p_{\leaf},p_{\node})$) +$$s(\leaf(c))\equiv p_{\leaf}(c)\qquad\text{and}\qquad +s(\node(f))\equiv p_{\node}(f,\lam{n}s(f\ n)),$$ +where applying $s$ to the recursive constructor leads to a recursive call of $s$. + +One important generalization of inductive types are families of inductive types. In this case, a family of types $P$ is being defined simultaneously indexed over some type $I$. In this case, constructors must have as target $P(t)$ where $t$ is a term of type $I$ formed by the (nonrecursive) arguments of the constructor. An example of an inductive family of types is the type of vectors in $A$ of some length $n:\N$. +\begin{inductive} + \texttt{inductive} $\Vector_A : \N \to \type \defeqp$ \\ + $\bullet\ \nil : \Vector_A(0);$\\ + $\bullet\ \cons : \{n : \N\} \to A \to \Vector_A(n) \to \Vector_A(n+1).$ +\end{inductive} +Note that the parameter $A$ remains fixed in the definition of $\Vector_A(n)$, while the index $n:\N$ is not: the constructor $\cons$ constructs a vector of length $n+1$ from a vector of length $n$. +The induction principle can again be extracted algorithmically. It is important that the motive also quantifies over all indices of the inductive family. For vectors it states that given a motive +$$P:\{n : \N\} \to \Vector_A(n)\to \type$$ +and induction steps +\begin{align*} + p_{\nil}&:P(\nil)\\ + p_{\cons}&:(n : \N) \to (a : A) \to (x : \Vector_A(n))\to P(x) \to P(\cons(a,x)), +\end{align*} +we get a section +$$\ind{\Vector}(p_{\nil},p_{\cons}):\{n : \N\} \to (x : \Vector_A(n)) \to P(x)$$ +with the expected computation rules. + +A very important inductive family of types is the \emph{identity type}.\footnote{also called \emph{path type}, \emph{identification type} or \emph{equality type}.} This is a family of types with parameters $A:\type$ and $a:A$ and is defined as +\begin{inductive} + \texttt{inductive} $\Id_A(a,{-}) : A \to \type \defeqp$ \\ + $\bullet\ \refl_a : \Id_A(a,a).$ +\end{inductive} +We also denote the type $\Id_A(a_1,a_2)$ by $a_1=_A a_2$ or $a_1=a_2$ and $\refl_a$ by $\refl$, $1_a$ or $1$. Its induction principle states that for a family $P:(a' : A) \to a = a' \to \type$ and a term $p_{\refl}: P(a,1_a)$ we find a section +$$\ind{=}(p_{\refl}):(a' : A) \to (p : a = a') \to P(a',p).$$ +In words: we may assume that a path with free right endpoint (that is, the right hand side of the equality is a variable) is reflexivity. + +Logically, the identity type corresponds to equality. Under this interpretation, a term of type $a_1=a_2$ is a proof that $a_1$ and $a_2$ are equal. Homotopically, the identity type corresponds to the path space of $A$, and we will explore this interpretation more in \autoref{sec:paths}. + +% We will use the following notation: +% \begin{itemize} +% \item We use $=$ for equality/identification/identity/``propositional'' equality and $\equiv$ for +% ``judgmental'' or ``definitional'' equality. We use $x\defeq t$ to define $x$ as $t$ and $\ap +% fp\defeqp q$ (or $\apd fp\defeq q$) to define a function $f$ on a path constructor $p$ of a higher inductive type. +% \item If $f:A\to B$ and $g:C\to D$, we define $f\times g:A\times C \to B\times D$, $f+g:A+C\to +% B+D$. +% \item If $f:A\to B$ and $g:A\to C$, we define $(f,g):A\to B\times C$. +% \item If $f:A\to C$ and $g:B\to C$, we define $\pair fg:A+B\to C$. +% \item We denote the maps $\emptyt\to A$ and $A\to\unit$ by $!$. +% \end{itemize} + + +\section{Homotopy Type Theory}\label{sec:homotopy-type-theory} + +We will now discuss in more detail the homotopical interpretation of types, and the basic concepts of homotopy type theory. + +\subsection{Paths}\label{sec:paths} + +Elements of an identity type form paths in the space. We can define the usual operations on paths. + +Given a path $p:a=_Ab$, we can define the \emph{inverse} $p\sy:b=_Aa$. We can do this by path induction. Define the family $$P\defeq \lam{x:A}{q:a=_Ax}x=_Aa:(x:A)\to a=_Ax \to \type.$$ +We now have $\refl_a:P(a,p)\equiv a =_A a$, and therefore we get +$$p\sy\defeq\ind{=}(\refl_a,b,p):b=_Aa.$$ +The computation rule gives that $\refl_a\sy\equiv \refl_a$. + +We can explain the proof in words more intuitively. Path induction states that we may assume that a path with a free endpoint is reflexivity. Since $p$ has a free endpoint ($b$ is a variable), we may assume that $b\equiv a$ and $p\equiv\refl_a$. In this case, we can define $$p\sy\equiv \refl_a\sy\defeq \refl_a:a=a.$$ +The map path inversion we have defined this way has type +$$\{a\ b : A\} \to a = b \to b = a.$$ +We can also define path \emph{concatenation}. Given $p:a=_Ab$ and $q:b=_Ac$, we define $p\tr q:a=_Ac$ again by path induction. We will only give the intuitive argument and leave the formal proof to the reader. Since $q$ has free endpoint $c$, we may assume that $c\equiv b$ and $q\equiv\refl_b$. In this case, we define $p\tr \refl_b\defeq p:a=b$. + +We can also define higher paths. For example, given $p:a=b$ and $q:b=c$ and $r:c=d$, we have a path +$$p\tr(q \tr r)=(p\tr q)\tr r,$$ +which is the \emph{associativity} of path concatenation. We can prove this by path induction on $r$: if $r$ is reflexivity, then both sides reduce to $p\tr q$. + +By using path induction, we can also prove the following equalities: +\begin{align*} + p\cdot 1 &= p & p\cdot p\sy &= 1\\ + 1\cdot p &= p & p\sy\cdot p &= 1. +\end{align*} +It is trickier to prove the Eckmann-Hilton property of equality, which states that given $a:A$ and $p,q:\refl_a=\refl_a$, we have $p\cdot q=q\cdot p$. +The problem is that cannot apply path induction to $p$ or $q$ directly. We omit the proof here and refer to~\cite[Theorem 2.1.6]{hottbook}. + +Given a map $f:A\to B$, we can prove that $f$ respects paths. Given a path +$p:a=_Aa'$, we define $\apfunc{f}(p):f(a)=_Bf(b)$ by path induction: for +reflexivity we define $\apfunc{f}(\refl_a)\defeq \refl_{f(a)}$. We will sometimes +abuse notation and write $f(p)$ for $\apfunc{f}(p)$. From a logical perspective this just states that functions respect equality, +but from a homotopical perspective, this states that functions respect paths, which is in line with our intuition that all functions are continuous in HoTT. + +We can compute what $\apfunc{}$ does when our map is the identity map, a constant map or a composition of maps: +\begin{align*} + \apfunc{\id_A}(p)&=p&\apfunc{\const_b}(p)&=\refl_b&\apfunc{g\o f}(p)=\apfunc{g}(\apfunc{f}(p)). + \end{align*} + All three of these properties are easily proven by path induction. Also, we can compute $\apfunc{}$ when we apply it to inverses or concatenations of paths: + \begin{align*} + \apfunc{f}(p\tr q)&=\apfunc{f}(p)\tr\apfunc{f}(q)&\apfunc{f}(p\sy)&=(\apfunc{f}(p))\sy. + %&\apfunc{f}(\refl_a)&\equiv\refl_{f(a)} + \end{align*} + + Given a dependent type $P:A\to\type$ and a path $p:a=_Aa'$, we can define the + \emph{transport} function $\transp^P(p): P(a)\to P(a')$. We define it by path induction; for reflexivity we define $\transp^P(\refl_a)\defeq\id_{P(a)}$. When $P$ is known from context we will write $p_*(b)$ for $\transp^P(p,b)$. + +By path induction we can prove basic equalities about transports. We have + +\begin{align*} + \transp^P(p \tr q,x) &= \transp^P(q,\transp^P(p,x))\\ + \transp^{\lam{a}B}(p,x) &= x\\ + \transp^{P\o f}(p,x) &= \transp^P(\apfunc{f}(p),x)\\ + f_{a'}(\transp^P(p,x)) &= \transp^Q(p,f_a(x))&&\text{for $f:(a:A)\to P(a)\to Q(a)$.} +\end{align*} + +\subsection{Equivalences}\label{sec:equivalences} % rename to functions? + +In this section we talk about maps between types that have an inverse in a suitable way. Before we can give the definition, we need to define homotopy. + +Given two dependent maps $f,g:(a:A)\to B(a)$, a \emph{homotopy} $h:f\sim g$ is a proof that $f$ and $g$ are pointwise equal: +$$(f\sim g)\defeq (a:A)\to f(a)=_{B(a)}g(a).$$ +Recall that all maps are considered continuous, so this actually gives a continuous deformation of $f$ to $g$, which is exactly what a homotopy is in topology. + +\begin{defn}Suppose given a function $f:A\to B$. + \begin{itemize} + \item A \emph{left-inverse} of $f$ is an inhabitant of $(g:B\to A)\times g\o f\sim\id_A$. + \item Similarly, a \emph{right-inverse} of $f$ is an inhabitant of $(h:B\to A)\times f\o h\sim\id_B$. + \item We say that $f$ is an \emph{equivalence} or $\isequiv(f)$ if $f$ has both a left and a right inverse. We will denote its left-inverse by $f\sy$. We can then show that $f\sy$ is also a right inverse of $f$. + \item The type of equivalences between $A$ and $B$ is $(A\simeq B)\defeq(f:A\to B)\to \isequiv(f)$. Given an element $f:A\simeq B$, we will also use $f$ to denote the underlying map $A\to B$. + \end{itemize} +\end{defn} +It is easy to show that the identity map $\id_A:A\simeq A$ is an equivalence. Moreover, if $g:B\to C$ and $f:A\to B$ are both equivalences, then $g\o f$ and $f\sy$ are also equivalences. This shows that equivalences are reflexive, symmetric and transitive. + +A very important property is that any two inhabitants of $\isequiv(f)$ are equal: if $p,q:\isequiv(f)$, then $p=q$. We will not prove this here, but it is shown in~\cite[Theorem 4.3.2]{hottbook}. This property is the reason that we define the notion of equivalences this way. +% that $f$ is an equivalence if $f$ has both a left inverse and a right inverse, which are not required to be the same (although we can prove that they are). +If we would define $\isequiv(f)$ by requiring a map that is both a left \emph{and} a right inverse of $f$, then this property would not hold. + +Given two equivalences $f,f':A\simeq B$, it does not matter whether we compare them as functions or equivalences: +$$(f=_{A\simeq B}f')\simeq (f=_{A\to B}f') \simeq (f\sim f').$$ + +By path induction we also get a map $(A=_{\type}B)\to(A\simeq B)$, because if the path $p:A=B$ is reflexivity, we can just take $\id_A:A\simeq A$ as our equivalence. In plain Martin L\"of type theory one cannot characterize what the type $A=B$ is. This is where the univalence axiom comes in. The \emph{univalence axiom} states that the map +$$(A=B)\to (A\simeq B)$$ +is an equivalence. In particular this means that we get a map in the other direction: given an equivalence $e:A\simeq B$, we get an equality $\ua(e):A=B$. + +\subsection{More on paths}\label{sec:more-paths} +In this section we will discuss dependent paths, or pathovers; higher paths, such as squares and cubes; and paths in type formers. + +\subsubsection*{Pathovers} +We will often need to relate elements in two different fibers of a dependent type. Suppose we have a family $P:A\to\type$ with $x:P(a)$ and $x':P(a')$. If we have a path $p:a=a'$, we can form the type $x =_p^P x'$ of \emph{dependent paths} or \emph{pathovers} over $p$. There are four equivalent ways to define this: +\begin{enumerate} + \item \label{item:pathover-tr} We can define $(x =_p^P x')\defeq(\transp^P(p,x)=x')$ + \item We can define $(x =_p^P x')\defeq(x=\transp^P(p\sy,x'))$ + \item We can define $(x =_p^P x')$ by path induction on $p$. If $p\defeq\refl_a$, we define + $(x =_p^P x')\equiv(x =_{\refl_a}^P x')\defeq(x=_{P(a)}x')$. + \item \label{item:pathover-ind} We can define $(x =_p^P x')$ by a family of inductive types. For fixed $A:\type$ and $P:A\to\type$ and $a:A$ and $x:P(a)$ we have the following family: + \begin{inductive} + \texttt{inductive} $x =_{(-)}^P {(-)} : \{a' : A\} \to a = a' \to P(a') \to \type \defeqp$ \\ + $\bullet\ \refl : x =_{\refl_a}^P x.$ + \end{inductive} +\end{enumerate} +It does not matter which of these definitions we pick, because we can prove that all of them are equivalent.\footnote{In Lean, we chose option \ref{item:pathover-ind}. Option \ref{item:pathover-tr} would probably be slightly more convenient to work with, because then this characterization becomes a definitional equality. In practice it will not matter much, though.} + +We have the following equivalences between pathovers:%\footnote{We will define equivalences in \autoref{sec:equivalences}. The only thing we need in this section is that if two types are equal, then they are equivalent. All equivalences in this section can be proved by applying path induction and then using reflexivity of equivalences.} +\begin{align*} + (x=^{\lam{a}B}_px') &\simeq (x =_B x') & (x =^{P \o f}_p x') \simeq (x =^P_{\apfunc{f}(p)}x'). +\end{align*} + +We can do operations on pathovers, similar to the operations on paths. We have concatenation and inversion, and we will abuse notation and denote them with the same notation. +\begin{align*} +({-})\cdot ({-})&: x_1 =^P_{p} x_2 \to x_2 =^P_{q} x_3 \to x_1 =^P_{p\tr q} x_3\\ +({-})\sy&: x_1 =^P_{p} x_2 \to x_2 =^P_{p\sy} x_1. +\end{align*} +We have a dependent version of $\apfunc{}$. Given a dependent map $f:(a:A)\to P(a)$, we get +$$\apd_f: (p : a = a') \to f(a) =^P_p f(a').$$ +A variant to $\apd$ is the following. Given $f:A\to B$, a family $P:B\to\type$ and a section $g:(a:A)\to P(f(a))$, we define +\begin{align}\apdtilde_g: (p : a = a') \to g(a) =^P_{\apfunc{f}(p)} g(a').\label{eq:apdtilde}\end{align} +The difference between $\apd$ and $\apdtilde$ is over which path they lie. + +Furthermore, if we have a map $f:A\to B$ and two families $P:A\to\type$ and $Q:B\to\type$ and a fiberwise map $g:(a:A)\to P(a)\to Q(f(a))$, then we get a fiberwise version of $\apfunc{}$: +\begin{equation}\apo_g: x =^P_p x'\to g_a(x) =^Q_{\apfunc{f}(p)} g_{a'}(x').\label{eq:apo}\end{equation} + +\subsubsection*{Squares} +For higher paths, it is convenient to define a separate notion of a square in a type: +\begin{center} + \begin{tikzpicture}[node distance=10mm] + \node (tl) at (0,0) {$a_{00}$}; + \node[right = of tl] (tr) {$a_{20}$}; + \node[below = of tl] (bl) {$a_{02}$}; + \node (br) at (tr |- bl) {$a_{22}$}; + %\node (m) at ($(tl)!0.5!(br)$) {$s_{11}$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node [above] {$p_{10}$} (tr) + edge[double equal sign distance] node [right] {$p_{01}$} (bl) + (bl) edge[double equal sign distance] node [above] {$p_{12}$} (br) + (tr) edge[double equal sign distance] node [right] {$p_{21}$} (br); + \end{tikzpicture} +\end{center} +Suppose given four paths as in the diagram above, that is +\begin{align*} + p_{10}&:a_{00}=a_{20}&p_{01}&:a_{00}=a_{02}\\ + p_{12}&:a_{02}=a_{22}&p_{21}&:a_{20}=a_{22}. +\end{align*} +We have a type of squares $\squaret(p_{10},p_{12},p_{01},p_{21})$, which we can define in either of the two following equivalent ways +\begin{enumerate} +\item We can define $\squaret(p_{10},p_{12},p_{01},p_{21})\defeq (p_{10}\tr p_{21} = p_{01} \tr p_{12})$. +\item $\squaret(p_{10},p_{12},p_{01},p_{21})$ is defined as an inductive family of types. For a fixed $a_{00}:A$ we define the family +\begin{inductive} + \texttt{inductive} $\squaret({-},{-},{-},{-}) : \{a_{20}\ a_{02}\ a_{22} : A\} \to a_{00} = a_{20} \to a_{02} = a_{22} \to a_{00} = a_{02} \to a_{20} = a_{22} \to \type \defeqp$ \\ + $\bullet\ \refl : \squaret(\refl_{a_{00}},\refl_{a_{00}},\refl_{a_{00}},\refl_{a_{00}}).$ +\end{inductive} +\end{enumerate} +We will usually write squares using diagrams as above. There are various operations on squares. For example, we can horizontally concatenate them. If we can fill each of the individual squares below, we can fill the outer rectangle (which has as top $p_{10}\cdot p_{30}$ and as bottom $p_{12}\cdot p_{32}$). +\begin{center} + \begin{tikzpicture}[node distance=10mm] + \node (tl) at (0,0) {$a_{00}$}; + \node[below = of tl] (bl) {$a_{02}$}; + \node[right = of tl] (tr) {$a_{20}$}; + \node (br) at (tr |- bl) {$a_{22}$}; + \node[right = of tr] (tr2) {$a_{40}$}; + \node (br2) at (tr2 |- bl) {$a_{42}$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node [above] {$p_{10}$} (tr) + edge[double equal sign distance] node [right] {$p_{01}$} (bl) + (bl) edge[double equal sign distance] node [above] {$p_{12}$} (br) + (tr) edge[double equal sign distance] node [above] {$p_{30}$} (tr2) + edge[double equal sign distance] node [right] {$p_{21}$} (br) + (br) edge[double equal sign distance] node [above] {$p_{32}$} (br2) + (tr2) edge[double equal sign distance] node [right] {$p_{41}$} (br2); + \end{tikzpicture} +\end{center} +We can also vertically concatenate squares, and horizontally or vertically invert squares. + +Given a homotopy $h:f\sim g$ between nondependent functions $f,g:A\to B$ and a path $p:a=_Aa'$, we get the following \emph{naturality square}. +\begin{center} + \begin{tikzpicture}[node distance=10mm] + \node (tl) at (0,0) {$f(a)$}; + \node[right = of tl] (tr) {$g(a)$}; + \node[below = of tl] (bl) {$f(a')$}; + \node (br) at (tr |- bl) {$g(a')$}; + %\node (m) at ($(tl)!0.5!(br)$) {$s_{11}$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node [above] {$h(a)$} (tr) + edge[double equal sign distance] node [left] {$\apfunc{f}(p)$} (bl) + (bl) edge[double equal sign distance] node [above] {$h(a')$} (br) + (tr) edge[double equal sign distance] node [right] {$\apfunc{g}(p)$} (br); + \end{tikzpicture} +\end{center} + +\subsubsection*{Squareovers and cubes} +Going up further, we have the type of \emph{squareovers}. A squareover is a square in a dependent type over a square. Suppose that we have a dependent type $P:A\to\type$, a square $s$ in $A$ and a dependent path over each of the sides of the square, as in the following diagram. +\begin{center} + \begin{tikzpicture}[node distance=10mm] + \node (tl) at (0,0) {$x_{00}$}; + \node[right = of tl] (tr) {$x_{20}$}; + \node[below = of tl] (bl) {$x_{02}$}; + \node (br) at (tr |- bl) {$x_{22}$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node [above] {$q_{10}$} (tr) + edge[double equal sign distance] node [right] {$q_{01}$} (bl) + (bl) edge[double equal sign distance] node (t) [above] {$q_{12}$} (br) + (tr) edge[double equal sign distance] node [right] {$q_{21}$} (br); + \node[below = of bl] (tl) {$a_{00}$}; + \node (tr) at (br |- tl) {$a_{20}$}; + \node[below = of tl] (bl) {$a_{02}$}; + \node (br) at (tr |- bl) {$a_{22}$}; + \node (m) at ($(tl)!0.5!(br)$) {$s$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node (b) [above] {$p_{10}$} (tr) + edge[double equal sign distance] node [left] {$p_{01}$} (bl) + (bl) edge[double equal sign distance] node [above] {$p_{12}$} (br) + (tr) edge[double equal sign distance] node [right] {$p_{21}$} (br) + (t) edge[->, shorten <= 3mm] (b); + \end{tikzpicture} +\end{center} +We have the type of \emph{squareovers} or \emph{dependent squares}, which fill the top square and lie over the bottom square. We can again define this using multiple methods, but the most convenient method here is to define it as an inductive family. We take as parameters the type $A$, the family $P$ and the points $a_{00}$ and $x_{00}$ and let all the other arguments be indices. We have a ``reflexivity squareover'' when the square $s$ is the reflexivity square and each of the four pathovers are reflexivity pathovers. + +We can also define a type of cubes. Given six squares in a type with twelve paths as sides, fitting together in a cube, we can define the type of fillers of the cube. This is again done using a family of inductive types, where we give a cube filler when all the six sides are reflexivity squares. Of course, we could continue by defining cubeovers and 4-cubes, but we will not need them in this dissertation. + +\subsection*{Paths in type formers} + +In each of the type formers of \autoref{sec:martin-lof-type} we can compute what the paths in that type are, and what the operations of paths are in that type. + +As a simple example, consider the cartesian product type $A\times B$. A path in the cartesian product is just a pair of paths. +$$(x=_{A\times B}y)\simeq(p_1x=_Ap_1y)\times (p_2x=_Bp_2y)$$ +In particular, given paths $r:p_1x=p_1y$ and $s:p_2x=p_2y$, we get a path +$x=y$, which we will denote $(r,s)$. Given maps $f:A\to A'$ and $g:B\to B'$, we get the map $f\times g:A\times B\to A'\times B'$ and we can compute +$$\apfunc{f\times g}(r,s)=(\apfunc{f}(r),\apfunc{g}(s))$$. +Given families $P,Q:A\to\type$, we can compute transport: +$$\transp^{\lam{a}P(a)\times Q(a)}(p,(x,y))=(\transp^P(p,x),\transp^Q(p,y))$$ +Pathovers in a family of cartesian products are also pairs of pathovers: +$$(x,y)=^{\lam{a}P(a)\times Q(a)}_p(x',y')\simeq(x=^P_px')\times (y=^Q_p y').$$ + +In sigma-types the relations are a bit more difficult, since the second component depends on the first. In the type $(a:A)\times B(a)$ paths are pairs of a path and a path over that path: +$$(x=_{(a:A) \times B(a)}y)\simeq(r:p_1x=_Ap_1y)\times (p_2x=^B_rp_2y)$$ +We will also denote in this case the map from right to left by $({-},{-})$. Given a map $f:A\to A'$ and a fiberwise map $g:(a:A)\to B(a)\to B'(f(a))$, we get a functorial action of the sigma type: $f\times g: ((a:A)\times B(a))\to ((a':A')\times B(a'))$. In this case, we can compute +$$\apfunc{f\times g}(r,s)=(\apfunc{f}(r),\apo_g(s)),$$ +where $\apo$ is defined in \eqref{eq:apo}. +We leave the rule for transports as an exercise to the reader, but the rule for pathovers in a family of sigma-types is the following. For +$B:A\to\type$ and $C:(a:A)\to B(a)\to\type$ we get:\footnote{We could define a new notion ``path over a pathover,'' but the rule given here suffices for all the cases we considered.} +$$((a,b)=^{\lam{a}(b:B(a))\times C(a,b)}_p(a',b'))\simeq(q:a=^P_pa')\times (y=^{\lam{x:(a{:}A)\times B(a)}Q(p_1x,p_2x)}_{(p,q)} y').$$ + +For dependent function types the situation is a bit more complicated. Given $f,g:(a:A)\to +B(a)$, by path induction we get a map +$$\happly:(f=g)\to f\sim g.$$ +However, we cannot show in plain Martin-L\"of type theory that this map gives +rise to an equivalence. In homotopy type theory we can use the univalence axiom +(see \autoref{sec:equivalences}) to show that $\happly$ is an equivalence. We +skip the proof here, but refer the reader to~\cite[Section 4.9]{hottbook}. Using univalence we can also prove the other properties. The general rule for pathovers in a dependent function type is complicated, but two important special cases are the following. In the first case, the domain does not depend on the path. We have types $A$ and $B$ and a family $C:A\to B\to \type$ and then we can prove: +$$(f=^{\lam{a}(b:B)\to C(a,b)}_pg)\simeq(b:B)\to f(b)=^{C({-},b)}_{p} g(b).$$ +The second case is for nondependent functions. Given a type $A$ and two families $B,C:A\to\type$, we have +$$(f=^{\lam{a}B(a)\to C(a)}_pg)\simeq(b:B(a))\to f(b)=^{C}_{p} g(p_*(b)).$$ + +We characterized paths in the universe in \autoref{sec:equivalences} using the univalence axiom. We will not need to do much path algebra in inductive types, except for the identity type, pathover type and square type. +A pathover in a family of identity types is a square. Suppose given types $A$ and $B$ and functions $f, g : A \to B$, a path $p:a=_Aa'$ and paths $q:f(a)=g(a)$ and $r:f(a')=g(a')$. Then the pathover type becomes equivalent to the square type shown below. +$$(q=^{\lam{a}f(a)=g(a)}_{p}r)\simeq +\begin{tikzpicture}[node distance=10mm,baseline=(l.base)] + \node (tl) at (0,0) {$f(a)$}; + \node[right = of tl] (tr) {$g(a)$}; + \node[below = of tl] (bl) {$f(a')$}; + \node (br) at (tr |- bl) {$g(a')$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node [above] {$q$} (tr) edge[double + equal sign distance] node [right] (l) {$\apfunc{f}(p)$} (bl) (bl) + edge[double equal sign distance] node [above] {$r$} (br) (tr) edge[double + equal sign distance] node [right] {$\apfunc{g}(p)$} (br); +\end{tikzpicture}$$ + +We also sometimes encounter a pathover in a dependent family of +pathovers. In that case we get a squareover. Suppose we are given +functions $f,g:A\to B$, and a homotopy $h:f\sim g$, a dependent family +$C:B\to\type$ and sections $c:(a:A)\to C(f(a))$ and $c':(a:A)\to +C(g(a))$. We want to characterize a pathover in the family +$P\defeq\lam{a}c(a)=^C_{h(a)}c'(a):A\to\type$. If we are also given a +path $p:a=_Aa'$ and two pathovers $q:c(a)=^C_{h(a)}c'(a)$ and +$q':c(a')=^C_{h(a')}c'(a')$, then the pathover $q=^P_pq'$ is equivalent to the +following squareover, where $\apdtilde$ is defined in \eqref{eq:apdtilde}, and the bottom square is a naturality square. + +\begin{center} + \begin{tikzpicture}[node distance=10mm] + \node (tl) at (0,0) {$c(a)$}; + \node[right = of tl] (tr) {$c'(a)$}; + \node[below = of tl] (bl) {$c(a')$}; + \node (br) at (tr |- bl) {$c'(a')$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node [above] {$q$} (tr) + edge[double equal sign distance] node [left] {$\apdtilde_b(p)$} (bl) + (bl) edge[double equal sign distance] node (t) [above] {$q'$} (br) + (tr) edge[double equal sign distance] node [right] {$\apdtilde_{b'}(p)$} (br); + \node[below = of bl] (tl) {$f(a)$}; + \node (tr) at (br |- tl) {$g(a)$}; + \node[below = of tl] (bl) {$f(a')$}; + \node (br) at (tr |- bl) {$g(a')$}; + \node (m) at ($(tl)!0.5!(br)$) {nat.}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node (b) [above] {$h(a)$} (tr) + edge[double equal sign distance] node [left] {$\apfunc{f}(p)$} (bl) + (bl) edge[double equal sign distance] node [below] {$h(a')$} (br) + (tr) edge[double equal sign distance] node [right] {$\apfunc{g}(p)$} (br) + (t) edge[->, shorten <= 3mm] (b); + \end{tikzpicture} +\end{center} +Lastly, we will mention that a pathover in a family of squares is a cube, but we will not explain the details here. + +\subsection{Truncated Types}\label{sec:truncatedness} + +In HoTT we can define iterated path spaces in any type. In certain types, if we iterate path spaces enough times, these path spaces do not contain any information. These types are called \emph{truncated}. The notion of an $n$-truncated type, was introduced in 2009 by Vladimir Voevodsky under the name ``a type of h-level $n+2$.'' + +We define the notion that $A$ is $n$-truncated, or that $A$ is an \emph{$n$-type} or $\istrunc{n}(A)$ recursively for $n\geq-2$. We say that a type $A$ is $(-2)$-truncated or \emph{contractible} if it has exactly one inhabitant, i.e. if we can prove +$$(a_0:A)\times (a:A)\to a=a_0.$$ +A type $A$ is $(n+1)$-truncated if for all $a\ a':A$ the type $a=_Aa'$ is $n$-truncated. + +We can show that $\unit$ is contractible and that every contractible type is equivalent to $\unit$. + +The $(-1)$-truncated types are called \emph{mere propositions} or \emph{propositions} for short. A type $A$ is a proposition precisely when any two of its inhabitants are equal, i.e. if we can prove +$$(a\ a':A)\to a=a'.$$ +We call these types propositions because these types correspond to truth values, and do not contain any further information. In particular, if a proposition is inhabited, then it is contractible. +It is easy to see that $\emptyt$ and $\unit$ are mere propositions, and in \autoref{sec:equivalences} we saw that the statement $\isequiv(f)$ is a mere proposition. + +One level up, the $0$-types are called \emph{sets}. These are the types for which uniqueness of identity proofs holds. Examples of sets are $\N$ and $\bool$. + +On the next level we have the $1$-types or \emph{groupoids}. Below we list some properties of truncated types, see~\cite[Section 7.1]{hottbook} for their proofs. +\begin{lem}\mbox{} +\begin{itemize} +\item If $A$ is $n$-truncated, then $A$ is $m$-truncated for all $m\geq n$. +\item If $A$ is $n$-truncated and $A\simeq B$, then $B$ is $n$-truncated. +\item If $A$ and $B$ are $n$-truncated types, then $A\times B$ and $A\simeq B$ are $n$-truncated. + If $n\geq0$, then $A+B$ is also $n$-truncated. +\item If $B:A\to\type$ is a family of $n$-truncated types (i.e. $(a:A)\to\istrunc{n}(B(a))$), then $(a:A)\to B(a)$ is $n$-truncated. If moreover $A$ is also $n$-truncated, then $(a:A)\times B(a)$ is also $n$-truncated. +\item Given $a_0:A$, the type $(a:A)\times (a_0=a)$ is contractible. +\item The type $\istrunc{n}A$ is a mere proposition. +\end{itemize} +\end{lem} + +We define the \emph{subuniverse of $n$-types} as $\type_{\leq n}\defeq (X:\type)\times\istrunc{n}(X)$. For $X:\type_{\leq n}$ we will also write $X$ for the underlying type of $X$. We write $\prop\defeq \type_{\leq -1}$ and $\set\defeq\type_{\leq0}$. + +We can do set-level mathematics in the subuniverse of sets. For example, we can define a \emph{group} to be a set with operations satisfying the following axiomatization:\footnote{From these equalities the fact that $e$ is a left-identity and $i$ is a left-inverse can be derived.} +\begin{align*} + \group&\defeq(G:\set)\times (m:G\to G \to G)\times (i:G\to G)\times (e:G)\times ((x\ y\ z: G) \to {}\\ + &\mathrel{\hphantom{\defeq}} m(x,m(y,z))=m(m(x,y),z)\times m(x,e)=x\times m(x,i(x))=e). +\end{align*} +A group $G$ is \emph{abelian} if it moreover satisfies $m(x,y)=m(y,x)$ for all $x,y:G$. This gives the usual notion of groups, and we can perform all basic group theory in this setting. + +\subsubsection*{Truncations} + +We can turn every type $A$ into an $n$-type $\|A\|_n$ in a universal way, which is called the \emph{$n$-truncation} of $A$. It comes with a map $|{-}|_n:A\to\|A\|_n$ and has the following induction principle. Suppose given $P:\|A\|_n\to\type$ such that $P(x)$ is $n$-truncated for all $x:\|A\|_n$. If we are given a dependent map $f:(a:A)\to P(|a|_n)$, we get a section +$$\ind{\|{-}\|}(f):(x:\|A\|_n)\to P(x)$$ +such that $\ind{\|{-}\|}(f,|a|_n)\equiv f(a)$. + +We will now state some properties of the $n$-truncation, for the proofs we refer to~\cite[Section 7.3]{hottbook}. +\begin{lem}\mbox{} + \begin{itemize} + \item The truncation is functorial. Given $f:A\to B$, we get a map $\|f\|_n : \|A\|_n \to \|B\|_n$. This map respects composition and identities: $\|g\o f\|_n\sim \|g\|_n \o \|f\|_n$ and $\|\id_A\|_n\sim \id_{\|A\|_n}$. + \item $A$ is an $n$-type iff $|{-}|_n : A \to \|A\|_n$ is an equivalence. + \item The equality type in the truncation is truncated equality, but shifted: $$(|a|_{n+1} =_{\|A\|_{n+1}} = |a'|_{n+1})\simeq \|a =_A a'\|_n.$$ + \item Truncating twice is the same as truncating once: $$\|\|A\|_n\|_k\simeq \|A\|_{\min(n,k)}.$$ + \end{itemize} +\end{lem} + +In particular the \emph{propositional truncation} $\|A\|\defeq\|A\|_{-1}$ of $A$ is a proposition stating that $A$ is \emph{merely inhabited}~\cite{awodey2004Propositions}. +We can use it to define \emph{proof irrelevant} versions of the disjunction or existential quantifier. We have the \emph{mere disjunction} +\begin{align*}(P\vee Q)&\defeq \|P+Q\| + \intertext{and the \emph{mere existential}} +(\exists(x:A).P(x))&\defeq \|(x:A)\times P(x)\|. +\end{align*} +We say that there \emph{merely exists} $x:A$ such that $P(x)$ holds if $\exists(x:A).P(x)$ is inhabited, to contrast with constructing an element in the untrucated dependent pair type. If we construct an element of $(x:A)\times P(x)$, we will sometimes say that there \emph{purely exists} an $x$ such that $P(x)$ holds, but often we will drop the adverb \emph{purely}. + +\subsubsection*{Connected types} +A type is truncated if the type contains no interesting information in a high enough dimension. Dually, a type is \emph{connected} if it contains no interesting information in a low enough dimension. + +We say that a type $A$ is \emph{$n$-connected} for $n\geq-2$ if $\|A\|_n$ is contractible. From the definition we see that every type is $(-2)$-connected. A type is $(-1)$-connected precisely when it is merely inhabited. A type is called 0-connected %or \emph{path connected} +when $A$ has exactly one \emph{connected component}. A 1-connected type is called \emph{simply connected}. + +\subsubsection*{Fibers} + +We can extend the notion of truncated types and connected types to functions. Given a function $f:A\to B$ and a point $b:B$, we define the \emph{fiber} of $f$ at $b$ to be +$$\fib_f(b)\defeq(a:A)\times f(a)=b.$$ +The fiber of the projection $p_1:((a:A)\times B(a))\to A$ at $a:A$ is equivalent to $B(a)$, which explains the terminology that $B(a)$ is the fiber of $B$ over $a$. +% Here are some properties of fibers: +% \begin{lem}\mbox{} +% \begin{itemize} + +% \item The fiber of the projection $p_1:((a:A)\times B(a))\to A$ at $a:A$ is equivalent to $B(a)$, which explains the terminology that $B(a)$ is the fiber of $B$ over $a$. +% \end{itemize} +% \end{lem} + +We say that a function $f:A\to B$ is $n$-truncated ($n$-connected) when for all $b:B$ the type $\fib_f(b)$ is $n$-truncated ($n$-connected). The function $f$ is $(-2)$-truncated precisely when it is an equivalence. The function $f$ is $(-1)$-truncated, or an \emph{embedding}, if for all $a\ a':A$ the map $\apfunc{f}:a=_Aa'\to f(a)=_Bf(a')$ is an equivalence. A map $f:A\to B$ between sets is an embedding iff it is \emph{injective}, i.e. if we have a map $f(a)=f(a')\to a=a'$ for all $a\ a':A$. On the other hand, a $(-1)$-connected map is called a \emph{surjection}, which means that for every $b:B$ there merely exists an $a:A$ such that $f(a)=b$. + +Every map can be factorized as an $n$-connected map followed by an $n$-truncated map in a unique way, which means that these classes form an \emph{orthogonal factorization system}~\cite{rijke2017modalities}. + +Similar to the universe of $n$-truncated types, we have a universe of $n$-connected types: $$\type_{>n}\defeq (A:\type)\times\isconn{n}(A).$$ + +\subsection{Pointed Types}\label{sec:pointed} +A lot of homotopy theory is done in the $(\infty,1)$-category of pointed types +where the morphisms are maps that preserve the basepoints of the types. Below are the +basic definitions for pointed types. +\begin{defn}\label{def:pointed-types-basic}\mbox{} + \begin{enumerate} + \item A type $A$ is \emph{pointed} if $A$ has a distinguished basepoint + $a_0:A$. For example, $\unit$ is pointed by $\star$ and $\bool$ is pointed + with $\bfalse$. We will also write $\pbool$ for the pointed type $\bool$. $A\times B$ is pointed if both $A$ and $B$ are + pointed,\footnote{More formally, we have to specify the basepoint of $A\times + B$, because being pointed is structure on a type, not a property of the + type, but there is only one choice of basepoint in this example and other + examples where we leave the basepoint implicit.} $(a:A)\to B(a)$ is + pointed if $B$ is a family of pointed types, and $(a:A)\times B(a)$ is + pointed if $A$ is pointed and $B(a_0)$ is pointed. + \item The type of \emph{pointed types} is $\type^*\defeq(A:\type)\times A$. + Given a pointed type $A:\type^*$, we will also write $A$ for its underlying + type. + \item Given two pointed types $A,B:\type^*$, a \emph{pointed map} $f:A\to^*B$ + is a pair consisting of a map $f:A\to B$ and a path $f_0$ stating that $f$ + preserves the basepoint, that is $f_0:f(a_0)=b_0$. The type $A\to^*B$ is + pointed with basepoint + $\const\equiv\const_{A,B}\defeq(\lam{a}b_0,\refl_{b_0})$. + \item We have an identity pointed map $\id\equiv\id_A:A\to^*A$ defined as + $(\lam{x}x,\refl_{a_0})$ and if $g:B\to^*C$ and $f:A\to^*B$ we have a + composite $g\o f:A\to^* C$ defined as $(\lam{x}g(f(x)),\mapfunc{g}(f_0)\cdot + g_0)$. + \item More generally, Given a pointed type $A:\type^*$ and a family of types + $B:A\to\type$ with a basepoint $b_0:B(a_0)$, a \emph{pointed dependent map} + $f:(a:A)\to^*B(a)$ is a pair consisting of a dependent map $f:(a:A)\to B(a)$ + and a path $f_0:f(a_0)=b_0$. If we require that $B$ is a family of pointed + types, i.e. $B:A\to\type^*$, then $(a:A)\to^* B(a)$ is pointed with + basepoint $(\lam{a}b_0(a),\refl_{b_0(a_0)}).$ + \item Given two pointed dependent maps $f,g:(a:A)\to^* B(a)$, a \emph{pointed + homotopy} $h:f\sim^* g$ is a pointed dependent map $(a:A)\to^* f(a)=g(a)$. + This is well-defined, since the type $f(a_0)=g(a_0)$ is pointed by $f_0\cdot g_0\sy$. + Expanding the definition, this means that $h$ is a pair of a homotopy + $h:f\sim g$ and a 2-path stating that $h$ relates the basepoint-preserving + paths of $f$ and $g$. This means that we have $h_0:h(a_0)=f_0\cdot g_0\sy$, or equivalently, + $h_0:h(a_0)\cdot g_0=f_0$. We say that a diagram of pointed types commutes + if there are pointed homotopies between the corresponding composites of + pointed maps. + \item A pointed map $e:A\to^* B$ is a \emph{pointed equivalence} if it has a + left-inverse and a right-inverse. That is, there is $\ell:B\to^* A$ such that + $\ell\o e\sim^* \id_A$ and $r:B\to^* A$ such that $e\o r\sim^* \id_B$. The + type of pointed equivalences between $A$ and $B$ is denoted $A\simeq^* B$. + The identity map is a pointed equivalence and pointed equivalences are + closed under composition. + \item Given $A:\type^*$, we define its \emph{loop space} $\Omega A:\type^*\defeq + (a_0=a_0,\refl_{a_0})$. We define the \emph{iterated loop space} $\Omega^nA$ by iteration as $\Omega^0A\defeq A$ and $\Omega^{n+1}A\defeq \Omega(\Omega^nA)$. + \item We define the $n$-th homotopy group of $A$ as the set-truncation of the iterated loop space, i.e. $\pi_n(A)\defeq\|\Omega^nA\|_0$. This is a group for $n\geq1$ that is abelian for $n\geq 2$. + \item Given a pointed map $f:A\to^* B$, we define the \emph{pointed fiber of + $f$} $\fib_f:\type^*$ as $\fib_f(b_0)\equiv(x:A)\times f(a)=b_0$ with basepoint $(a_0,f_0)$. + There is a pointed map $p_1:\fib_f\to^*A$ defined as + $(\lam{x}p_1(x),\refl_{a_0})$. + \end{enumerate} +\end{defn} +Here are some basic properties of pointed types. We omit the proofs. +\begin{lem}\label{lem:pointed-types-basic}\mbox{} + \begin{enumerate} + \item Suppose given a pointed map $f:A\to^* B$. The type of proofs that $f$ is an + equivalence is equivalent to the type that $f$ is a pointed equivalence. In + particular, being a pointed equivalence is a property. Also, we can define a + pointed equivalence $X\simeq^* Y$ by giving a map $e:X\to Y$ that is both + an equivalence and pointed. + \item Suppose given $A,B:\type^*$. Univalence implies \emph{univalence for pointed + types}: the canonical map $(A = B)\to (A\simeq^* B)$ is an equivalence. + \item Suppose given pointed maps $f,g:(a:A)\to^* B(a)$. Function extensionality + implies \emph{function extensionality for pointed maps:} the canonical + map $(f = g) \to (f \sim^* g)$ is an equivalence. + \item We have the usual categorical laws: + \begin{align*} + f\o\id &\sim^* f&\id\o f&\sim^* f&(h\o g)\o f&\sim^* h\o (g\o f)\\ + f\o\const &\sim^* \const&\const\o f&\sim^*\const + \end{align*} + The two homotopies showing $\const\o\id\sim^*\const$ are equal. This is also + true for the two homotopies of $\id\circ\const\sim^*\const$ and of $\id\o\id\sim^*\id$ + and of $\const\o \const\sim^*\const$. + \item + We can form iterated pointed maps $(A\to^* B \to^* C)\defeq (A\to^* + (B\to^* C))$. To show that such a map preserves the basepoint, we need to + give an equality between pointed maps, or equivalently, we can give a + pointed homotopy between pointed maps. For example, the above homotopies + involving $\const$ imply that we have precomposition and postcomposition maps. + For $f:A\to^* B$ we have a pointed map $({-})\o f:(B\to^* C)\to^* A \to^* C$ + and for $g:B\to^* C$ we have a pointed map $g\o({-}):(A\to^* B)\to^* A \to^* + C$. We will also write $f\to C$ resp. $A \to g$ for these maps. + Precomposition and postcomposition commute, which means that the + following square commutes. + \begin{center} + \begin{tikzpicture}[node distance=10mm] + \node (tl) at (0,0) {$(A\to^* B)$}; + \node[right = of tl] (tr) {$(A\to^* B)$}; + \node[below = of tl] (bl) {$(A'\to^* B)$}; + \node (br) at (tr |- bl) {$(A'\to^* B')$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above] {$g\o({-})$} (tr) + edge[->] node [right] {$({-})\o f$} (bl) + (bl) edge[->] node [above] {$g\o({-})$} (br) + (tr) edge[->] node [right] {$({-})\o f$} (br); + \end{tikzpicture} + \end{center} + Moreover, if $f$ or $g$ are constant, then these maps are pointed homotopic + to constant maps, which gives a pointed map + $$({-})\o({-}):(B\to^* C)\to^*(A\to^* B)\to^* A\to^* C.$$ + \item\label{item:fiber-composition} + There are also dependent versions of these composition maps. In particular, + if $g:(a:A)\to B(a)\to^* C(a)$, then we have a map + $$g\o({-}):((a:A)\to^* B(a))\to^* (a:A)\to^* C(a).$$ + We have an equivalence $$\fib_{g\o({-})}\simeq^* ((a:A)\to^* \fib_{ga}).$$ + \item $\Omega$ and $\Omega^n$ are pointed functors. For $\Omega$ this means + that given a pointed map $f:A\to^* B$, we can define $\Omega f:\Omega A \to^* + \Omega B$, with pointed homotopies $\Omega(g\o f)\sim^* \Omega g \o \Omega + f$ and $\Omega\id\sim^* \id$ and $\Omega\unit\simeq^*\unit$. This also + implies that $\Omega\const\sim^*\const$ and that if $e:A\simeq^* B$ then + $\Omega e:\Omega A\simeq^* \Omega B$. + \item \label{item:pointed-function-extensionality} + There is a pointed version of function extensionality for pointed types. + If $B$ is a family of pointed types, we have a pointed equivalence + $$e_B:\Omega((a:A)\to^* B(a))\simeq^* ((a:A)\to^* \Omega B(a)).$$ + This equivalence is natural in $B$. This means that given a fiberwise + pointed map $f:(a:A)\to B(a)\to^* C(a)$, the following square commutes. + \begin{center} + \begin{tikzpicture} + \node (tl) at (0,0) {$\Omega((a:A)\to^* B(a))$}; + \node[right = of tl] (tr) {$((a:A)\to^* \Omega B(a))$}; + \node[below = of tl] (bl) {$\Omega((a:A)\to^* C(a))$}; + \node (br) at (tr |- bl) {$((a:A)\to^* \Omega C(a))$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above] {$e_B$} (tr) + edge[->] node [right] {$\Omega(f\o({-}))$} (bl) + (bl) edge[->] node [above] {$e_C$} (br) + (tr) edge[->] node [right] {$\Omega f \o ({-})$} (br); + \end{tikzpicture} + \end{center} + \item The fiber of a pointed map is functorial. This means that given a commuting square, we get a pointed map from the fiber of the top map to the fiber of the bottom map. + \begin{center} + \begin{tikzpicture} + \node (tl) at (0,0) {$\fib_f$}; + \node[right = of tl] (t) {$A$}; + \node[right = of t] (tr) {$B$}; + \node[below = of tl] (bl) {$\fib_{f'}$}; + \node (b) at (t |- bl) {$A'$}; + \node (br) at (tr |- bl) {$B'$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above] {$p_1$} (t) + edge[->, dashed] (bl) + (t) edge[->] node [above] {$f$} (tr) + edge[->] node [right] {$g$} (b) + (bl) edge[->] node [above] {$p_1$} (b) + (b) edge[->] node [above] {$f'$} (br) + (tr) edge[->] node [right] {$h$} (br); + \end{tikzpicture} + \end{center} + Moreover, if the left and the right sides of the squares are equivalences, then the functorial action is an equivalence. Lastly, $p_1$ is natural, which means that the left square commutes. + \item Given a pointed map $f:A\to^*B$, we have a equivalence $\Omega\fib_f\simeq^*\fib_{\Omega f}$ that is natural in $f$. This means that if we have a commuting square with top $f$ and bottom $f'$, then the following square commutes (the left and the right side come from the functorial action of $\fib$). + \begin{center} + \begin{tikzpicture} + \node (tl) at (0,0) {$\Omega\fib_f$}; + \node[right = of tl] (tr) {$\fib_{\Omega f}$}; + \node[below = of tl] (bl) {$\Omega\fib_{f'}$}; + \node (br) at (tr |- bl) {$\fib_{\Omega f'}$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above] {$\sim$} (tr) + edge[->] (bl) + (bl) edge[->] node [above] {$\sim$} (br) + (tr) edge[->] (br); + \end{tikzpicture} + \end{center} + \item We have a pointed equivalence $(\pbool\to^* X)\simeq^* X$ natural in $X$. + \item A pointed type $A$ is $n$-connected iff $\pi_k(A)$ is trivial (contractible) for all $k\leq n$. If a type $A$ is $n$-truncated, then $\pi_k(A)$ is trivial for $k>n$ (however, the converse is not true in general). + \end{enumerate} +\end{lem} + +\subsection{Higher Inductive Types} \label{sec:high-induct-types} + +Higher inductive types are a generalization of inductive types where we specify not only the generating points in the type by constructors, but also the generating paths and higher paths. +The idea is that the type together with its (higher) path spaces are freely generated by these constructors. + +A simple example is the \emph{interval}. The interval $I$ is generated by two points $0,1:I$ and a path $\seg:0=1$. Using a syntax similar to that of inductive types, we could write +\begin{inductive} + \texttt{HIT} $I:\type \defeqp$ \\ + $\bullet\ 0, 1 : I$; \\ + $\bullet\ \seg : 0=_I1$. +\end{inductive} +Note that this is not an inductive type, since the last constructor does not specify an element in $I$, but an element in the path space of $I$. We get an induction principle for higher inductive types, similar to the induction principle for inductive types. We first give a special case, the nondependent induction principle, also called the \emph{recursion principle}. For the interval this states the following. Given a type $X$, if we have points $x_0\ x_1:X$ and a path $p:x_0=_Xx_1$, then we get a map $\rec{I}(x_0,x_1,p):I\to X$. On the points this has the expected computation rules: +$$\rec{I}(x_0,x_1,p,0)\equiv x_0\qquad\text{and}\qquad\rec{I}(x_0,x_1,p,1)\equiv x_1.$$ +We want a similar computation rule on paths. We can apply the induction principle to $\seg$ using $\apfunc{}$. The resulting computation rule is +$$\apfunc{\rec{I}(x_0,x_1,p)}(\seg)=p.$$ +Note that for this case we postulate a member of the identity type instead of making this a definitional equality. There are various reasons for this. Firstly, in this type theory, there is no justification for this equality to be definitional. There are various ways to define $\apfunc{}$, and there is no good reason for the computation rules to favor this definition. Secondly, in the early proof assistants for HoTT there was no support for definitional computation rules on path constructors, but there was a trick to get it for the point constructors~\cite{licata2011trick}. In fact, calling this rule a ``computation rule'' is not quite accurate, since there is no computation going on. We will still keep using this terminology, so that we have the same terminology as for inductive types. In the cubical type theories mentioned in the introduction we can make these terms reduce judgmentally, making them convenient for working with higher inductive types. + +The induction principle for the interval is the following. Suppose given a family $P:I\to\type$ with elements $x_0:P(0)$ and $x_1:P(1)$. We need to relate $x_0$ and $x_1$ in some way, but we cannot ask that they are equal, since they live in different types. Instead, we require a pathover $p:x_0=^P_{\seg}x_1$. In this case we get a dependent map +$\ind{I}(x_0,x_1,p):(i:I)\to P(i)$ with computation rules on points +$$\ind{I}(x_0,x_1,p,0)\equiv x_0\qquad\text{and}\qquad\ind{I}(x_0,x_1,p,1)\equiv x_1.$$ +For the computation rule on paths, we need to use $\apd$ to apply the induction principle to $\seg$, and we get +$$\apd_{\ind{I}(x_0,x_1,p)}(\seg)=p.$$ + +A more interesting example of a higher inductive type is the \emph{(graph) quotient} which we will call a \emph{quotient} in this dissertation. Given $A : \U$ and $R : A \to A \to \U$, the quotient is the following higher inductive type. +\begin{inductive} +\texttt{HIT} $\quotient_A(R) \defeqp$ \\ +$\bullet\ i : A \to \quotient_A(R)$; \\ +$\bullet\ \glue : (a\ a' : A) \to R(a,a')\to i(a) = i(a')$. +\end{inductive} +We will sometimes use the notation $[{-}]_0$ for $i$ and $[{-}]_1$ for $\glue$. + +A very similar higher inductive type is the \emph{homotopy pushout}, or \emph{pushout} for short. Given two maps $f:A\to B$ and $g:A\to C$, their pushout is the following HIT. +\begin{inductive} +\texttt{HIT} $\pushout(f,g) \defeqp$ \\ +$\bullet\ \inl : B \to \pushout(f,g)$ \\ +$\bullet\ \inr : C \to \pushout(f,g)$ \\ +$\bullet\ \glue : (a : A) \to \inl(f(a)) = \inr(g(a))$ +\end{inductive} +We denote $\pushout(f,g)$ by $B+_AC$ if $f$ and $g$ are clear from the context. In this section, we will define other higher inductive types in terms of the pushout. However, we could also start with the quotient, by the following lemma. + +\begin{lem} + The pushout and quotient are interdefinable in MLTT. +\end{lem} +\begin{proof} + We will only give the definitions of the pushout and the quotient in terms of the other. + Showing that these definitions are correct is easy, and we omit it here. + + If we have quotients, we can define the pushout of $f: A \to B$ and $g:B\to C$ as the quotient of + $B+C$ under the relation $R:B+C\to B+C\to\U$, which is inductively generated by + $\mk : (a : A) \to R (f(a),g(a))$. + + On the other hand, if we have pushouts, we can define the quotient of $A$ under $R$ as + follows. Let $T\defeq(a\ a' : A)\times R(a,a')$ be the total space of $R$. Then the quotient of + $A$ under $R$ is the pushout of $f\defeq\pair{\pi_1}{\pi_2} : T+T \to A$ and + $g\defeq\pair\id\id:T+T\to T$. +\end{proof} + +Many higher inductive types can be defined in terms of the homotopy pushout (or equivalently, the quotient): +\begin{itemize} +\item The \emph{cofiber} of a map $f : A \to B$ is defined as $C_f\defeq B+_A\unit$. +The maps are $f$ and $!$. +\item The \emph{suspension} $\susp A$ of type $A$ is defined as $\susp A \defeq + \unit+_A\unit$, i.e. as the cofiber of the map $A \to \unit$. The points are + called $\north$ and $\south$ and $\glue$ is called $\merid$. +\item The \emph{wedge sum} of a family of pointed types $A : I \to \pU$ is defined as the + cofiber of the map $I \to (i : I) \times A(i)$, which sends $i$ to the pair + $(i,\pt_{A(i)})$. The binary wedge $A\vee B$ of two pointed types $A\ B:\pU$ can + equivalently be described as the pushout of $A+_\unit B$ where the maps come + from the basepoints of $A$ and $B$. +\item The \emph{smash product} $A\wedge B$ of $A$ and $B$ can be defined as the cofiber + of the map $A\vee B\to A\times B$, which sends $\inl(a)$ to $(a,b_0)$ and $\inr(b)$ + to $(a_0,b)$ and $\glue(\star)$ to $\refl_{(a_0,b_0)}$. We will discuss the smash product in~\cref{sec:smash-product}. +\item The \emph{$n$-sphere} $\S^n$ is defined inductively for $n\geq0$: $\S^0\defeq\bool$ and + $\S^{n+1}\defeq\susp\S^n$. The $n$-sphere is pointed with point $\north$ for $n\geq1$ and with $\bfalse$ for $n=0$. + We could also start counting at $n=-1$, defining $\S^{-1}=\emptyt$, but we often only want to consider the pointed spheres. +\end{itemize} + +Another higher inductive type that we will study is the \emph{sequential colimit} or \emph{colimit} for short. This is the following HIT for $A:\N\to\type$ and $f:(n:\N)\to A(n)\to A(n+1)$: +\begin{inductive} + \texttt{HIT} $\colim(A,f) \defeqp$ \\ + $\bullet\ \iota : (n : \N) \to A(n) \to \colim(A,f)$; \\ + $\bullet\ \kappa : (n : \N) \to (a : A(n)) \to i_{n+1}(f_n(a))=i_n(a)$. +\end{inductive} + +We can define $\colim(A,f)$ using quotients, namely as $\quotient(B,R)$ where + $B=(n : \N)\times A(n)$ is the total space of $A$ and $R:B\to B\to\U$ is inductively generated by + $\mk:(n : \N)\to(a : A(n))\to R(f_n(a),a)$. We will discuss the colimit more in \autoref{sec:colimits} + +We will use the following properties of these higher inductive types. For the proof we refer to~\cite[Chapter 8]{hottbook} +\begin{lem}\mbox{} +\begin{itemize} +\item If $A$ is $n$-connected, then $\Sigma A$ is $(n+1)$-connected. +\item The suspension is left-adjoint to the loop space: $\Sigma\dashv\Omega$. That means that for any two pointed types $A$ and $B$ there is a pointed equivalence +$$(\Sigma A\to^* B)\simeq^*(A\to^*\Omega B)$$ +that is natural in $A$ and $B$. +\item We have the following equivalence: $\Omega \S^1\simeq \Z$. Therefore $\S^1$ is1-truncated and $\pi_1(\S^1)\simeq \Z$. +\end{itemize} +\end{lem} +In particular, by the above lemma we know that $\S^n$ is $(n-1)$-connected, and hence that $\pi_k(\S^n)$ is trivial for $k] (1.3,-3.5); + \end{tikzpicture} + \end{center} + We will first focus on the left side of the squareover. We compute + \begin{align*} + \apdtilde_{f_0\circ u_t}(\lp) + &= \apdtilde_{f_0}(\apfunc{u_t}(\lp))\\ + &= \apdtilde_{f_0}(\overline{[t]_1})\\ + &= \widetilde{\overline{\widetilde{s_1}}(t)}\\ + &= \widetilde{\overline{s_1}(t)}\\ + &= \widetilde{(\br{q}_2\sy)_*1} && \text{(using $s_2$)} \\ + &\equiv\vcentcolon \widetilde{1}. + \end{align*} + Here with $\widetilde{1}$ we mean the pathover $1:s_0(a) =_{1_{\br{a}_0}}^{P}s_0(a)$ but transported along the path + $$1_{\br{a}_0} \stackrel{\br{q}_2}= \overline{\br{t}_1} = \apfunc{[{-}]_0}\overline{[t]_1} = + \apfunc{[{-}]_0}(\apfunc{u_t}(\lp))=\apfunc{[u_t({-})]_0}(\lp).$$ + By unfolding the definition of $\br{q}_2$ this can be simplified to the following concatenation: + $$1_{\br{a}_0} \stackrel{e}= \apfunc{[{-}]_0}\overline{[t]_1} = + \apfunc{[{-}]_0}(\apfunc{u_t}(\lp))=\apfunc{[u_t({-})]_0}(\lp).$$ + The right side of the squareover is easier to manipulate: + $$\apdtilde_{\const_{[q,b]_{1*}(s_0(a))}}(\lp)=\widetilde{1},$$ + where in this case we mean the pathover $1:s_0(a) =_{1_{\br{a}_0}}^{P}s_0(a)$ transported along the path + $$1_{\br{a}_0} = \apfunc{\const_{[[\inr(q)]_0]_0}}(\lp).$$ + Now in both the left and the right side these transports only act on the path they lie over. This means that we can ``push them down'' to the base square. %% TODO: maybe general lemma + + After we do that, we have a vertically degenerate squareover, and we only have to show that the square over which it lies is also vertically degenerate, which is a straightforward calculation. + % TODO + + This finishes the definition of $f$. The computation rule $f\br{a}_0\equiv s_0(a)$ follows directly from the computation rule for the quotient. Furthermore, we have + $$\apd_f\br{r}_1\equiv\apd_f\mapfunc{[{-}]_0}[r]_1=\apdtilde_{f\circ[{-}]_0}[r]_1\equiv\apdtilde_{f_0}[r]_1=s_1(r).$$ + \textbf{Recursion Principle.}\\ + For the recursion principle, suppose given $P, p_0, p_1, p_2$ as in the theorem statement. We first define $g_0 : C \to P$ by + \begin{align*} + g_0[\inl a]_0&\defeq p_0(a)\\ + g_0[\inr (a,t,q)]_0&\defeq p_0(a)\\ + \apfunc{g_0}[r]_1&\defeqp p_1(r). + \end{align*} + We define $g:D\to P$ by $g[c]_0\defeq g_0(c)$ and then we need to define + $\apfunc{g}[q,x]_1 : g_0(u_t(x)) = p_0(a)$, which we do by induction to $x$. For $x\equiv b$, this can be done by reflexivity, so $\apfunc{g}[q,b]_1\defeqp 1_{p_0(a)}$. When $x$ varies over $\lp$, we need to fill the following square. + \begin{center} + \begin{tikzpicture}[thick, node distance=3cm] + \node (tl) at (0,0) {$p_0(a)$}; + \node[right of = tl] (tr) {$p_0(a)$}; + \node (bl) at (0,-2) {$p_0(a)$}; + \node (br) at (tr |- bl) {$p_0(a)$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[double equal sign distance] node {$1$} (tr) + edge[double equal sign distance] node[left] {$\apfunc{g_0\circ u_t}(\lp)$} (bl) + (tr) edge[double equal sign distance] node {$\apfunc{\const_{p_0(a)}}(\lp)$} (br) + (bl) edge[double equal sign distance] node {$1$} (br); + \end{tikzpicture} + \end{center} + This can be done by the following calculation. + $$\apfunc{g_0\circ u_t}(\lp)=\apfunc{g_0}\apfunc{u_t}(\lp)=\apfunc{g_0}\overline{[t]_1}= + \overline{p_1}(t)\stackrel{p_2}=1=\apfunc{\const_{p_0(a)}}(\lp).$$ + This completes the definition of $g$. The computation rule $g\br{a}_0\equiv p_0(a)$ follows from the computation rule for quotients on points. We can define the computation rule on paths as the composite + $$\iota_1 : \apfunc{g}\br{r}_1 \equiv \apfunc{g}\apfunc{[{-}]_0}[r]_1 = \apfunc{g_0}[r]_1 = p_1(r).$$ + The fact that $g$ has the correct computation rule for 2-paths requires some complicated path algebra, which we will omit here. + % TODO: apply on 2-paths +\end{proof} + +We can now define the general version of the 2-quotient, $\twoquotient(A,R,S)$, to be equal to +$\simpletwoquotient(A,R,Q)$ where $Q$ is the inductive family +\begin{inductive} +\texttt{inductive} $Q : \{a : A\} \to \words_R(a, a) \to \type \defeqp$ \\ +$\bullet\ (a\ a' : A) \to (t\ t' : \words_R(a,a')) \to (s : S(t,t')) \to Q(t \cdot t\sy).$ +\end{inductive} +We then show that $\twoquotient(A,R,S)$ and $\|\twoquotient(A,R,S)\|_n$ have the right elimination +principles and computation rules (it requires some work to show that the eliminator of the truncated 2-quotient has the right computation rules on 2-paths). %% TODO + +This allows us to define all nonrecursive HITs with point, 1-path and 2-path constructors. +For example, we define the torus $T^2 := \twoquotient(\unit,R,S)$ where +$R(⋆,⋆) = \bool$ (giving two path constructors $p$ and $q$ from the basepoint to itself) and +$Q$ is generated by the constructor +$s_0 : S (\bfalse \cdot \btrue) (\btrue \cdot \bfalse)$, which determines a path $p \cdot q = q \cdot p$. +We also define the \emph{groupoid quotient}: For a groupoid $G$ we define its quotient as +$\|\twoquotient(G, \homm_G, S)\|_1$ where: +\begin{inductive} +\texttt{inductive} $S \defeqp$ \\ +$\bullet\ (a\ b\ c : G) \to (g : \homm(b,c)) \to (f : \homm(a,b)) \to S (g \circ f) (f \cdot g)$ +\end{inductive} +If $G$ is just a group (considered as a groupoid with a single object), then the groupoid quotient +of $G$ is exactly the Eilenberg-MacLane space $K(G,1)$. For more information, see \autoref{sec:eilenb-macl-spac}. + +\section{Colimits}\label{sec:colimits} +\lstDeleteShortInline" + +We can ask whether we can use the construction of \Cref{sec:prop-trunc} can be generalized to +construct other higher inductive types.\footnote{The work in this section is joint work with Egbert Rijke and Kristina Sojakova.} +The general idea is that we can construct a recursive higher +inductive type as a sequential colimit of repeatedly applying a nonrecursive version of the +HIT. This does not work in general: if a constructor is infinitary, there is no reason why the type after $\omega$ many steps is the desired type. However, this does work for a general class of higher inductive types, the \emph{$\omega$-compact localizations}. In this section we will show various properties of colimits that are used in the proof of this fact. The full proof will appear in an upcoming preprint. + +\begin{defn} + Suppose given a type $A$, families $P, Q : A \to \type$ and $F : \{a : A\} \to P(a) \to Q(a)$. + + A type $X$ is \emph{$F$-local} if for all $a : A$ the map + $$\psi_X(a)\defeq \lam{f}f \o F(a) : (Q(a) \to X) \to (P(a) \to X)$$ + is an equivalence. + +% \begin{center}\begin{tikzcd} +% P(a) \ar[d]\ar[r] & X \\ +% Q(a) \ar[ur, dashed] +% \end{tikzcd}\end{center} + + The \emph{$F$-localization} $L_FX$ or $LX$ of $X$ turns $X$ into a $F$-local type in a universal + way. This means there is a map $\ell_X : X \to LX$ such that for any $F$-local type $Y$ there is an + equivalence of maps $(LX \to Y) \to (X \to Y)$ given by precomposition with $\ell_X$. $L_FX$ can + be given as a higher inductive type with the following constructors: + \begin{lstlisting}[gobble=4] + HIT L F X : Type := + | incl : X → L X + | rinv : Π{a} (f : P a → L X), Q a → L X + | isri : Π{a} (f : P a → L X) (x : P a), rinv f (F x) = f x + | linv : Π{a} (f : P a → L X), Q a → L X + | isli : Π{a} (f : Q a → L X) (x : Q a), linv (f ∘ F) x = f x. + \end{lstlisting} +\end{defn} + +For a sequence $(A_n,f_n)_n$ we denote the colimit by $\colim(A)$ or $A_\infty$. Also, for any type $X$, we can define a new sequence $(X\to A_n, f_n \o ({-}))_n$. Note that there is a canonical map +$$\xi_X : \colim(X\to A_n)\to (X\to A_\infty).$$ +It is defined by $\xi_X(i_n(f))\defeq i_n\o f$ and $\xi_X(\kappa(f))\defeq \kappa_n\o f$, where $\kappa$ is the path constructor of the colimit. + +\begin{defn} + A type $X$ is said to be \emph{$\omega$-compact} if the map $\xi_X$ is an equivalence for all sequences $(A_n,f_n)_n$. +\end{defn} + +Examples of $\omega$-compact types are the finite types. Moreover, the $\omega$-compact types are closed under dependent pair types and pushouts. A non-example of an $\omega$-compact type is $\N$. We will omit the details here. + +\begin{thm}\label{thm:localization} + Assume that for all $a : A$ the types $P(a)$ and $Q(a)$ are $\omega$-compact. Then we can + construct the $F$-localization in MLTT$+$quotients. +\end{thm} + +We will not prove this theorem here, but defer it to an upcoming preprint. However, we will develop machinery here that is crucial to prove this theorem. In particular we prove that sigma-types commute with sequential colimits. + +% Suppose given a sequence $(A_n,f_n)$, where $A : \N \to \U$ and $f : (n : \N) \to A(n) \to A(n+1)$. We +% define $f_n^k : A(n) \to A(n+k)$ by repeatedly composing $f$'s, i.e. $f_n^0(a)\defeq a$ and +% $f_n^{k+1}(a)\defeq f_{n+k}(f_n^k(a))$. This type checks if addition on the natural numbers is +% defined by induction on the second arguments, which we assume it is. + +% \begin{lem}\label{lem:rep-succ} +% We have $f_{n+1}^k(f_n(x))=^A_{\opr{succ\_add}(n,k)}f_n^{k+1}(x)$ for all $x : A(n)$ where +% $\opr{succ\_add}(n,k):(n+1)+k=n+(k+1)$ +% \end{lem} +% \begin{proof} +% Trivial by induction on $k$. +% \end{proof} + +% \begin{lem}\label{lem:shift-equiv} +% For a sequence $(A_n,f_n)$, we have $\shift : A_\infty\simeq\colim(A_{n+1},f_{n+1})_n$. +% \end{lem} +% \begin{proof} +% Straightforward. We will spell out the proof, because we will need underlying maps later. +% We define for $a : A(n)$ and $a' : A(n+1)$ +% $$\shift(i_n(a))\defeq i_n(f_n(a))\qquad \shift(\glue_n(a))\defeqp \glue_n(f_n(a)).$$ +% $$\shift\sy(i_n(a'))\defeq i_{n+1}(a')\qquad \shift\sy(\glue_n(a'))\defeqp \glue_{n+1}(a').$$ +% To show that $p : (x : A_\infty) \to \shift\sy(\shift(x))=x$ we define +% $p(i_n(a))\defeq\glue_n(a)$. If $x$ varies over $\glue_n(a)$ we need to fill the following square: +% \begin{center} +% \begin{tikzpicture}[thick] +% \node (tl) at (0,0) {$i_{n+2}(f^2(a))$}; +% \node (tr) at (4.5,0) {$i_{n+1}(f(a))$}; +% \node[below = of tl] (bl) {$i_{n+1}(f(a))$}; +% \node (br) at (tr |- bl) {$i_n(a)$}; +% \path[every node/.style={font=\sffamily\small}] +% (tl) edge[double equal sign distance] node {$\glue_{n+1}(f(a))$} (tr) +% edge[double equal sign distance] node[left] {$\ap{\shift\sy\o\shift}{\glue_n(a)}$} (bl) +% (tr) edge[double equal sign distance] node {$\ap\id{\glue_n(a)}$} (br) +% (bl) edge[double equal sign distance] node {$\glue_n(a)$} (br); +% \end{tikzpicture} +% \end{center} +% which is straightforward path algebra. To show that +% $q : (x : \colim(A_{n+1})_n) \to \shift(\shift\sy(x))=x$ we define $q(i_n(a))\defeq\glue_n(a)$ and +% we have to fill a similar square which is also straightforward. +% \end{proof} + +% \begin{lem}\label{lem:colimit-natural} +% Given a natural transformation $\alpha$ of sequences $(A_n,f_n)\to (B_n,g_n)$, i.e. a family of +% maps $\alpha_n : A_n \to B_n$ such that the obvious squares commute. Then there is a map +% $\colim(\alpha):\colim(A)\to\colim(B)$. Moreover, if $\alpha$ is a natural isomorphism +% (i.e. $\alpha_n$ is an equivalence for all $n$), then $\colim(\alpha)$ is an equivalence. +% \end{lem} +% \begin{proof} +% Tedious, but standard proof that higher inductive types are functorial. +% \end{proof} + +% \begin{defn} \label{def:seq-over} +% A \emph{sequence $(P_n,g_n)$ over $(A_n,f_n)$} consists of $P : \{n : \N\} \to A(n) \to \U$ and +% $g : \{n : \N\} \to (a : A(n)) \to P(a) \to P(f(a))$. We $(P_n,g_n)_n$ \emph{equifibered} if $g_a$ +% is an equivalence for all $a : A(n)$. +% \end{defn} +% Note that a equifibered sequence $(P_n,g_n)$ corresponds exactly to a family $A_\infty \to \U$. + +% Given a sequence $P\equiv(P_n,g_n)$ over $(A_n,f_n)$ we can define a new +% sequence $$\Sigma_AP\defeq((a : A_n)\times P_n(a), (f_n,g_n))_n.$$ We can ask what the colimit of +% this sequence is. If $P$ is equifibered, the answer is quite easy: the sequence determines a family +% $A_\infty \to \U$, and the colimit of $\Sigma_AP$ is the total space of this family, by the +% flattening lemma. In the case that $P$ is not equifibered, it is more complicated, because we need +% to turn $P$ into an equifibered family first. + +% We can turn it $P$ into an equifibered sequence as follows using the \emph{equifibrant replacement}, +% which we can define as +% $$\eqf(P)_n(a)\defeq\colim(P_{n+k}(f^k(a)),g_{n+k})_k.$$ +% Then there is an equivalence $e_n(a):\eqf(P)_n(a)\simeq \eqf(P)_{n+1}(f(a))$ given by +% \begin{align*} +% \colim(P_{n+k}(f^k(a)),g_{n+k})_k&\simeq\colim(P_{n+(k+1)}(f^{k+1}(a)),g_{n+(k+1)})_k\\ +% &\simeq\colim(P_{(n+1)+k}(f^k(f(a))),g_{(n+1)+k})_k +% \end{align*} +% The first equivalence is given by \cref{lem:shift-equiv} and the second equivalence is given by +% \cref{lem:colimit-natural} applied to the natural isomorphism +% $P_{n+k+1}(f^{k+1}(a))\simeq P_{n+1+k}(f^k(f(a)))$ which is given by transporting along the pathover +% of \cref{lem:rep-succ}. Naturality is an instance of the general fact that transporting is +% natural. $\eqf(P)$ determines a family $P_\infty:A_\infty \to \U$ by $P_\infty(i_n(a))\defeq +% \eqf(P)_n(a)$ and $P_\infty$ respects the path constructor $\glue_n(a)$ by $e_n(a)$. + +% Now the total space of $P_\infty$ is the colimit of $\Sigma_AP$, as captured by the following +% theorem. +% \begin{thm}\label{thm:sigma-colim} +% There is an equivalence $$\colim(\Sigma_AP)\simeq (x : A_\infty) \times P_\infty(x).$$ +% \end{thm} + + + + + + + + + + + + + + + + + + + + + + +\subsubsection*{Type Sequences} + +\begin{defn} +A \emph{type sequence} $\sequence{A}{f}$ consists of a diagram of the form +\begin{equation*} +\begin{tikzcd} +A_0 \arrow[r,"f_0"] & A_1 \arrow[r,"f_1"] & A_2 \arrow[r,"f_2"] & \cdots +\end{tikzcd} +\end{equation*} +Thus, the type of all sequences of types is +\begin{equation*} +\mathrm{Seq} \defeq (A:\nat\to\type)\times(n:\nat) \to A_n\to A_{n+1} +\end{equation*} +\end{defn} + +Recall that the relation $\leq$ on the natural numbers is defined as an inductive family of types $\leq\mathop{:}\N\to\N\to\UU$ with +\begin{align*} +r & : (n:\N) \to n\leq n \\ +s & : (n,m:\N)\to n\leq m \to n\leq m+1. +\end{align*} +It follows that $n\leq m$ is a a mere proposition for each $n,m:\N$. + +\begin{defn} +Let $\sequence{A}{f}$ be a type sequence. For any $n,m:\nat$, we define +\begin{equation*} +f^{n\leq m} : A_n\to A_m. +\end{equation*} +where we leave the proof that $n\leq m$ implicit. +\end{defn} + +\begin{proof}[Construction] +We define $f^{n\leq m}$ by induction on the proof that $n\leq m$ by taking +\begin{align*} +f^{n\leq n} & \defeq \idfunc[A_n] \\ +f^{n\leq m+1} & \defeq f_m\circ f^{n\leq m} \qedhere +\end{align*} +\end{proof} + +\begin{defn} +Let $\sequence{A}{f}$ be a type sequence. For any $n,k:\nat$, we define +$f_n^k:A_n\to A_{n+k}$ to be $f^{n\leq n+k}(p)$, where $p$ is the canonical proof that $n\leq n+k$. +\end{defn} + +\begin{defn} +A \emph{sequence $\sequence{B}{g}$ of types over $\sequence{A}{f}$} consists of a diagram of the form +\begin{equation*} +\begin{tikzcd} +B_{0} \arrow[r,"g_0"] \arrow[d,->>] & B_{1} \arrow[r,"g_1"] \arrow[d,->>] & B_{2} \arrow[r,"g_2"] \arrow[d,->>] & \cdots \\ +A_0 \arrow[r,"f_0"] & A_1 \arrow[r,"f_1"] & A_2 \arrow[r,"f_2"] & \cdots +\end{tikzcd} +\end{equation*} +where each $g_n$ has type $(a:A_n)\to B_n(x)\to B_{n+1}(f_n(a))$, implicitly rendering the +squares commutative. + +We say that a sequence $\sequence{B}{g}$ over $\sequence{A}{f}$ is \emph{equifibered} if each $g_n$ is a family of equivalences. +\end{defn} + +\begin{defn} +Let $\sequence{A}{f}$ and $\sequence{A'}{f'}$ be type sequences. +A \emph{natural transformation} $\sequence{A}{f}\to\sequence{A'}{f'}$ +is a pair $\sequence{\tau}{H}$ consisting of a family of maps +\begin{equation*} +\tau : (n:\N) \to A_n \to A'_n +\end{equation*} +and a family $H_n$ of homotopies witnessing that the diagram +\begin{equation*} +\begin{tikzcd} +A_{0} \arrow[r,"f_0"] \arrow[d,"\tau_0"] & A_{1} \arrow[r,"f_1"] \arrow[d,"\tau_1"] & A_{2} \arrow[r,"f_2"] \arrow[d,"\tau_2"] & \cdots \\ +A'_0 \arrow[r,"{f'_0}"] & A'_1 \arrow[r,"{f'_1}"] & A'_2 \arrow[r,"{f'_2}"] & \cdots +\end{tikzcd} +\end{equation*} +commutes. +\end{defn} + +\begin{defn} +A \emph{natural equivalence} is a natural +transformation $(\tau,H)$ such that each $\tau_n$ is an equivalence. +The type of natural equivalences from $\sequence{A}{f}$ to $\sequence{A'}{f'}$ +is called $\mathsf{NatEq}(\sequence{A}{f},\sequence{A'}{f'})$. +\end{defn} + +\begin{lem} +The canonical dependent function $\mathsf{idtonateq}$ +\begin{equation*} +(\sequence{A}{f}=\sequence{A'}{f'})\to \mathsf{NatEq}(\sequence{A}{f},\sequence{A'}{f'}) +\end{equation*} +that sends $\refl_{\sequence{A}{f}}$ to the identity natural transformation, is +an equivalence. +\end{lem} + +\begin{proof} +Straightforward application of univalence. +\end{proof} + +Every type sequence $\sequence{B}{g}$ over $\sequence{A}{f}$ gives rise to a natural transformation, by the following definition. + +\begin{defn} +Let $\sequence{B}{g}$ be a sequence over $\sequence{A}{f}$. Then we define the +sequence $\msm{\sequence{A}{f}}{\sequence{B}{g}}$ to consist of the diagram +\begin{equation*} +\begin{tikzcd}[column sep=large] +(a:A_0)\times B_0(a) \arrow[r,"\pairr{f_0,g_0}"] & (a:A_1)\times B_1(a) \arrow[r,"\pairr{f_1,g_1}"] +& (a:A_2)\times B_2(a) \arrow[r,"\pairr{f_2,g_2}"] & \cdots +\end{tikzcd} +\end{equation*} +where we take the usual definition +\begin{equation*} +\pairr{f_n,g_n} \defeq \lam{\pairr{a,b}}\pairr{f_n(a),g_n(a,b)}. +\end{equation*} +Furthermore, we define a natural transformation +\begin{equation*} +\sequence{\pi}{\theta}:\msm{\sequence{A}{f}}{\sequence{B}{g}}\to \sequence{A}{f} +\end{equation*} +by taking +\begin{align*} +\pi_n & \defeq \proj1 & & : ((a:A_n)\times B_n(a))\to A_n \\ +\theta_n(a,b) & \defeq\refl_{f_n(a)} & & : f_n(\proj 1(a,b))= \proj 1(f_n(a),g_n(b)). +\end{align*} +\end{defn} + +% \begin{defn} +% A natural transformation is said to be \emph{cartesian} if all the naturality squares are pullback squares. +% \end{defn} + +% Of course, every natural equivalence is also cartesian. Recall that for any $f:A\to A'$ and any $g:(a:A)\to B(a)\to B'(f(a))$, the square +% \begin{equation*} +% \begin{tikzcd} +% (a:A)\times B(a) \arrow[r,"\pairr{f,g}"] \arrow[d,->>] & (a':A')\times B'(a') \arrow[d,->>] \\ +% A \arrow[r,swap,"f"] & A' +% \end{tikzcd} +% \end{equation*} +% is a pullback square if and only if $g$ is a family of equivalences. Thus, we have the following: + +% \begin{lem} +% A family $\sequence{B}{g}$ of sequences over $\sequence{A}{f}$ is equifibered if and only if the projection transformation $(\pi,\theta):\msm{(A,f)}{(B,g)}\to (A,f)$ is cartesian. \qed +% \end{lem} + +We will now look at the shift operation on type sequences, in particular to bring up subtleties that come up in the formalization of mathematics in homotopy type theory. The issue we face is that equality in the natural numbers is not always strict. For instance, when addition is defined by induction on the second argument, then $n+0$ is judgmentally equal to $n$, while $0+n$ is not. This implies that sometimes we might have to \emph{transport} along the equalities in the natural numbers (such as $n=0+n$), and this complicates the formalization process. + +We define the shift operation. +\begin{defn} +For any type sequence $\sequence{A}{f}$ we define a new type sequence $(S(A),S(f))$ by taking +\begin{align*} +S(A)_n & \defeq A_{n+1} \\ +S(f)_n & \defeq f_{n+1}. +\end{align*} +\end{defn} + +Of course we can iterated the shift operation, defining a type sequence $(S^k(A),S^k(f))$ for every $k:\N$. However, while the type $S^k(A)_n$ is $A_{n+k}$, the function $S^k(f)_n$ is some function $A_{n+k}\to A_{(n+1)+k}$ that is not judgmentally equal to a function of the form $f_m$ for some $m:\N$. Therefore, we make an alternative definition of the $k$-shift that is different from $S^k$, the type sequence obtained from iterating the shift $S$. + +\begin{defn} +Given a type sequence $(A,f)$, we define $S_k(A,f)\jdeq(S_k(A),S_k(f))$ to be the type sequence given by +\begin{align*} +S_k(A)_n & \defeq A_{k+n} \\ +S_k(f)_n & \defeq f_{k+n}. +\end{align*} +Given a dependent sequence $(B,g)$ over $(A,f)$, we also define $S_k(B,g)\jdeq (S_k(B),S_k(g))$ by +\begin{align*} +S_k(B)_n & \defeq B_{k+n} \\ +S_k(g)_n & \defeq g_{k+n}. +\end{align*} +\end{defn} + +Note that the sequence $(S_{k+1}(A),S_{k+1}(f))$ is not judgmentally equal to the sequence $S(S_k(A),S_k(f))$, since in general we do not have $(k+1)+n\jdeq (k+n)+1$. Therefore we have the following lemma. + +\begin{lem}\label{lem:iterate_succ} +For any $k,n:\nat$ and $a : A_k$, one has $q_{k,n}(a):\dpath{A}{p(k,n)}{f_k^{n+1}(a)}{f_{k+1}^n(f_k(a))}$ where $p(k,n):(k+n)+1=(k+1)+n$ is the canonical path in $\nat$. +\end{lem} + +\begin{proof} +By induction on $n:\N$. +\end{proof} + +\begin{cor} +For any type sequence $\sequence{A}{f}$, the type sequence $(S_{k+1}(A),S_{k+1}(f))$ is naturally equivalent to the type sequence $(S(S_k(A)),S(S_k(f)))$. +\end{cor} + +\subsubsection*{Sequential Colimits} +\begin{rmk} +The induction principle for sequential colimits tells us how to construct a dependent function $f:(a:A_\infty)\to P(a)$ for a type family $P:A_\infty\to\type$. + +Given $s:(a:A_\infty)\to P(a)$, we get +\begin{align*} +\lam{n}{a} s(\iota_n(a)) & : (n:\N)(a:A_n)\to P(\iota_n(a)) \\ +\lam{n}{a} \apd{s}{\kappa_n(a)} & : (n:\N)(a:A_n)\to s(\iota_n(a)) =_{\kappa_n(a)}^P s(\iota_{n+1}(f_n(a))) +\end{align*} +In other words, we have a canonical map +\begin{align*} +&\Big((a:A_\infty)\to P(a)\Big)\to\\ +&\Big((h:(n:\N)(a:A_n)\to P(\iota_n(a)))\times(n:\N)(a:A_n)\to h_n(a) =_{\kappa_n(a)}^P h_{n+1}(f_n(a))\Big) +\end{align*} +Now we can state the induction principle and computation rule concisely: the canonical map described above comes equipped with a section. We assume that that the computation rule is strict on the point constructors. +\end{rmk} + +The universal property of sequential colimits is a straightforward consequence of the induction principle. + +\begin{thm} +Let $\sequence{A}{f}$ be a type sequence, and let $X$ be a type. Then the canonical map +\begin{equation*} +(A_\infty\to X)\to (h:(n:\N) \to A_n\to X)\times(n:\N)\to h_n\htpy h_{n+1}\circ f_n +\end{equation*} +is an equivalence. +\end{thm} + +The following theorem is a descent theorem for sequential colimits. + +\begin{thm}\label{thm:descent} +Consider a sequence $\sequence{A}{f}$. The type $A_\infty\to\UU$ is equivalent to the type of equifibered type sequences over $\sequence{A}{f}$. +\end{thm} + +\begin{proof} +By the universal property of $A_\infty$ and by univalence we have +\begin{align*} +(A_\infty\to \UU) & \eqvsym (B:(n:\N)\to A_n\to \UU)\times (n:\N) \to B_n\htpy B_{n+1}\circ f_n \\ +& \eqvsym (B:(n:\N) \to A_n\to \UU)\times(n:\N)(x:A_n)\to B_n(x)\eqvsym B_{n+1}(f_n(x))\qedhere +\end{align*} +\end{proof} + + +\begin{lem}\label{lem:seq_colim_functor} + Suppose given a natural transformation $(\tau,H):\sequence{A}{f}\to\sequence{A'}{f'}$. + \begin{enumerate} + \item\label{part:seq_colim_functor} We get a function $\tfcolim(\tau,H)$ or $\tau_\infty:A_\infty\to A'_\infty$. + \item\label{part:1_functoriality} The sequential colimit is 1-functorial. This means the following three things. If $(\sigma,K):\sequence{A'}{f'}\to\sequence{A''}{f''}$, then $(\tau\circ\sigma)_\infty\sim\tau_\infty\circ\sigma_\infty$. Moreover, $1_\infty\sim \idfunc$, where $1$ is the identity natural transformation. Lastly, if $(\tau',H'):\sequence{A}{f}\to\sequence{A'}{f'}$ and $q:(n:\N)\to \tau_n\sim\tau'_n$ and we can fill the following square for all $a:A_n$ + \begin{center}\begin{tikzcd}[column sep=25mm] + \tau_{n+1}(f_na) + \ar[r,equal,"{q_{n+1}(f_na)}"] + \ar[d,equal,"{H_n(a)}"] & + \tau'_{n+1}(f_na) + \ar[d,equal,"{H'_n(a)}"]\\ + f'_n(\tau_n(a)) + \ar[r,equal,"{\apfunc{f'_n}(q_n(a))}"] & + f'_n(\tau'_n(a)) + \end{tikzcd}\end{center} + then $\tau_\infty\sim\tau'_\infty$. + \item\label{part:functor_equivalence} If $\tau$ is a natural equivalence, then $\tau_\infty$ is an equivalence. + \end{enumerate} +\end{lem} +\begin{proof}\mbox{} + \begin{enumerate} + \item + We define $\tau_\infty(\iota_n(a))\defeq\iota_n(\tau_n(a))$ and + $$\apfunc{\tau_\infty}(\kappa_n(a))\vcentcolon=\apfunc{\iota_{n+1}}(H(a))\cdot\kappa_n(\tau_n(a)): + \iota_{n+1}(\tau_{n+1}(f_na))=\iota_n(\tau_n(a)).$$ + \item All three parts are by induction on the element of $A_\infty$, and all parts are straightforward. + % todo? We might want to give an explicit proof + \item We define $(\tau_\infty)^{-1}\defeq(\tau^{-1})_\infty$ where $\tau^{-1}$ is the natural transformation by inverting $\tau_n$ for each $n$. Now we can check that this is really the inverse by using all three parts of the 1-functoriality. + $$\tau_\infty^{-1}\circ\tau_\infty\sim (\tau^{-1}\circ\tau)_\infty\sim 1_\infty\sim\idfunc[A_\infty].$$ + For the second homotopy we need to show that we can fill a certain square, which is straightforward. + The other composite is homotopic to the identity by a similar argument.\qedhere + \end{enumerate} +\end{proof} + +The following lemma states that $\iota_0$ is an equivalence if all maps in the sequence are an equivalence. We will have a more general result in \autoref{cor:trunc_colim}\ref{part:iota_is_trunc_conn}, but in that proof we will use some special cases of this lemma. +\begin{lem}\label{lem:equiv_equiseq} + Suppose given a sequence $\sequence{A}{f}$ where $f_n$ is an equivalence for all $n$. + Then $\iota_0:A_0\to A_\infty$ is an equivalence. +\end{lem} +\begin{proof} + First note that the map $f^{0\le n}:A_0\to A_n$ is an equivalence, which is an easy induction on the proof that $0\le n$, because $f^{0\le0}\jdeq\idfunc$ is an equivalence and $f^{0\le n+1}\jdeq f_n\circ f^{0\le n}$ is a composition of two equivalences. + + Also note that we have paths $\kappa^{n\le m}(a):\iota_m(f^{n\le m}(a))=\iota_n(a)$ for $a:A_n$. + + Now we define $\iota_0^{-1}:A_\infty\to A_0$ as + $$\iota_0^{-1}(\iota_n(a))\defeq (f^{0\le n})^{-1}(a)$$ + and we define + $$\apfunc{\iota_0^{-1}}(\kappa_n(a)):(f^{0\le n})^{-1}(f_n^{-1}(f_n(a)))=(f^{0\le n})^{-1}(a)$$ + as $\apfunc{(f^{0\le n})^{-1}}(\ell_n(a))$, where $\ell_n(a):f_n^{-1}(f_n(a))$ is the canonical path. + + Now $\iota_0^{-1}\circ\iota_0\sim\idfunc$ is true by definition. To show that for $x:A_\infty$ we have + $p(x):\iota_0(\iota_0^{-1}(x))=x$, we use induction on $x$. If $x\jdeq\iota_n(a)$, we have + \begin{align*} + \iota_0(\iota_0^{-1}(\iota_n(a))) + &\jdeq \iota_0((f^{0\le n})^{-1}(a))\\ + &=\iota_n(f^{0\le n}((f^{0\le n})^{-1}(a)))\\ + &=\iota_n(a). + \end{align*} + If we write $r^{0\le n}: f^{0\le n} \circ (f^{0\le n})^{-1}\sim\idfunc$ for the canonical homotopy, then we explicitly define $p(\iota_n(a))$ as + $$p(\iota_n(a))\defeq(\kappa^{0\le n}((f^{0\le n})^{-1}(a)))^{-1}\cdot \apfunc{\iota_n}(r^{0\le n}(a)).$$ + + If $x$ varies over $\kappa_n(a)$, then we need to fill the following square. + \begin{center}\begin{tikzcd}[column sep=25mm] + \iota_0(\iota_0^{-1}(\iota_{n+1}(f_na))) + \ar[r,equal,"{p(\iota_{n+1}(f_na))}"] + \ar[d,equal,"{\mapfunc{\iota_0\circ\iota_0^{-1}}(\kappa_n(a))}"] & + \iota_{n+1}(f_na) + \ar[d,equal,"{\kappa_n(a)}"]\\ + \iota_0(\iota_0^{-1}(\iota_n(a))) + \ar[r,equal,"{p(\iota_n(a))}"] & + \iota_n(a) + \end{tikzcd}\end{center} + If we unfold the definitions of $\iota_0^{-1}$ and $p$, we can fill this as the horizontal concatenation of the following two squares (where we have left out some arguments to the paths) + \begin{center}\begin{tikzcd}[column sep=15mm] + \iota_0((f^{0\le n})^{-1}(f_n^{-1}(f_na))) + \ar[r,equal,"{(\kappa^{0\le n})^{-1}}"] + \ar[d,equal,"{\mapfunc{\iota_0\circ(f^{0\le n})^{-1}}(\ell)}"] & + \iota_n(f^{0\le n}((f^{0\le n})^{-1}(f_n^{-1}(f_na)))) + \ar[d,equal,"{\mapfunc{\iota_n\circ f^{0\le n}\circ (f^{0\le n})^{-1}}(\ell)}"] \\ + \iota_0((f^{0\le n})^{-1}(a)) + \ar[r,equal,"{(\kappa^{0\le n})^{-1}}"] & + \iota_n(f^{0\le n}((f^{0\le n})^{-1}(a))) + \end{tikzcd}\end{center} + \begin{center}\begin{tikzcd}[column sep=15mm] + \iota_n(f^{0\le n}((f^{0\le n})^{-1}(f_n^{-1}(f_na)))) + \ar[rr,equal,"{\kappa^{-1}\cdot\apfunc{\iota_{n+1}}(\apfunc{f}(r^{0\le n})\cdot r)}"] + \ar[dd,equal,"{\mapfunc{\iota_n\circ f^{0\le n}\circ (f^{0\le n})^{-1}}(\ell)}"] + \ar[rd,equal,"{\mapfunc{\iota_n}(r^{0\le n})}"] & & + \iota_{n+1}(f_na) + \ar[dd,equal,"{\kappa}"]\\ + & \iota_0(f_n^{-1}(f_na)) + \ar[rd,equal,"{\mapfunc{\iota_n}(\ell)}"] & \\ + \iota_n(f^{0\le n}((f^{0\le n})^{-1}(a))) + \ar[rr,equal,"{\apfunc{\iota_n}(r^{0\le n})}"] & & + \iota_n(a) + \end{tikzcd}\end{center} + The first square is a naturality square, as is the bottom-left part of the second square. + We can use the triangle equalities of $f$ to rewrite the $r$ in the top part to $\apfunc{f}(\ell)$. After doing that, the top-right square becomes the following naturality square. + + \begin{center}\begin{tikzcd}[column sep=20mm] + \iota_n(f_n(f^{0\le n}((f^{0\le n})^{-1}(f_n^{-1}(f_na))))) + \ar[r,equal,"{\apfunc{\iota_{n+1}\circ f}(r^{0\le n}\cdot \ell)}"] + \ar[d,equal,"{\kappa}"] & + \iota_{n+1}(f_na) + \ar[d,equal,"{\kappa}"]\\ + \iota_0(f_n^{-1}(f_na)) + \ar[r,equal,"{\mapfunc{\iota_n}(r^{0\le n}\cdot\ell)}"] & + \iota_n(a) + \end{tikzcd}\end{center} +\end{proof} + + +\begin{lem}\label{lem:colim_shift_one} +For any type sequence $(A,f)$, the colimits of $(A,f)$ and $S(A,f)$ are equivalent. +\end{lem} + +\begin{proof} +We construct a map $\varphi:A_\infty \to S(A)_\infty$ by induction on $A_\infty$, by taking +\begin{align*} +(x:A_n) & \mapsto \iota^{S(A),S(f)}_n(f_n(x)) \\ +(x:A_n) & \mapsto \kappa^{S(A),S(f)}_n(f_n(x)). +\end{align*} + +Next, we construct a map $\psi:S(A)_\infty\to A_\infty$ by induction on $S(A)_\infty$, by taking +\begin{align*} +(x:S(A)_n) & \mapsto \iota^{A,f}_{n+1}(x) \\ +(x:S(A)_n) & \mapsto \kappa^{A,f}_{n+1}(x). +\end{align*} + +Then we prove that $\psi\circ \varphi\htpy \idfunc$ by induction on $A_\infty$, by taking +\begin{align*} +(x:A_n) & \mapsto \kappa^{A,f}_n(x) +\end{align*} +Now we compute +\begin{align*} +\mathsf{ap}_{\psi\circ\varphi}(\kappa^{A,f}_n(x))) & = \mathsf{ap}_\psi(\mathsf{ap}_\varphi(\kappa^{A,f}_n(x))) \\ +& = \mathsf{ap}_\psi(\kappa^{S(A),S(f)}_n(f_n(x))) \\ +& = \kappa^{A,f}_{n+1}(f_n(x)) +\end{align*} +from the computation rules of $A_\infty$ and $S(A)_\infty$. + +We construct the homotopy $\varphi\circ\psi\htpy\idfunc$ by induction on $A_\infty$, by taking +\begin{align*} +(x:S(A)_n) & \mapsto \kappa_{n+1}(S(f)_n(x)) +\end{align*} +Now we compute +\begin{align*} +\mathsf{ap}_{\varphi\circ\psi}(\kappa^{S(A),S(f)}_n(x)) & = \mathsf{ap}_\varphi(\mathsf{ap}_\psi(\kappa^{S(A),S(f)}_n(x))) \\ +& = \mathsf{ap}_\varphi(\kappa^{A,f}_{n+1}(x)) \\ +& = \kappa^{S(A),S(f)}_{n+1}(f_{n+1}(x)).\qedhere +\end{align*} +\end{proof} + +\begin{lem}\label{lem:colim_shift_k} +For any type sequence $(A,f)$, we have an equivalence +\begin{equation*} +\kshiftequiv_{k} : \tfcolim(A,f)\eqvsym\tfcolim (S_k(A,f)). +\end{equation*} +\end{lem} + +The shift operations and the corresponding equivalences on the sequential colimits can be used to turn an arbitrary sequence $\sequence{B}{g}$ over $\sequence{A}{f}$ into an equifibered sequence over $\sequence{A}{f}$. + +\begin{defn} +Given a dependent sequence $(B,g)$ over $(A,f)$ and $x:A_0$, we define a type sequence $(B[x],g[x])$ by +\begin{align*} +B[x]_n & \defeq B_n(f^n(x)) \\ +g[x]_n & \defeq g_n(f^n(x),\blank). +\end{align*} +\end{defn} + +\begin{defn} +Given any sequence $\sequence{B}{g}$ over $\sequence{A}{f}$, we define an equifibered sequence +$\sequence{\square B}{\square g}$ over the sequence $\sequence{A}{f}$. +\end{defn} + +\begin{constr} +For $x:A_n$ we define +\begin{equation*} +(\square B)_n(x) \defeq S_n(B)[x]_{\infty} \jdeq \tfcolim_m(B_{n+m}(f^m(x))). +\end{equation*} +Now note that +\begin{align*} + (\square B)_{n+1}(f(x)) & \jdeq \tfcolim_m(B_{(n+1)+m}(f^m(f(x))) \\ + & \simeq \tfcolim_m(B_{n+(m+1)}(f^{m+1}(x))\\ + & \simeq \tfcolim_m(B_{n+m}(f^m(x))\\ + & \jdeq (\square B)_n(x) +\end{align*} +The first equivalence $u_{n,m}$ is given by transporting along the dependent path in \autoref{lem:iterate_succ} in the family $B$. This forms a natural equivalence, because $\mathsf{transport}$ is natural. The second equivalence is given by applying \autoref{lem:colim_shift_one}. We call the composite equivalence $F$, which shows that $\square B$ is an equifibered sequence. +\end{constr} + +\begin{defn} +Let $\sequence{B}{g}$ be a sequence over $\sequence{A}{f}$. Then we define +\begin{equation*} +B_\infty : A_\infty\to\UU +\end{equation*} +to be the family over $A_\infty$ associated to the equifibered sequence $(\square B,\square g)$ via the equivalence of \autoref{thm:descent}. +\end{defn} + +By construction of $B_\infty$ we get the equality +$$r(y):\transfib{B_\infty}{\kappa_n(x)}{y}=F(y)$$ +for $y:B_\infty(\iota_{n+1}(f_n(x)))$ witnessing that $B_\infty$ is defined by the equivalence $F$ on the path constructor. + +We now state our main result, which could be seen as a flattening lemma for sequential colimits, +with the added generality that the sequence $\sequence{B}{g}$ over +$\sequence{A}{f}$ is not required to be equifibered. + +\begin{thm}\label{thm:colim_sm} + Let $P\defeq\sequence{P}{f}$ be a sequence over $A\defeq\sequence{A}{a}$. Then we have a + commuting triangle + \begin{equation*} + \begin{tikzcd}[column sep=huge] + \tfcolim(\msm{A}{P}) \arrow[rr,"{\alpha}"] \arrow[dr,swap,"{p\defeq\mathsf{rec}(\iota_n\circ \proj 1,\blank)}"] + & & (x:A_\infty)\times P_\infty(x) \arrow[dl,"{\proj 1}"] \\ + & A_\infty + \end{tikzcd} + \end{equation*} + in which $\alpha$ is an equivalence. + \end{thm} + + The strategy of the proof is to first show that $(x:A_\infty)\times P_\infty(x)$ has the induction principle of $\colim((x:A_n)\times P_n(x),(a_n,f_n))_n$. This simplifies giving the equivalence, because + $(x:A_\infty)\times P_\infty(x)$ is a 2-HIT, being a sigma-type of two 1-HITs, while $\colim((x:A_n)\times P_n(x),(a_n,f_n))_n$ is a 1-HIT. Before we continue, we first define $\alpha$. + + The map $\alpha$ is defined by induction on $\tfcolim(\msm{A}{P})$. On the point constructors we define + \begin{equation*} + \alpha(\iota_n(x,y))\defeq \pairr{\iota_n(x),\iota_0(y)}. + \end{equation*} + For the path constructor we need to define + $$\kappa'_n(x,y):(\iota_{n+1}(a_nx),\iota_0(f_n(x,y)))=(\iota_n(x),\iota_0(y))$$ + The first components are equal by $\kappa_n(x)$. By the definition of $P_\infty$, transporting along $\kappa_n(x)$ takes $\iota_0(f_n(x,y))$ to $\iota_1(f_n(x,y))$, which is equal to $\iota_0(y)$ by $\kappa_0(y)$. Explicitly, we define + $$\apfunc{\alpha}(\kappa_n(x,y))\vcentcolon=\kappa'_n(x,y)\defeq (\kappa_n(x),r(\iota_0(f_n(x,y)))\cdot\kappa_0(y)).$$ + + \begin{thm}\label{thm:sigma-colim-induction} + Let $E: (x:A_{\infty})\to P_{\infty}(x) \to \UU$ such that + \begin{enumerate} + \item For each $n:\N$, $x:A_n$, $y:P_n(x)$, a term $e_n(x,y):E(\iota_n(x),\iota_0(y))$. + \item For each $n:\N$, $x:A_n$, $y:P_n(x)$, a path + \begin{equation*} + w_{n}(x,y):e_{n+1}(a_nx,f_n(x,y))=_{\kappa'_n(x,y)}^Ee_n(x,y). + \end{equation*} + \end{enumerate} + Then there exists a function $s:(x:A_\infty)(y:P_\infty(x))\to E(y)$. + \end{thm} + + \begin{proof} + We define the function $s$ by induction on both $x$ and $y$. We need to consider four cases, since both $x$ and $y$ can be a point constructor or vary over a path constructor. + + \emph{(point-point)} + Fix $x:A_n$, we first define $g(n,x):(p:P_{\infty}(\iota_n(x))) \to E(\iota_n(x),p)$. To obtain $g(n,x)$, we do induction on $p:P_\infty(\iota_n(x))$. Fix $y:P_{n+k}(a_n^k(x))$, we need to construct a term of type $g_\ast(k,n,x,y) : E(\iota_n(x),\iota_k(y))$. Proceed by induction on $k$. We can define + $$g_\ast(0,n,x,y)\defeq e_n(x,y) : E(\iota_n(x),\iota_0(y)).$$ + Assume that $g_\ast(k)$ is defined. + We need to define $g_\ast(k+1,n,x,y):E(\iota_n(x),\iota_{k+1}(y))$, where $y:P(n+(k+1),a_n^{k+1}(x))$. However, the type of $y$ is equivalent to the type $P((n+1)+k,a_{n+1}^k(a_n(x)))$ via the equivalence $u_{n,k}$. Therefore, it suffices to define for $z:P_{(n+1)+k}(a_{n+1}^k(a_n(x)))$ + \begin{equation*} + g_\ast(k+1,n,x,u_{n,k}(z)) : E(\iota_n(x),\iota_{k+1}(u_{n,k}(z))). + \end{equation*} + + By induction hypothesis we have $g_\ast(k,n+1,a_n(x),z):E(\iota_{n+1}(a_n(x)),\iota_k(z))$, so it suffices to show that + + $$\kappa^\ast_{n,k}(x,z):(\iota_{n+1}(a_n(x)),\iota_k(z))=(\iota_n(x),\iota_{k+1}(u_{n,k}(z))).$$ + This construction is similar to that of $\kappa'_n(x,y)$. + The first components are equal by $\kappa_n(x)$, and for the second components we need to show that $\transfib{P_\infty}{\kappa_n(x)}{\iota_k(z)}=\iota_{k+1}(u_{n,k}(z))$. This follows from the computation rule of $P_\infty$ on paths, since the equivalence used to define $P_\infty$ sends $\iota_k(z)$ to $\iota_{k+1}(u_{n,k}(z))$. Specifically, + $$\kappa^\ast_{n,k}(x,z)\defeq(\kappa_n(x),r(\iota_k(y))).$$ + This finishes the construction of $g_\ast$, hence also of $g$ on points. By construction, we get the following equation: + $$\mu_{n,k}(x,z):\dpath{E}{\kappa^\ast_{n,k}(x,z)}{g_\ast(k,n+1,a_n(x),z)}{g_\ast(k+1,n,x,u_{n,k}(z))}$$ + + \emph{(point-path)} To define $g$ on paths $\kappa_k(y):\iota_{k+1}(f(y))=\iota_k(y)$, we need to give a dependent path + $$\nu(k,n,x,y):g_\ast(k+1,n,x,f(y))=^{E(\iota_n(x))}_{\kappa_k(y)}g_\ast(k,n,x,y).$$ + We do this by induction on $k$. For $k=0$ note that $u_{n,0}$ is the identity function, and the goal definitionally reduces to + $$\nu(k,n,x,y):\transfib{E}{\kappa^\ast_{n,0}(x,f_n(x,y)}{e_{n+1}(a_n(x),f_n(x,y)}=^{E(\iota_n(x))}_{\kappa_0(y)}e_n(x,y).$$ + Note that $\kappa'_n(x,y)=\kappa^\ast_{n,0}(f_n(x,y)\cdot(1,\kappa_0(y))$, which means we get this from $w_n(x,y)$. Now suppose that $\nu(k)$ is defined. We need to define for $y:P(n+(k+1),a_n^{k+1}(x))$ + $$\nu(k+1,n,x,y):g_\ast(k+2,n,x,f(y))=^{E(\iota_n(x))}_{\kappa_k(y)}g_\ast(k+1,n,x,y).$$ + Now we again write $y=u_{n,k}(z)$ for $z:P((n+1)+k,a_{n+1}^k(a_n(x)))$ and we equivalently need to give + $$\nu(k+1,n,x,u_{n,k}(z)):g_\ast(k+2,n,x,f(u_{n,k}(z)))=^{E}_{(1,\kappa_k(y))}g_\ast(k+1,n,x,u_{n,k}z).$$ + We will define this as the composition of a square that we will give later in the proof. + + \emph{(path-point)} We have defined $s$ on points constructors of $A_\infty$. To define it on the path $\kappa_n(x):\iota_{n+1}(a_n(x))=\iota_n(x)$ we need a path $g(n+1,a_n(x))=g(n,x)$ over $\kappa_n(x)$. By function extensionality, we can characterize dependent paths in a function type, which means we need to show: + \begin{equation*} + (p:P_\infty(\iota_{n+1}(a_n(x))))\to g(n+1,a_n(x),p) =^E_{(\kappa_n(x),1)} g(n,x,\transfib{P_\infty}{\kappa_n(x)}{p}). + \end{equation*} + Now for $p:P_\infty(\iota_{n+1}(a_n(x)))$, we can apply the path $r(p)$, which means we need to construct the following path (note that $r(p)$ is added to the path, since $g$ is a dependent function): + \begin{equation*} + g(n+1,a_n(x),p) =^E_{(\kappa_n(x),r(p))} g(n,x,F(p)). + \end{equation*} + We proceed by induction on $p$. If $p\jdeq\iota_k(y)$ for $k:\N$, $y:P_{(n+1)+k}(a_{n+1}^k(a_n(x)))$, then $F(p)\jdeq \iota_{k+1}(u_{n,k}(y))$ and we need a path + \begin{equation*} + g_\ast(k,n+1,a_n(x),y) =^E_{(\kappa_n(x),r(\iota_k(y))))} g_\ast(k+1,n,x,u_{n,k}(y)). + \end{equation*} + Now the path $(\kappa_n(x),r(\iota_k(y)))\jdeq\kappa^\ast_{n,k}(x,y)$, hence this dependent path is given by $\mu_{n,k}(x,y)$. + + \emph{(path-path)} If $p$ varies over $\kappa_k(y)$, we need to give a dependent path in a family of dependent paths. + This is equivalent to filling the following dependent square in the family $E$, + which lies over the naturality square form by applying $\lam{p}(\kappa_n(x),r(p))$ to the path $\kappa_k(y)$.\footnote{The + left and right sides of the square are not quite correct, + the dependent function applied to $\kappa_k(y)$ are pathovers lying over $\kappa_k(y)$, and not $(1,\kappa_k(y))$. + However, pathovers lying over $\kappa_k(y)$ in the family $E(\iota_n(x))$ are equivalent to pathovers lying over $(1,\kappa_k(y))$ in the family $E$, + and this equivalence commutes with all operations we perform, therefore we omit them in this proof. + The following calculations are only type correct when these equivalences are inserted back. + Furthermore, we omit some other details. For example, if $p = q$, + then $\mapdep{g}{p}$ and $\mapdep{g}{q}$ have different types: + the former is a dependent path over $p$ and the latter one over $q$. + However, if you modify the path over which they lie, they become equal. + These ``modifications'' can be pushed down to the square in $(x:A_\infty)\times P_\infty(x)$, + and the proof still goes through. For the full details, consult the formal proof.} + \begin{xcenter}\begin{tikzcd}[column sep=25mm] + g_\ast(k+1,n+1,a_n(x),f_{(n+1)+k)}(y)) + \ar[r,equal,"{\mu_{n,k+1}(x,f_{(n+1)+k}(y))}","{\kappa^\ast_{n,k+1}(x,f_{n+k}(y))}" swap] + \ar[d,equal,"{\mapdep{g(n+1,a_n(x))}{\kappa_k(y))})}","{(1,\kappa_k(y)}" swap] & + g_\ast(k+2,n,x,u_{n,(k+1)}(f_{(n+1)+k}(y))) + \ar[d,equal,"{\mapdep{g(n,x)\circ F}{\kappa_k(y))})}","{(1,\kappa_k(y)}" swap]\\ + g_\ast(k,n+1,a_n(x),y) + \ar[r,equal,"{\mu_{n,k}(x,y)}","{\kappa^\ast_{n,k}(x,y)}" swap] & + g_\ast(k+1,n,x,u_{n,k}(y)) + \end{tikzcd}\end{xcenter} + Below and to the left of each equal sign we give the path in $(x:A_\infty)\times P_\infty(x)$ over which the pathover lie. Above and to the right of each equal sign we give the value of the dependent path. + + Now $\mapdep{g(n+1,a_n(x))}{\kappa_k(y)}$ (occurring in the left pathover) is equal to $\nu(k,n+1,a_n(x),y)$ by definition of $g$. On the right, we have ($\delta$ is the naturality of $u_{n,k}$) + \begin{align*} + \mapdep{g(n,x)\circ F}{\kappa_k(y)} &= + \mapdep{g(n,x)}{\map{F}{\kappa_k(y)}}\\ + &= \mapdep{g(n,x)}{\map{\iota_{k+2}}{\delta(y)}\cdot\kappa_{k+1}(u_{n,k}(y))}\\ + &= \mapdep{g_\ast(k+2,n,x)}{\delta(y)} \cdot \mapdep{g(n,x)}{\kappa_{k+1}(u_{n,k}(y))}\\ + &= \mapdep{g_\ast(k+2,n,x)}{\delta(y)} \cdot \nu(k+1,n,x,u_{n,k}(y))\\ + \end{align*} + Now we can move the first part of the expression to the top of the square, which means we need to fill the following squareover (where we made some arguments implicit). + \begin{center}\begin{tikzcd}[column sep=25mm] + g_\ast(f(y)) + \ar[r,equal,"{\mu(f(y))}","{\kappa^\ast(f(y))}" swap] + \ar[d,equal,"{\nu(k,n+1,a_n(x),y)}","{(1,\kappa(y))}" swap] & + g_\ast(u(f(y))) + \ar[r,equal,"{\mapdep{g_\ast(k+2,n,x)}{\delta(y)}}","{(1,\delta(y))}" swap] & + g_\ast(f(u(y))) + \ar[d,equal,"{\nu(k+1,n,x,u_{n,k}(y))}","{(1,\kappa(u_{n,k}(y)))}" swap]\\ + g_\ast(y) + \ar[rr,equal,"{\mu(y)}","{\kappa^\ast(y)}" swap] & & + g_\ast(u_{n,k}(y)) + \end{tikzcd}\end{center} + Note that in this squareover $g$ does not occur, and $\nu$ only occurs on the left side (applied to $k$) and on the right side (applied to $k+1$). Therefore, the top, bottom and left side form a valid open box, and we define $\nu(k+1,n,x,u_{n,k}(y))$ to be the composition of \emph{this} open box. This inductively defines $\nu$, and makes the filler for this square automatic. This finishes the proof. + \end{proof} + \begin{proof}[Proof (of Theorem \ref{thm:colim_sm})] + We first define a map $$\beta: \big((x:A_\infty)\times P_\infty(x)\big) \to \tfcolim(\msm{A}{P}).$$ + We do this by induction on $x:A_\infty$ and $p:P_\infty(x)$ individually, so we get four cases again + (we do not use our newly defined induction principle, because we have not proven a computation rule for it). + + \emph{(point-point)} Suppose $x:A_n$ and $y:P_{n+k}(a_n^k(x))$. We define + $$\beta(\iota_n(x),\iota_k(y))\defeq\iota_{n+k}(a_n^k(x),y).$$ + \emph{(point-path)} To show that the second argument respects $\kappa_k(y)$, we define + $$\mapfunc{\beta(\iota_n(x))}(\kappa_k(y))\vcentcolon=\kappa_{n+k}(a_n^k(x),y):\iota_{n+(k+1)}(a_n^{k+1}(x),f(y))=\iota_{n+k}(a_n^k(x),y).$$ + \emph{(path-point)} To show that the first argument respects $\kappa_n(x)$, we need to give a dependent path + $$\beta(\iota_{n+1}(a_n(x)))=^{P_\infty({-})\to\tfcolim(\msm{A}{P})}_{\kappa_n(x)}\beta(\iota_n(x)).$$ + By function extensionality, this is equivalent to showing for $p:P_\infty(\iota_{n+1}(a_n(x)))$ that + $$\beta(\iota_{n+1}(a_n(x)),p)=\beta(\iota_n(x),\transfib{P_\infty}{\kappa_n(x)}{p}).$$ + We apply $\apfunc{\beta(\iota_n(x))}\inv{(r(p))}$ on the right, so that we have to show + $$\beta(\iota_{n+1}(a_n(x)),p)=\beta(\iota_n(x),F(p)).$$ + Now we apply induction on $p$. If $p\equiv\iota_k(y)$, then $F(p)\jdeq \iota_{k+1}(u_{n,k}(y))$ and we need to show + $$\mu_{n,k}(x,y):\iota_{(n+1)+k}(a_{n+1}^k(a_n(x)),y)=\iota_{n+(k+1)}(a_n^{k+1},u_{n,k}(y)).$$ + But the triples $((n+1)+k,a_{n+1}^k(a_n(x)),y)$ and $(n+(k+1),a_n^{k+1},u_{n,k}(y))$ are equal: the first two components by \autoref{lem:iterate_succ} and the last component because $u_{n,k}$ was defined by transporting along the equality of the first components. Let us call this equality $s$. So we define $\mu_{n,k}(x,y)$ by applying $\iota$ to $s$. + + \emph{(path-path)} Suppose $p$ varies along $\kappa_k(y)$, we need to construct a proof of a pathover in an equality type. This is equivalent to filling the following square. + \begin{center}\begin{tikzcd}[column sep=25mm] + \beta(\iota_{n+1}(a_n(x)),\iota_{k+1}(f(y))) + \ar[r,equal,"{\mu_{n,k+1}(x,f(y))}"] + \ar[d,equal,"{\mapfunc{\beta(\iota_{n+1}(a_nx))}(\kappa_k(y)})"] & + \beta(\iota_n(x),\iota_{k+2}(u_{n,k+1}(f(y)))) + \ar[d,equal,"{\mapfunc{\beta(\iota_n(x))\circ F}(\kappa_k(y))}"]\\ + \beta(\iota_{n+1}(a_n(x)),\iota_k(y)) + \ar[r,equal,"{\mu_{n,k}(x,y)}"] & + \beta(\iota_n(x),\iota_{k+1}(u_{n,k}(y))) + \end{tikzcd}\end{center} + By simplifying the left and right path, this reduces to + \begin{center}\begin{tikzcd}[column sep=10mm] + \beta(\iota(a(x)),\iota(f(y))) + \ar[r,equal,"{\mu(f(y))}"] + \ar[d,equal,"{\kappa(a^k(a(x)),y)}"] & + \beta(\iota(x),\iota(u(f(y)))) + \ar[r,equal,"{\mapfunc{\beta(\iota(a(x)),\iota({-}))}(\kappa(y))}"] &[15mm] + \beta(\iota(x),\iota(f(u(y)))) + \ar[d,equal,"{\kappa(a^k(x),u(y))}"]\\ + \beta(\iota(a(x)),\iota_k(y)) + \ar[rr,equal,"{\mu(y)}"] & & + \beta(\iota(x),\iota(u(y))) + \end{tikzcd}\end{center} + Now the concatenation of the two paths on the top reduces to the function $i\defeq\lam{n}{x}{y}\iota_{n+1}(a(x),f(y))$ applied to $s$. Then the square is exactly the naturality square of the homotopy $\kappa:i\sim\iota$ applied to the path $s$. This finishes the definition of $\beta$. + + Now we need to show that $\beta\circ\alpha\sim\idfunc$. Take $p:\tfcolim(\msm{A}{P})$, we apply induction to $p$. If $p\jdeq\iota_n(x,y)$, then the equality holds by reflexivity: + $$\beta(\alpha(p))\jdeq\beta(\iota_n(x),\iota_0(y))\jdeq\iota_n(x,y)\jdeq p.$$ + If $p$ varies over $\kappa_n(x,y)$, we need to fill a square with two degenerate sides, so we need to prove that + $\apfunc{\beta\circ\alpha}(\kappa_n(x,y))=\kappa_n(x,y).$ + We can show this as follows. + \begin{align*} + &\mathrel{\hphantom{=}}\apfunc{\beta\circ\alpha}(\kappa_n(x,y))\\ + &=\apfunc{\beta}(\kappa_n(x),r(\iota_0(f(y)))\cdot\kappa_0(y))\\ + &=\apfunc{\beta}(\kappa'_n(x,y))\\ + &=\mu_{n,1}(x,f(y))\cdot\apfunc{\beta(\iota_n(x))}\inv{(r(\iota_0(f(y))))}\cdot\apfunc{\beta(\iota_n(x))}(r(\iota_0(f(y)))\cdot\kappa_0(y))\\ + &=\mu_{n,1}(x,f(y))\cdot\apfunc{\beta(\iota_n(x))}(\kappa_0(y))\\ + &=\apfunc{\beta(\iota_n(x))}(\kappa_0(y))\\ + &=\kappa_n(x,y) + \end{align*} + In the third step we use that $\apfunc{\beta}(p,q)=\mapdep{\beta}{p}(f(y))\cdot\mapfunc{\beta{\iota_n(x)}(q)}$ and in the fifth step that $\mu_{n,k}(x,y)=1$ for any \emph{numeral} $k$. + + Finally we need to show that $\alpha\circ\beta\sim\idfunc$. Take $p:(x:A_\infty)\times P_\infty(x)$. We apply the induction principle proven in \autoref{thm:sigma-colim-induction} to $p$. Suppose that $p\jdeq(\iota_n(x),\iota_0(y))$. Then the equality holds by reflexivity: + $$\alpha(\beta(p))\jdeq\alpha(\iota_n(x,y))\jdeq(\iota_n(x),\iota_0(y))\jdeq p.$$ + If $p$ varies over $\kappa'_n(x,y)$, then we have to show (similar to the proof $\beta\circ\alpha\sim\idfunc$) that + $$\apfunc{\alpha\circ\beta}(\kappa'_n(x,y))\kappa'_n(x,y).$$ + But by the previous computation, $\apfunc{\beta}(\kappa'_n(x,y))=\kappa_n(x,y)$, so we have + $$\apfunc{\alpha\circ\beta}(\kappa'_n(x,y))=\apfunc{\alpha}(\kappa_n(x,y))=\kappa'_n(x,y).$$ + This finishes the proof. + \end{proof} + + \begin{cor}\label{cor:eq_colim} + Consider a sequence $(A_n,f_n)_n$. Then for any $a,a':A_n$ there is an equivalence + \begin{equation*} + \eqv{(\iota_n(a)=_{A_\infty}\iota_n(a'))}{\tfcolim(f^k(a)=_{A_{n+k}}f^k(a'))}. + \end{equation*} + \end{cor} + \begin{proof}%% TODO (Floris): check this proof, I think some complications arose during the formalization. + We first prove this for $n \jdeq 0$. + % By \autoref{lem:colim_shift}, we only need to show this for $n\jdeq 0$. + Note that for any $a:A_0$, we have the diagram + \begin{equation*} + \begin{tikzcd} + \lam{a':A_0}a=a' \arrow[r] \arrow[d,->>] & \lam{a':A_1} f(a)=a' \arrow[r] \arrow[d,->>] & \lam{a':A_2} f^2(a)=a' \arrow[r] \arrow[d,->>] & \cdots \\ + A_0 \arrow[r,"f_0"] & A_1 \arrow[r,"f_1"] & A_2 \arrow[r,"f_2"] & \cdots + \end{tikzcd} + \end{equation*} + This defines a type family $P:A_\infty\to\type$ with + $$P(\iota_n(a'))\defeq\tfcolim_k(f^{0\le n+k}(a)=_{A_{n+k}}f^k(a')).$$ + Now we use \autoref{thm:colim_sm} to see that the total space of $P$ is contractible. + \begin{align*} + (a':A_\infty)\times P(a') + & \eqvsym \tfcolim_n((a':A_n)\times f^{n}(a)=a') \\ + & \eqvsym \tfcolim_n(\unit) \\ + & \eqvsym \unit. + \end{align*} + Since $\iota_0(\refl{a}):P(\iota_0(a))$ and noting that $f^{0\le 0+k}(a)\jdeq f^k(a)$ we can now conclude by the total space method to characterize the identity type that + $$(\iota_0(a)=_{A_\infty}\iota_0(a'))\simeq P(\iota_0(a'))\jdeq\tfcolim(f^k(a)=_{A_{0+k}}f^k(a')).$$ + + For general $n$, we use \autoref{lem:colim_shift_k}, which gives us an equivalence + $\kshiftequiv_n: A_\infty\simeq \tfcolim(S_n(A,f))$. For $a,a':A_n$ we can now compute: + \begin{align*} + (\iota_n(a)=_{A_\infty}\iota_n(a')) + & \eqvsym (\kshiftequiv_n(\iota_n(a))=_{\tfcolim(S_n(A,f))}\kshiftequiv_n(\iota_n(a'))) \\ + & \eqvsym (\iota_0(a)=_{\tfcolim(S_n(A,f))}\iota_0(a')) \\ + & \eqvsym \tfcolim(S_n(f)^k(a)=_{S_n(A)_{0+k}}S_n(f)^k(a')) \\ + & \eqvsym \tfcolim(f^k(a)=_{A_{n+k}}f^k(a')).. + \end{align*} + The last equivalence comes from a natural equivalences of the sequences, because there is a dependent path between + $S_n(f)^k(a)$ and $f^k(a)$ over the canonical path that $n+(0+k)=n+k$. + \end{proof} + + \begin{cor}\label{cor:fiber_functor} + Suppose given a natural transformation $\tau : \sequence{A'}{f'}\to\sequence{A}{f}$ and a point $a:A_n$. Then + $$\hfib{\tau_\infty}{\iota_n(a)}\simeq\colim(\square\mathsf{fib}_\tau[a])\jdeq\colim_k(\hfib{\tau_{n+k}}{f^k(a)}).$$ + \end{cor} + \begin{proof} + Consider the following diagram, where the equivalences on the top are given by \autoref{thm:colim_sm} and the fact that the total space of the fiber of a function is the domain of that function. + \begin{center}\begin{tikzcd}[column sep=10mm] + (x:A_\infty)\times (\mathsf{fib}_\tau)_\infty(x) \ar[r,"{\sim}"] \ar[dr,"{\pi_1}"] & + \colim_k((x:A_n))\times \hfib{\tau_n}{x}) \ar[r,"{\sim}"]\ar[d,"p"] & + A'_\infty \ar[dl,"{\tau_\infty}"] \\ + & A_\infty & + \end{tikzcd}\end{center} + This diagram commutes: the left triangle commutes by \autoref{thm:colim_sm} and the right triangle commutes by the 1-functoriality of the colimit, \autoref{lem:seq_colim_functor}. Therefore, + $$\hfib{\tau_\infty}{\iota_n(a)}\simeq\hfib{\pi_1}{\iota_n(a)}\simeq (\mathsf{fib}_\tau)_\infty(\iota_n(a))\jdeq \colim(\square\mathsf{fib}_\tau[a]).$$ + \end{proof} + + \begin{cor}\label{cor:trunc_colim} + Consider a sequence $\sequence{A}{f}$ and some $k\geq-2$. + \begin{enumerate} + \item\label{part:colim_is_trunc} If $A_n$ is $k$-truncated for all $n:\nat$, then $A_\infty$ is $k$-truncated. + \item\label{part:colim_trunc} We have an equivalence $$\trunc{k}{A_\infty}\simeq\colim(\trunc{k}{A_n},\trunc{k}{f_n})_n.$$ + \item\label{part:colim_is_connected} If $A_n$ is $k$-connected for all $n:\nat$, then $A_\infty$ is $k$-connected. + \item\label{part:functor_is_trunc_conn} Given a natural transformation $(\tau,H):\sequence Af \to \sequence{A'}{f'}$ such that $\tau_n$ is $k$-truncated ($k$-connected) for all $n$, then $\tau_\infty$ is $k$-truncated ($k$-connected). + \item\label{part:iota_is_trunc_conn} If $f_n$ is $k$-truncated ($k$-connected) for all $n$, then $\iota_0$ is $k$-truncated ($k$-connected). + \end{enumerate} + \end{cor} + + \begin{rmk} + By \autoref{lem:colim_shift_k} we can generalize the quantification ``for all $n:\nat$'' in this Corollary to the weaker ``there exists an $m:\nat$ such that for all $n\ge m$''. In part \ref{part:iota_is_trunc_conn} the conclusion then becomes that $\iota_m$ is $k$-truncated ($k$-connected). + \end{rmk} + + \begin{proof}\mbox{} + \begin{enumerate} + \item We prove this by induction on $k$. Suppose $k=-2$, then $f_n$ is an equivalence for all $n$. Therefore $A_\infty\simeq A_0$ by \autoref{lem:equiv_equiseq}, hence $A_\infty$ is contractible. + + Now suppose $k\jdeq k'+1$. Take $x,x' : A_\infty$, we need to show that $x=x'$ is $k'$-truncated. Since being truncated is a mere proposition, by induction on $x$ and $x'$ we may assume that $x\jdeq\iota_n(a)$ and $x'\jdeq\iota_m(a')$. Now + $\iota_n(a)=\iota_{\max(n,m)}(f^{n\le \max(n,m)}(a))$ and $\iota_m(a')=\iota_{\max(n,m)}(f^{m\le \max(n,m)}(a'))$, therefore the type $\iota_n(a)=\iota_m(a')$ is equivalent to + $$\iota_{\max(n,m)}(f^{n\le \max(n,m)}(a))=\iota_{\max(n,m)}(f^{m\le \max(n,m)}(a')).$$ Therefore it suffices to show that the latter equality type is $k'$-truncated. By \autoref{cor:eq_colim} we need to show that + $$\tfcolim(f^\ell(f^{n\le \max(n,m)}(a))=f^\ell(f^{m\le \max(n,m)}(a')))_\ell$$ + is $k'$-truncated, which follows from the induction principle and the fact that $A_{\max(n,m)+\ell}$ is $(k'+1)$-truncated. + \item From the functoriality of the sequential colimit, we get a function + $$A_\infty\to\colim_n(\trunc{k}{A_n},\trunc{k}{f_n}).$$ + Because the right hand side is $k$-truncated, this induces a map + $$g:\trunc{k}{A_\infty}\to\colim(\trunc{k}{A_n},\trunc{k}{f_n})_n.$$ + For the other direction, we define the function + $$h:\colim(\trunc{k}{A_n},\trunc{k}{f_n})_n\to\trunc{k}{A_\infty}$$ + by + $$h(\iota_n(\tproj{k}{a}))\defeq\tproj{k}{\iota_n(a)}$$ + and + $$\mapfunc{h}(\kappa_n(\tproj{k}{a}))\vcentcolon=\mapfunc{\tprojf{k}}(\kappa_n(a)).$$ It is straightforward to show that both $h\circ g$ and $g\circ h$ are homotopic to the identity. + \item Since $A_n$ is $k$-connected, $\trunc{k}{A_n}$ is contractible, and therefore $\colim_n(\trunc{k}{A_n})\simeq \trunc{k}{A_\infty}$ is contractible. + \item A function is $k$-truncated ($k$-connected) whenever its fibers are $k$-truncated ($k$-connected). + Let $x:A_\infty$. We need to show a proposition, so we may assume that $x\jdeq \iota_n(a)$ for some $a:A_n$. Now + $\hfib{\tau_\infty}{\iota_n(a)}\simeq\tfcolim(\square\mathsf{fib}_\tau[a])$ by \autoref{cor:fiber_functor}. Since $\hfib{\tau_n}{x}$ is $k$-truncated ($k$-connected) for all $n$, we know that $\tfcolim(\square\mathsf{fib}_\tau[a])$ is $k$-truncated ($k$-connected) for all $n$, by part \ref{part:colim_is_trunc} or \ref{part:colim_is_connected}. + \item Consider the natural transformation + \begin{center}\begin{tikzcd}[column sep=10mm] + A_0 \ar[r,equal] \ar[d,equal] & + A_0 \ar[r,equal] \ar[d,equal,"f"] & + A_0 \ar[r,equal] \ar[d,equal,"{f^{0\le2}}"] & + A_0 \ar[r,equal] \ar[d,equal,"{f^{0\le3}}"] & + \cdots \ar[r,equal] & + \colim(A_0)_n \ar[d] \\ + A_0 \ar[r] & + A_1 \ar[r] & + A_2 \ar[r] & + A_3 \ar[r] & + \cdots \ar[r] & + A_\infty + \end{tikzcd}\end{center} + The maps $f^{0\le n}:A_0\to A_n$ are $k$-truncated ($k$-connected) and form a natural transformation. Therefore, by part \ref{part:functor_is_trunc_conn} the map $f^{0\le\infty}:\colim_n(A_0)\to A_\infty$ is $k$-truncated ($k$-connected). The fiber of $\iota_0$ over $x:A_\infty$ is the same as the fiber of $f^{0\le\infty}$ over $x$, and therefore $\iota_0$ is $k$-truncated ($k$-connected). + \end{enumerate} + \end{proof} + +We can use this machinery, in particular \autoref{thm:colim_sm}, to define the localization for maps between $\omega$-compact types. We will omit the construction here, but this will be published in an upcoming preprint. + + + + + + + + + + + + + + + + +\chapter{Homotopy Theory}\label{cha:homotopy-theory} + +As discussed in the introduction, one very useful application of HoTT is synthetic homotopy +theory. Many results in homotopy theory have been stated and proven in HoTT in a synthetic way. Most +of these results have also been formalized in a proof assistant. This is important, +because one of the advantages of HoTT is to make verification of proofs by a proof assistant +practically possible. Formalizing results that have been proved internally in HoTT provides more evidence for this. + +In this chapter we will look at various topics in homotopy theory and give proofs for them in HoTT +that are fully checked by the Lean proof assistant. In \cref{sec:computing-pi3s2} we will describe +a formalization of the proof that $\pi_3(\S^2)=\Z$. This was already known to be provable in HoTT, but no fully +formalized proof has been given before. We will discuss some new properties proven about +Eilenberg-MacLane spaces in HoTT in \cref{sec:eilenb-macl-spac}, namely that the Eilenberg-MacLane +space functor induces an equivalence of categories. In \cref{sec:smash-product} we prove the +adjunction of the smash product and pointed maps, from which we can conclude that the smash product +is associative. + +None of these results have been formalized before, even including formalization in foundations other than HoTT. +In fact, not much homotopy theory has been formalized in other foundations. +The most notable examples of formalizations are the formalization of basic properties of the fundamental group~\cite{zhan2017auto2} +and the formalization of singular homology theory~\cite[\texttt{Multivariate/homology.ml}]{hollight}. + + +%Finally, in \cref{cha:serre-spectr-sequ} we construct the Serre Spectral Sequence. + +% \section{Basics}\label{sec:basics} + +% \begin{lem} +% The following results will be used below. +% \begin{itemize} +% \item $\Omega^k\|X\|_{n+k}\simeq \|\Omega^k X\|_n$ (as pointed types). +% \item If $X \simeq Y$ as pointed types, then $\pi_n(X) \simeq \pi_n(Y)$. +% \item $\pi_k(\Omega^n X)\simeq \pi_{n+k}(X)$. +% \end{itemize} +% \end{lem} + +% \begin{thm} +% The suspension is left adjoint to loop spaces: $\Sigma \dashv \Omega$. +% This means that for pointed types $X$ and $Y$ there is a pointed equivalence $\Sigma X \to^* Y +% \equiv^* X \to^* \Omega Y$ which is natural in both $X$ and $Y$. +% \end{thm} + +\section{Computing \texorpdfstring{$\pi_3(\S^2)$}{pi\textunderscore3(S\textasciicircum2)}}% +\label{sec:computing-pi3s2} +Computing that $\pi_3(\S^2)=\Z$ has been done before in Homotopy Type Theory, but it has not been +formalized in a proof assistant before. In this section we will discuss some considerations of +formalizing the proof that $\pi_3(\S^2)=\Z$. The Hopf fibration was formalized in Lean by Ulrik +Buchholtz and was formalized before in Agda by Guillaume Brunerie. The remaining results are formalized by the author. + +% Recall the following classical notion of exactness. +% \begin{defn}\label{def:exact} +% Suppose $A$, $B$ are sets and $C$ is a pointed set. We call a sequence +% $A \xrightarrow{f} B \xrightarrow{g} C$ \emph{exact at $B$} if $\image(f)=\ker(g)$. More explicitly, +% it is exact at $B$ if $(a : A) \to g(f(a))=c_0$ and +% $(b : B) \to g(b)=c_0 \to \|(a : A) \times f(a)=b\|$. + +% We call any sequence of maps between pointed types \emph{exact} if it is exact at every type which +% is not an endpoint of the sequence. +% \end{defn} + +\subsection{The long exact sequence of homotopy groups}\label{sec:les-homotopy} +We start with an important result in homotopy theory, the long exact sequence of homotopy +groups. + +This has been proven before in HoTT. Two different proofs are given in~\cite[Section 8.4]{hottbook} +and~\cite[Section 2.5.1]{brunerie2016spheres}, +although these proofs have not been formalized. There have been previous +formalizations of parts of this +result~\cite{avigad2015limits,voevodsky2015lecture,unimath}; however none of +these formalizations are complete in the sense that they can be used to deduce +the results in this section. + +The statement is as follows. +\begin{thm}[Long exact sequence of homotopy groups]\label{thm:les-homotopy} + Suppose $f : X \to Y$ is a pointed map. Then the following is an exact sequence + + \begin{center}\begin{tikzpicture}[node distance=3cm, thick, node + distance=12mm] + \node (Y) at (0,0) {$\pi_0(Y)$}; + \node[left = of Y] (X) {$\pi_0(X)$}; + \node[left = of X] (F) {$\pi_0(F)$}; + \node[above = of Y] (OY) {$\pi_1(Y)$}; + \node[above = of X] (OX) {$\pi_1(X)$}; + \node[above = of F] (OF) {$\pi_1(F)$}; + \node[above = of OY] (O2Y) {$\pi_2(Y)$}; + \node[above = of OX] (O2X) {$\pi_2(X)$}; + \node[above = of OF] (O2F) {$\pi_2(F)$}; + \node[above = 5mm of O2X] (dots) {$\vdots$}; + % \node[above = 4mm of O2X] (O3X) {$\vdots$}; + \path[every node/.style={font=\sffamily\small}] + (X) edge[->] node [above] (f){$\pi_0(f)$} (Y) + (F) edge[->] node [below] (f){$\pi_0(p_1)$} (X) + (OY) edge[->] node [left] (f){$\pi_0(\delta)\quad\mbox{}$} (F) + (OX) edge[->] node [above] (f){$\pi_1(f)$} (OY) + (OF) edge[->] node [below] (f){$\pi_1(p_1)$} (OX) + (O2Y) edge[->] node [left] (f){$\pi_1(\delta)\quad\mbox{}$} (OF) + (O2X) edge[->] node [above] (f){$\pi_2(f)$} (O2Y) + (O2F) edge[->] node [below] (f){$\pi_2(p_1)$} (O2X); +\end{tikzpicture} +\end{center} +Here $F\defeq\fib_f$ is the fiber of $f$, $p_1:F\to X$ is the first projection, and +$\delta:\Omega Y \to F$ is defined in the proof. +\end{thm} +First of all, we have to carefully formulate the statement of this theorem in type theory. The naive +thing to do is to say that there is a sequence $A : \N \to \set^*$ and maps $f : (n : \N) \to +A_{n+1} \to A_n$ such that +$$A_0\defeq\pi_0(Y),\quad A_1\defeq\pi_0(X),\quad A_2\defeq\pi_0(F),$$ +and so forth. Continuing, this means that +$$A_{3n}=\pi_n(Y),\quad A_{3n+1}=\pi_n(X),\quad A_{3n+2}=\pi_n(F).$$ +However, there is no way to make these equalities definitional, the elimination principle for the +natural numbers does not allow for computation rules like that. This means that the map +$f_{3n}:A_{3n+1}\to A_{3n}$ cannot be compared directly to $\pi_n(f)$ since the domain and codomain +are note definitionally equal. Setting things up this way is possible, but makes reasoning about it +unnecessarily complicated. Instead, we change the indexing set, using $\N\times\fin_3$ instead of $\N$. We will work with a general +notion of sequences with a flexible choice of indexing set. +\begin{defn}\label{def:chain-complex} + A \emph{successor structure} is a type $I$ with endomap $S : I \to I$ called the + \emph{successor}. We will write $i+n$ for $i:I$ and $n:\N$ to mean iterated application of the + successor function, $i+n\defeq S^n(i)$. + + A \emph{chain complex} indexed by a successor structure $I$ is a family of pointed sets + $A : I \to \set^*$ and maps $f : (i : I) \to A_{i+1} \to A_i$ with the property that + $(i : I) \to (a : A_{i+2}) \to f_i(f_{i+1}(a))=a_0^i$ where $a_0^i$ is the basepoint of $A_i$. We call a chain complex \emph{exact} or a \emph{long exact sequence} if + $$(i : I) \to (a : A_{i+1}) \to f_i(a) = a_0^i \to \|(a' : A_{i+2}) \times f_{i+1}(a')=a\|.$$ + A \emph{type-valued chain complex} is the same, except that $A_i$ is only required to be a pointed type (not a pointed set). A type-valued chain complex is \emph{exact} or a \emph{type-valued exact sequence} if the above property holds without any propositional truncation, i.e. if + $$(i : I) \to (a : A_{i+1}) \to f_i(a) = a_0^i \to (a' : A_{i+2}) \times f_{i+1}(a')=a.$$ +\end{defn} + +\begin{rmk} +Note that a type-valued exact sequence gives part of the structure of a \emph{fiber sequence}. A \emph{fiber sequence} is a sequence where $A_{i+2}$ ``is'' the fiber of $f_i$. This means that $(A_{i+2},f_{i+1})=(\fib_{f_i},p_1)$ for all $i$. Using univalence this can be unpacked in an equivalence and a commuting triangle. +In a type-valued exact sequence we just require two maps back and forth $A_{i+2}\leftrightarrow \fib_{f_i}$ such that the corresponding triangles commute, but we do not require that these maps are mutual inverses. In the text below we will have sequences that are not fiber sequences, so we require this additional generality. +\end{rmk} + +\begin{ex}\label{ex:succ_structure} + Some useful examples of successor structures are $(\N,\lam{n}n+1)$ and $(\Z,\lam{n}n+1)$. Sequences over these successor structures correspond to one-sided and two-sided infinite sequences. We can also mimic one-sided infinite sequences in the other direction using the successor structure $(\N,\lam{n}n-1)$ (with the convention that $0-1=0$). This has the disadvantage that there is one extra map $A_0\to A_0$. Whenever we use $\N$ as successor structure in this section, we use $\lam{n}n+1$ as its successor. + + Furthermore, if $N$ is a successor structure and $k : \N$, then we define a successor structure on + $N \times \fin_{k+1}$ by defining + $$S(n,i)\defeq\begin{cases}(n+1,0) & \text{if $i=k$}\\ + (n,i+1) & \text{otherwise}\end{cases}$$ + Note that $n+1$ is addition in the successor structure $N$. +\end{ex} + +We now build the long exact sequence of homotopy groups in five steps. The order of these steps is somewhat arbitrary and can be altered. +We perform the 0-truncation of the sequence as the last step, so that the intermediate sequences contain as much information as possible. +\begin{enumerate}[(1)] +\item First we define the fiber sequence of $f$. +\item Then we show that this sequence is equivalent to a sequence involving iterated loop spaces. +\item We fix some negation signs in the exact sequence. +\item We index the sequence over $\N\times\fin_3$. +\item We 0-truncate the sequence to obtain the sequence in \cref{thm:les-homotopy}. +\end{enumerate} + +We first need some lemmas about fibers. + +\begin{lem}\label{lem:fibers} + Suppose given a pointed map $f : A \to^* B$. Let $p_1 : \fib_f \to^* A$ be the first projection. Then there is a pointed natural equivalence $e_f:\fib_{p_1} \simeq^* \Omega B$. + % Naturality means that if we have a square of pointed maps + % \begin{center}\begin{tikzpicture}[node distance=2cm, thick, node + % distance=12mm] + % \node (A) at (0,0) {$A$}; + % \node (B) at (2,0) {$B$}; + % \node (A') at (0,-2) {$A'$}; + % \node (B') at (2,-2) {$B'$}; + % \node (h) at (1,-1) {$H$}; + % \path[every node/.style={font=\sffamily\small}] + % (A) edge[->] node [above] {$f$} (B) + % edge[->] node [right] {$g$} (A') + % (B) edge[->] node [right] {$h$} (B') + % (A') edge[->] node [above] {$f'$} (B'); + % \end{tikzpicture} + % \end{center} + % then this induces a square of pointed maps + % \begin{center}\begin{tikzpicture}[node distance=2cm, thick, node + % distance=12mm] + % \node (OB) at (0,0) {$\Omega B$}; + % \node[right = of OB] (Ff) {$\fib_{p_1}$}; + % \node[below = of OB] (OB') {$\Omega B'$}; + % \node (Ff') at (Ff |- OB') {$\fib_{p_1'}$}; + % \path[every node/.style={font=\sffamily\small}] + % (OB) edge[->] node [above] {$e_f$} (Ff) + % edge[->] node [right] {$\Omega h$} (OB') + % (Ff) edge[->] node [right] {$\fib_H$} (Ff') + % (OB') edge[->] node [above] {$e_{f'}$} (Ff'); + % \end{tikzpicture} + % \end{center} + % Here $\fib_H$ is the functorial action of the fiber applied to the commuting square $H$. + + Furthermore, if $q_1:\fib_{p_1}\to\fib_f$ is the first projection, we get a commuting square + \begin{center}\begin{tikzpicture}[node distance=2cm, thick, node + distance=12mm] + \node (tl) at (0,0) {$\Omega A$}; + \node[right = of tl] (tr) {$\Omega B$}; + \node[below = of tl] (bl) {$\fib_{q_1}$}; + \node (br) at (tr |- bl) {$\fib_{p_1}$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node {$-\Omega f$} (tr) + edge[<-] node {$e_{p_1}$} (bl) + (tr) edge[<-] node {$e_f$} (br) + (bl) edge[->] node {$r_1$} (br); + \end{tikzpicture} + \end{center} + where $r_1$ is (also) the first projection. We write $-\Omega f$ for the map $\Omega f \o ({-})\sy$. +\end{lem} +\begin{proof} + The underlying equivalence is the following composite + \begin{align*} + \fib_{p_1}&\simeq((a,p):\fib_f)\times a = a_0\\ + &\simeq (a:A)\times a = a_0 \times f(a)=b_0\\ + &\simeq f(a_0)=b_0\\ + &\simeq b_0=b_0 \equiv \Omega B + \end{align*} + This equivalence sends $((a, p), q):\fib_{p_1}$ (with $p : fa=b_0$ and $q:a=a_0$) to + $f_0\sy \tr \ap fq \tr p$. So there is a path + $$r(a,p,q): e_f((a, p), q)=f_0\sy \tr f(q\sy) \tr p.$$ + This path satisfies $r(a_0,f_0,1)=1$ (equality is type correct since $e_f((a_0, f_0), q)\equiv f_0\sy \tr q$). We also have $e_f\sy(p)=((a_0,f_0\cdot p),1)$ for $p:\Omega B$. + + Now $e$ respects the basepoint, because $$e(a_0,f_0,1)=f_0\sy\cdot f_0=1.$$ + We will not prove naturality here, since it is not required for the results in this section. + For the commuting square, we will prove that + $$h:e_f \o r_1 \o e_{p_1}\sy\sim^*-\Omega f$$ + For the underlying homotopy, we compute for $p:\Omega A$ + \begin{align*} + e_f(r_1(e_{p_1}\sy p))&=e_f(r_1(((a_0,f_0),1 \tr p),1))\\ + &=e_f((a_0,f_0),p)\\ + &=f_0\sy \cdot f(p\sy) \cdot f_0 \equiv -\Omega f(p). + \end{align*} + To show that $h$ respects the basepoint, suppose that $p\equiv1$. In that case, the first two steps of the above equation becomes definitional equalities. Since we know that $r(a_0,f_0,1)=1$, the last equality is also reflexivity. Since the maps $e_f\o r\o e_{p_1}\sy$ and $-\Omega f$ respect the basepoints using the same path, this shows that $h$ is a pointed homotopy, which finishes the proof. +\end{proof} + +\subsubsection{Step 1} +Denote $\arrow^*\defeq(X\ Y : \U^*_i)\times (X \to^* Y)$. We define $F:\arrow^*\to\arrow^*$ by +$F(X,Y,f)\defeq(\fib_f,X,p_1)$. +Given a pointed map $f:X\to^* Y$, we define its fiber sequence $A:\N\to\U$ by $A_n\defeq p_2(F^n(X,Y,f))$, and we define $f_n:A_{n+1}\to A_n$ by $p_3(F^n(X,Y,f))$ (which is well-typed, since $A_{n+1}\equiv p_1(F^n(X,Y,f))$ by unfolding the definition of $F$). It is easy to show that $(A_n,f_n)_n$ is a type-valued exact sequence, since $A_{n+2}$ is (definitionally) the fiber of $f_n$. + +Note that by \autoref{lem:fibers} there is a pointed equivalence $e_f:A_3\simeq^* \Omega Y$. We define the diagonal map $\delta\defeq p_1\o e_f\sy : \Omega Y \to \fib_f$. + +\subsubsection{Step 2} +Define the sequence $B:\N\to\U$ and $g_n:B_{n+1}\to B_n$ by +\begin{align*} + B_0&\defeq Y&&\\ + B_1&\defeq X & g_0&\defeq f\\ + B_2&\defeq \fib_f & g_1&\defeq p_1\\ + B_{n+3}&\defeq \Omega B_n & g_2&\defeq \delta\\ + && g_{n+3}&\defeq -\Omega g_n +\end{align*} +Note that $g_2$ has the correct type, since $A_3\equiv B_3$. + +Now we can show that $(B,g)$ is a type-valued exact sequence by showing that it is equivalent to $(A,f)$. + +\begin{lem} + There is a natural equivalence $(A_n, f_n)_n\simeq(B_n,g_n)_n$. This means that there are pointed equivalences $\eta_n:A_n\simeq^* B_n$ such that for all $n:\N$ we have $$\eta_n\o f_n\sim^*g_n\o \eta_{n+1}.$$ +\end{lem} +\begin{proof} + We define the equivalence $\eta_n$ by induction on $n$. Note that $A_k\equiv B_k$ for $k=0,1,2$. Now suppose we have an equivalence $\eta_k: A_k\simeq B_k$. Then by \autoref{lem:fibers} we have + $$A_{k+3}\equiv\fib_{f_{k+1}}\stackrel{e_{f_k}}\simeq \Omega A_k\stackrel{\Omega\eta_k}\simeq \Omega B_k\equiv B_{k+3}.$$ + We also show the naturality by induction on $n$.\\ + For $n\equiv0$ we have $\idfunc[Y]\o f\sim^* f\o\idfunc[X].$\\ + For $n\equiv1$ we have $\idfunc[X]\o p_1\sim^* p_1\o\idfunc[\fib_f].$\\ + For $n\equiv2$ we have + $$\idfunc[\fib_f]\o p_1\equiv p_1\sim^* (p_1 \o e_f^{-1}) \o e_f \sim^* \delta\o (\Omega\idfunc[Y] \o e_f).$$ + Now suppose the naturality holds for $k$, then we get the following diagram. + \begin{center}\begin{tikzpicture}[node distance=2cm, thick, node + distance=12mm] + \node (tl) at (0,0) {$\Omega B_{k+1}$}; + \node[right = of tl] (tr) {$\Omega B_k$}; + \node[below = of tl] (l) {$\Omega A_{k+1}$}; + \node (r) at (tr |- l) {$\Omega A_k$}; + \node[below = of l] (bl) {$A_{k+4}$}; + \node (br) at (r |- bl) {$A_{k+3}$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node {$-\Omega g_k$} (tr) + edge[<-] node {$\Omega\eta_{k+1}$} (l) + (tr) edge[<-] node {$\Omega\eta_k$} (r) + (l) edge[->] node {$-\Omega f_k$} (r) + edge[<-] node {$e_{f_{k+1}}$} (bl) + (r) edge[<-] node {$e_{f_k}$} (br) + (bl) edge[->] node {$f_{k+3}$} (br); + \end{tikzpicture}\end{center} + The bottom square can be filled by the second part of \autoref{lem:fibers}. The top square can be filled by applying the functor $\Omega$ to the naturality for $k$ and then noticing that $({-})\sy \o \Omega\eta_k\sim^* \Omega\eta_k \o ({-})\sy,$ which is easily proven for an arbitrary pointed map. +\end{proof} + +\subsubsection{Step 3} +We now remove the inverses in our sequence. More precisely, we define a second sequence $h_n:B_{n+1}\to B_n$ by +$$h_0\defeq f\qquad h_1\defeq p_1\qquad h_2\defeq \delta\qquad h_{n+3}\defeq \Omega h_n.$$ + +To show that $(B,h)$ is a type-valued exact sequence we use the following lemma. +\begin{lem}\label{lem:LESstep3} + Suppose $N$ is a successor structure and $(B,g)$ is a type-valued exact sequence over $N$. Suppose $h_n:B_{n+1}\to^* B_n$ is another sequence of maps, and suppose that there are pointed maps $e_n, \ell_n, r_n : B_n \to^* B_n$ such that $e_n$ is an equivalence and the following diagrams commute as homotopies (not necessarily pointed): + \begin{center}\begin{tikzpicture}[node distance=2cm, thick, node distance=12mm] + \node (tl) at (0,0) {$B_{n+1}$}; + \node[below = of tl] (bl) {$B_{n+1}$}; + \node[right = of bl] (br) {$B_n$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node {$h_n$} (br) + (bl) edge[->] node {$e_{n+1}$} (tl) + edge[->] node {$g_n$} (br); + \node[right = of tr] (tl) {$B_{n+1}$}; + \node[right = of tl] (tr) {$B_n$}; + \node[below = of tl] (bl) {$B_{n+1}$}; + \node (br) at (tr |- bl) {$B_n$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node {$h_n$} (tr) + edge[->] node {$\ell_{n+1}$} (bl) + (tr) edge[->] node {$e_n$} (br) + (bl) edge[->] node {$h_n$} (br); + \node[right = of tr] (tl) {$B_{n+1}$}; + \node[right = of tl] (tr) {$B_n$}; + \node[below = of tl] (bl) {$B_{n+1}$}; + \node (br) at (tr |- bl) {$B_n$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node {$h_n$} (tr) + edge[->] node {$e_{n+1}$} (bl) + (tr) edge[<-] node {$r_n$} (br) + (bl) edge[->] node {$h_n$} (br); + % \node (tl) at (0,0) {$B_{n+2}$}; + % \node[right = of tl] (t) {$B_{n+1}$}; + % \node[right = of t] (tr) {$B_n$}; + % \node[below = of tl] (bl) {$B_{n+2}$}; + % \node (b) at (t |- bl) {$B_{n+1}$}; + % \node (br) at (tr |- b) {$B_n$}; + % \path[every node/.style={font=\sffamily\small}] + % (tl) edge[->] node {$h_{n+1}$} (t) + % edge[->] node {$\ell_{n+2}$} (bl) + % (t) edge[->] node {$h_n$} (tr) + % edge[->] node {$e_{n+1}$} (b) + % (tr) edge[->] node {$r_n$} (br) + % (bl) edge[->] node {$h_{n+1}$} (b) + % (b) edge[->] node {$h_n$} (br); + \end{tikzpicture}\end{center} + Then $(B,h)$ is a type-valued exact sequence over $N$. +\end{lem} +\begin{proof} + First we need to show that for $x:B_{n+2}$ we have $h_n(h_{n+1}(x))=b_0^n$. We compute + \begin{align*} + h_n(h_{n+1}(x))&=r_n(h_n(e_{n+1}(h_{n+1}(x))))\\ + &=r_n(g_n(h_{n+1}(x)))\\ + &=r_n(g_n(g_{n+1}(e_{n+2}\sy(x))))\\ + &=r_n(b_0^n)\\ + &=b_0^n. + \end{align*} + For exactness, suppose that $y:B_{n+1}$ such that $h_n(y)=b_0^n$. Then + $g_n(e_{n+1}\sy(y))=h_n(y)=b_0^n$, therefore, by exactness of $g$ there (purely) exists + an $x:B_{n+2}$ such that $g_{n+1}(x)=e_{n+1}\sy(y)$. Now we compute + \begin{align*} + h_{n+1}(\ell_{n+2}(e_{n+2}(x)))&=e_{n+1}(h_{n+1}(e_{n+2}(x)))\\ + &=e_{n+1}(g_{n+1}(x))\\ + &=e_{n+1}(e_{n+1}\sy(y))\\ + &=y. + \end{align*} + This finishes the proof. +\end{proof} + +\begin{lem} + The sequence $(B,h)$ is a type-valued exact sequence. +\end{lem} +\begin{proof} + We first define for $k\ge2$ we the pointed equivalence + $\epsilon_n^k:B_n\simeq^* B_n$ by induction on $n$. For $n\le k$ + $\epsilon_n^k\defeq\idfunc:B_n\simeq^* B_n$ we define + $\epsilon_{n+3}^k\defeq -\Omega\epsilon_n^k:B_{n+3}\simeq^* B_{n+3}$ for $n+3>k$. + Now define $e_n\defeq\epsilon_n^3$ and $\ell_n\defeq\epsilon_n^4$ and $r_n\defeq\epsilon_n^2$. + We apply \autoref{lem:LESstep3} using these equivalences to obtain the desired result. To do this we need to check three commuting triangles. We will check $h_n\o e_{n+1}\sim g_n$, the other two proofs are similar. Apply induction on $n$. For $n=0,1,2$ it is trivial, reducing to $g_n\o\id \sim g_n$. Suppose the homotopy is true for $n=k$. Then + $$h_{k+3}\o e_{k+4}\equiv \Omega h_k \o -\Omega e_{k+1} \sim -\Omega (h_k \o e_{k+1}) \sim -\Omega g_k\equiv g_{k+3}.$$ +\end{proof} + +\subsubsection{Step 4} +We now define a type-valued chain complex over $\N\times\fin_3$, which has a successor structure by \autoref{ex:succ_structure}. Let $\rho_X$ be the equivalence $\Omega^{n+1}X \simeq^* \Omega^n(\Omega X)$. We now define the sequence $C:\N\times\fin_3$ and $k_n:C_{n+1}\to C_n$ by +\begin{align*} + C_{(n,0)}&\defeq \Omega^nY& k_{(n,0)}&\defeq \Omega^nf\\ + C_{(n,1)}&\defeq \Omega^nX& k_{(n,1)}&\defeq \Omega^np_1\\ + C_{(n,2)}&\defeq \Omega^n\fib_f & k_{(n,2)}&\defeq \Omega^n\delta \o \rho_X +\end{align*} +In a diagram, $(C,k)$ looks like the following. +\begin{center}\begin{tikzpicture}[node distance=3cm, thick, node + distance=12mm] +\node (Y) at (0,0) {$Y$}; +\node[left = of Y] (X) {$X$}; +\node[left = of X] (F) {$F$}; +\node[above = of Y] (OY) {$\Omega Y$}; +\node[above = of X] (OX) {$\Omega X$}; +\node[above = of F] (OF) {$\Omega F$}; +\node[above = of OY] (O2Y) {$\Omega^2Y$}; +\node[above = of OX] (O2X) {$\Omega^2X$}; +\node[above = of OF] (O2F) {$\Omega^2F$}; +\node[above = 5mm of O2X] (dots) {$\vdots$}; +% \node[above = 4mm of O2X] (O3X) {$\vdots$}; +\path[every node/.style={font=\sffamily\small}] +(X) edge[->] node [above] (f){$f$} (Y) +(F) edge[->] node [below] (f){$p_1$} (X) +(OY) edge[->] node [left] (f){$\delta\quad\mbox{}$} (F) +(OX) edge[->] node [above] (f){$\Omega f$} (OY) +(OF) edge[->] node [below] (f){$\Omega p_1$} (OX) +(O2Y) edge[->] node [left] (f){$\Omega\delta\quad\mbox{}$} (OF) +(O2X) edge[->] node [above] (f){$\Omega^2f$} (O2Y) +(O2F) edge[->] node [below] (f){$\Omega^2p_1$} (O2X); +\end{tikzpicture} +\end{center} +There is an equivalence $e:\N \simeq \N\times\fin_3$ that sends $n$ to its quotient and remainder when dividing $n$ by 3. The proof of the following lemma is straightforward and omitted. +\begin{lem} + The sequence $(B,h)$ is naturally equivalent to $(C,k)$ over the equivalence $e$. + Therefore, $(C,k)$ is a type-valued exact sequence. +\end{lem} + +\subsubsection{Step 5} +If we 0-truncate the sequences at step 4, we get the sequence $(D,\ell)\defeq(\|C\|_0,\|k\|_0)$. This is exactly the sequence in \autoref{thm:les-homotopy}. It is now easy to show that this is a long exact sequence. +\begin{proof}[Proof of \autoref{thm:les-homotopy}] +First note that it is a chain complex by the following computation: +$$\ell_n \o \ell_{n+1}\sim \|k_n \o k_{n+1}\|_0\sim \|0\|_0 \sim 0.$$ +To show that it is exact, suppose given $x : D_{n+1}$ and $p:\ell_n(x)=d_0^n$. We need to construct an element in a proposition, so we may assume by induction that $x\equiv |y|_0$. Now the type of $p$ reduces to $|k_n(y)|_0=|c_0^n|_0$, which is equivalent to $\|k_n(y)=c_0^n\|_{-1}$ by the characterization of the identity type in truncations. Therefore, the latter type is inhabited, and by induction, we may assume that we have a path $k_n(y)=c_0^n$. By exactness of $(C,k)$ we get an element $z:C_{n+2}$ such that $q:k_{n+1}(z)=y$. Now we can find $|z|_0:D_{n+2}$ and the path $\mapfunc{|{-}|_0}(q):\ell_{n+1}(|z|_0)=x$, showing exactness. +\end{proof} + +\subsection{Computation of homotopy groups} + +An important application of the long exact sequence of homotopy groups comes in combination with the Hopf fibration. Combining these tools, we can compute more homotopy groups of spheres. The Hopf fibration was constructed in~\cite[Theorem 8.5.1]{hottbook} and has been formalized by Ulrik Buchholtz. We will not give the construction here. + +\begin{thm}[Hopf Fibration] \label{thm:hopf} + There is a pointed map $\S^3\to\S^2$ with fiber $\S^1$. +\end{thm} + +The quaternionic Hopf fibration has also been constructed in HoTT and formalized in Lean~\cite{buchholtz2016cayleydickson}. This gives a fibration $\S^7\to \S^4$ with fiber $\S^3$. + +\begin{cor}\label{cor:homotopy-group-spheres-1} + $\pi_2(\S^2)=\Z$ and $\pi_n(\S^3)=\pi_n(\S^2)$ for $n\geq 3$. +\end{cor} +\begin{proof} + We know by the connectedness of spheres that $\pi_1(\S^3)$ and $\pi_2(\S^3)$ are trivial, and by the truncatedness of the circle that $\pi_k(\S^1)$ is trivial for $k>1$ and $\Z$ for $k=1$. We now get the following long exact sequence, from which the result immediately follows. + \begin{center}\begin{tikzpicture}[node distance=3cm, thick, node + distance=12mm] + \node (Y) at (0,0) {$0$}; + \node[left = of Y] (X) {$0$}; + \node[left = of X] (F) {$\Z$}; + \node[above = 8mm of Y] (OY) {$\pi_2(\S^2)$}; + \node (OX) at (X |- OY) {$0$}; + \node (OF) at (F |- OY) {$0$}; + \node[above = 8mm of OY] (O2Y) {$\pi_3(\S^2)$}; + \node (O2X) at (X |- O2Y) {$\pi_3(\S^3)$}; + \node (O2F) at (F |- O2Y) {$0$}; + \node[above = 8mm of O2Y] (O3Y) {$\pi_4(\S^2)$}; + \node (O3X) at (X |- O3Y) {$\pi_4(\S^3)$}; + \node (O3F) at (F |- O3Y) {$0$}; + \node[above = 5mm of O3X] (dots) {$\vdots$}; + % \node[above = 4mm of O2X] (O3X) {$\vdots$}; + \path[every node/.style={font=\sffamily\small}] + (X) edge[->] (Y) + (F) edge[->] (X) + (OY) edge[->] (F) + (OX) edge[->] (OY) + (OF) edge[->] (OX) + (O2Y) edge[->] (OF) + (O2X) edge[->] (O2Y) + (O2F) edge[->] (O2X) + (O3Y) edge[->] (O2F) + (O3X) edge[->] (O3Y) + (O3F) edge[->] (O3X); +\end{tikzpicture} +\end{center} +\end{proof} + +The last ingredient we need is the Freudenthal Suspension +Theorem. This has been formalized before by Dan Licata in Agda, and our +formalization is a direct port of that proof to Lean. For the proof we refer to~\cite[Section 8.6]{hottbook}. +\begin{thm}[Freudenthal Suspension Theorem] \label{thm:freudenthal} + Suppose that $X$ is $n$-connected. Then $\|X\|_{2n}\simeq \|\Omega\Sigma X\|_{2n}$. +\end{thm} + +We can combine these results to compute the following homotopy groups. +\begin{cor}\label{cor:homotopy-group-spheres-2} + $\pi_n(\S^n)=\Z$ and $\pi_3(\S^2)=\Z$ +\end{cor} +\begin{proof} + Note that $\S^n$ is $(n-1)$-connected. Therefore, by the Freudenthal suspension theorem we have + $$\|\S^n\|_{2(n-1)}\simeq \|\Omega\S^{n+1}\|_{2(n-1)}.$$ + For $n\geq 2$ we have $2(n-1)\geq n$, and therefore we also have + $$\|\S^n\|_{n}\simeq \|\Omega\S^{n+1}\|_{n}.$$ + Taking the $n$-th homotopy group, we get + $$\pi_n(\S^n)\simeq \pi_{n+1}(\S^{n+1}).$$ + Combining this with \autoref{cor:homotopy-group-spheres-1}, we also get $\pi_3(\S^2)\simeq\Z$, as desired. +\end{proof} + +\section{Eilenberg-MacLane Spaces}\label{sec:eilenb-macl-spac} + +In this section we give an important equivalence between groups and Eilenberg-MacLane spaces~\cite{eilenberg1945spaces}.\footnote{Some of the contents of this section have been published in~\cite{buchholtz2018groups}. +The work in this section is joint work with Ulrik Buchholtz and Egbert Rijke.} Eilenberg-MacLane space are play an important role in homotopy theory, since they are spaces with simple homotopy groups. Therefore, they can be used to build up more complicated spaces with complicated homotopy groups. Also, they can be used to define homology and cohomology in HoTT, see Sections \ref{sec:spectral-sequence-cohomology} and \ref{sec:spectral-sequence-homology}. + +We prove in this section that the category of $n$-connected $(n+1)$-truncated pointed types is equivalent to the +category of groups for $n = 0$ and the category of abelian groups for $n \geq 1$. + +If $G$ is a (pre-)groupoid, the groupoid quotient is a higher inductive type with constructors +\begin{inductive} +\texttt{HIT} $\groupoidquotient(G) :=$ \\ +$\bullet\ i : G_0 \to \groupoidquotient(G)$; \\ +$\bullet\ p : (x\ y : G_0) \to \homm(x,y)\to x=y$; \\ +$\bullet\ q : (x\ y\ z : G_0) \to (g : \homm(y,z)) \to (f : \homm(x,y)) \to +p(g \circ f) = p(f) \cdot p(g)$; \\ +$\bullet\ \eps : \istrunc{1}(\groupoidquotient(G))$. +\end{inductive} +%As with the $n$-truncation, the fact that the groupoid quotient is 1-truncated can be encoded using +%hubs and spokes (or we could first define the HIT without truncation, and then truncate afterwards). + +The groupoid quotient can be constructed purely from homotopy pushouts. +The untruncated version was constructed in \autoref{sec:non-recursive-2}. +Then we can apply the 1-truncated afterwards, and we can also construct truncations from homotopy pushouts~\cite{rijke2017join}. + +In~\cite{licata2014em} the authors define Eilenberg-MacLane spaces. We use the same approach as in +that paper. We first quickly review the results in that paper. + +\subsection{Construction of Eilenberg-MacLane spaces} + +If $G$ is any group, the 1-dimensional Eilenberg-MacLane space $K(G,1)$ can be defined by viewing +$G$ as a groupoid, and taking the groupoid quotient of $G$. It is not hard to see that $K(G,1)$ is +0-connected and 1-truncated. Using an encode-decode proof, we can show that $\Omega K(G,1) \simeq G$ +and that this equivalence sends concatenation to multiplication. Hence the composite +$\pi_1K(G,1)\simeq \|G\|_0\simeq G$ is a group isomorphism. + +If $G$ is abelian, the higher Eilenberg-MacLane spaces can be defined recursively +as $$K(G,n+1):\equiv \|\Sigma K(G,n)\|_{n+1}$$ for $n\geq 1$. This definition is slightly different +than the one given in~\cite{licata2014em}, where $K(G,n+1)$ was defined using the iterated +suspension as $\|\Sigma^n K(G,1)\|_{n+1}$. We chose to modify the definition, since a lot of +properties of Eilenberg-MacLane spaces are proven by induction on $n$, so it is more convenient to +have $K(G,n+1)$ defined directly in terms of $K(G,n)$. + +It is easy to show that $K(G,n)$ is $(n-1)$-connected and $n$-truncated. It is trickier to show that +$\Omega K(G,n+1)\simeq K(G,n)$. This is done separately for $n=1$ and for $n\geq 2$. + +For $n=1$ we need the result that for every type $X$ with a coherent h-structure, the type $\|\Sigma +X\|_2$ is a \emph{delooping} of $X$, which means that $\Omega \|\Sigma X\|_2 \simeq X$. If $G$ is abelian, then $K(G,1)$ can be equipped with a coherent h-structure, +showing that $\Omega K(G,2)\simeq K(G,1)$. + +For $n\geq 2$, this can be done using the Freudenthal suspension theorem, +\cref{thm:freudenthal}. Then the equivalence follows from the following chain of equivalences: +$$\Omega K(G,n+1)\equiv \Omega\|\Sigma K(G,n)\|_{n+1}\simeq \|\Omega\Sigma K(G,n)\|_n\simeq +\|K(G,n)\|_n\simeq K(G,n).$$ The Freudenthal Suspension Theorem is applied in the third step, which +is allowed since $K(G,n)$ is $(n-1)$-connected and $n \leq 2(n-1)$ for $n\geq2$. + +This finishes the proof sketch that $\Kloop(G,n):\Omega K(G,n+1)\simeq K(G,n)$. By induction, +$\Omega^n K(G,n+1)\simeq K(G,1)$, hence we get the following group isomorphism +$\pi_{n+1}(K(G,n+1))\simeq \pi_1(K(G,1))\simeq G$. + +\subsection{Uniqueness} + +In this section we prove that Eilenberg-MacLane spaces are unique, which means that if $X$ and $Y$ +are both $(n-1)$-connected, $n$-truncated pointed types such that $\pi_n(X)\simeq \pi_n(Y)$, then +$X\simeq Y$. Note that from these assumptions one can show that $\pi_k(X)\simeq 1\simeq \pi_k(Y)$ +for $k < n$ since $X$ and $Y$ are $(n-1)$-connected, but also for $k > n$ since $X$ and $Y$ are +$n$-truncated. Hence from the assumptions we actually have that $\pi_k(X)\simeq\pi_k(Y)$ for all +natural numbers $k$. + +This is similar to Whitehead's Theorem, which states that if $f : X \to Y$ is a pointed map +that induces an equivalence on all homotopy groups, then $f$ is an equivalence. Whitehead's Theorem +is not true in general, but it is true under the assumption that both $X$ and $Y$ are $n$-truncated +for some $n$. For the special case that $X$ and $Y$ are both $(n-1)$-connected and $n$-truncated one +does not need to find a map between $X$ and $Y$ to show that they are equivalent, as long as they +have isomorphic homotopy groups. + +We first give an elimination principle for $K(G,n)$. +\begin{defn}\label{def:Kelim} +Suppose that $X$ is an $n$-truncated pointed type, and suppose that for some group +$G$ there is an map $\phi : G \to \Omega^n X$ that sends multiplication to +concatenation. Then there is a pointed map $\Kelim(\phi, n) : K(G,n)\to X$. +\end{defn} +\begin{proof}[Construction] + We construct this by induction on $n$. + For $n=1$ this follows directly from the induction principle of $K(G,1)$. For $n=k+1>1$ we can define the group homomorphism $\widetilde\phi$ as the composite $G \xrightarrow{\phi} \Omega^{k+1} X \simeq \Omega^k(\Omega X)$, and apply the induction hypothesis to get a map + $\Kelim(\widetilde\phi, k):K(G,k)\to^* \Omega X$. By the adjunction $\Sigma\dashv\Omega$ we get a pointed map $\Sigma K(G,k)\to^* X$, and by the elimination principle of the truncation we get a map + $K(G,k+1)\equiv\|\Sigma K(G,k)\|_{k+1}\to^* X$. +\end{proof} + +\begin{lem}\label{lem:Kelim1} +There is a pointed homotopy making the following diagram commute. +\begin{center} + \begin{tikzpicture}[node distance=3cm, + thick,main node/.style={font=\sffamily\bfseries}] + \node[main node] (Kn) at (0,0) {$K(G,n)$}; + \node[main node] (OK) at (4,0) {$\Omega K(G,n+1)$}; + \node[main node] (OX) at (2,-2) {$\Omega X$}; + \path[every node/.style={font=\sffamily\small}] + (Kn) edge [->] node {$\sim$} (OK) + edge [->] node [below left] {$\Kelim(\widetilde\phi,n)$} (OX) + (OK) edge [->] node [below right] {$\Omega(\Kelim(\phi,n+1))$} (OX); + \end{tikzpicture} +\end{center} +\end{lem} +\begin{proof} + This follows by unwinding the definition of the function $\Kelim(\phi,n+1)$ in terms of $\Kelim(\phi,n)$. +\end{proof} + +\begin{lem}\label{lem:Kelim2} +The following diagram commutes. %TODO: this can be done easier in formalization. +\begin{center} + \begin{tikzpicture}[node distance=3cm, + thick,main node/.style={font=\sffamily\bfseries}] + \node[main node] (OK) at (0,0) {$\Omega^nK(G,n)$}; + \node[main node] (G) at (4,0) {$G$}; + \node[main node] (OX) at (2,-2) {$\Omega^n X$}; + \path[every node/.style={font=\sffamily\small}] + (OK) edge [->] node {$\sim$} (G) + edge [->] node [below left] {$\Omega^n(\Kelim(\phi,n))$} (OX) + (OX) edge [<-] node [below right] {$\phi$} (G); + \end{tikzpicture} +\end{center} +\end{lem} +\begin{proof} + This follows by repeatedly applying \autoref{lem:Kelim1}. +\end{proof} + +\begin{thm}\label{thm:em-unique} +Suppose that $X$ is an $(n-1)$-connected $n$-truncated pointed type, and suppose that for some group +$G$ there is an equivalence $\phi : G \simeq \Omega^n X$ that sends multiplication to +concatenation. Then the map $\Kelim(\phi, n) : K(G,n)\to X$ is an equivalence. +In particular this means that if $X$ is an $(n-1)$-connected $n$-truncated pointed type, and there is a group isomorphism $e : \pi_n(X) \simeq G$, then $X\simeq^* K(G,n)$. +\end{thm} +\begin{proof} + We apply Whitehead's principle for truncated types. This states that a \emph{weak equivalence} (a map inducing an isomorphism on all homotopy groups) between truncated types is an equivalence. The proof can be found in~\cite[Theorem 8.8.3]{hottbook}. Since both $K(G,n)$ and $X$ are $(n-1)$-connected and $n$-truncated, the map $\Kelim(\phi,n)$ trivially induces an isomorphism on all homotopy groups for all levels other than $n$. It also induces an isomorphism on level $n$ by \autoref{lem:Kelim2}. This finishes the proof. +\end{proof} + +\begin{cor} +The type of $(n-1)$-connected, $n$-truncated pointed types is equivalent to the type of groups for +$n=1$ and equivalent to the type of abelian groups for $n \geq 2$. +\end{cor} +\begin{proof} + The maps back and forth are $K({-},n)$ and $\pi_n$. The composites are homotopic to the identity map, since $\pi_n(K(G,n))\simeq G$ and $K(\pi_n(X),n)\simeq^* X$ (the last equivalence comes from \autoref{thm:em-unique}). +\end{proof} + +\subsection{Equivalence of categories} +\begin{defn} + If $\phi : G \to H$ is a homomorphism between groups, then there is a pointed map $K(\phi, n) : + K(G, n) \to K(H, n)$. This action is functorial, i.e. it respects composition and identity maps. +\end{defn} +\begin{proof}[Construction] + The functorial action comes from \autoref{def:Kelim}. We omit the proof of the other properties. +\end{proof} + +To show that we get the desired equivalence of categories, we need to fill the following naturality squares. We will omit the proofs here. + +\begin{center} + \begin{tikzpicture}[node distance=3cm, + thick,main node/.style={font=\sffamily\bfseries}] + \node[main node] (KG) at (0,0) {$\pi_n(K(G,n))$}; + \node[main node] (KH) at (4,0) {$\pi_n(K(H,n))$}; + \node[main node] (G) at (0,-4) {$G$}; + \node[main node] (H) at (4,-4) {$H$}; + \path[every node/.style={font=\sffamily\small}] + (KG) edge [->] node {$\pi_n(K(\phi,n))$} (KH) + (G) edge [->] node [left] {$\sim$} (KG) + edge [->] node {$\phi$} (H) + (H) edge [->] node [right] {$\sim$} (KH); + \end{tikzpicture} +\qquad + \begin{tikzpicture}[node distance=3cm, + thick,main node/.style={font=\sffamily\bfseries}] + \node[main node] (X) at (0,0) {$X$}; + \node[main node] (Y) at (4,0) {$Y$}; + \node[main node] (KX) at (0,-4) {$K(\pi_n(X),n)$}; + \node[main node] (KY) at (4,-4) {$K(\pi_n(Y),n)$}; + \path[every node/.style={font=\sffamily\small}] + (X) edge [->] node {$f$} (Y) + (KX) edge [->] node [left] {$\sim$} (X) + edge [->] node {$K(\pi_n(f),n)$} (KY) + (KY) edge [->] node [right] {$\sim$} (Y); + \end{tikzpicture} +\end{center} +These diagrams show the following result. +\begin{thm}\label{thm:EM-equiv-categories} + $K({-},n)$ is an equivalence from the category of $(n-1)$-connected $n$-truncated pointed types to + the category of groups (for $n=1$) or abelian groups (for $n\geq2$). +\end{thm} +\begin{rmk} + In particular this shows that the type of pointed maps between two $(n-1)$-connected $n$-truncated types is a set. + This is a special case of the more general fact that the type of pointed maps from an $n$-connected type to a $(n+k+1)$-truncated type is $k$-truncated (for $n\geq-1$). +\end{rmk} +\begin{rmk} + It would be interesting, but a lot more work, to do this one level up. In that case, it should be possible to show that crossed modules or 2-groups correspond to pointed connected 2-types. Furthermore, pointed $(n-2)$ connected $n$-types should correspond to braided 2-groups for $n=3$ and to symmetric 2-groups for $n\geq 4$. A start of this project was given in~\cite{raumer2016doublegroupoids}. +\end{rmk} +\section{The Smash Product}\label{sec:smash-product} + +In this section we will discuss the smash product and its +properties.\footnote{The work in this section is joint work with Stefano +Piceghello. Parts of this section are based on ideas from Robin Adams, Marc +Bezem, Ulrik Buchholtz and Egbert Rijke.} The smash product has many uses in +homotopy theory. It can be used to define generalized homology theory (see +\autoref{sec:spectral-sequence-homology}) and it is used to define the cup +product for cohomology~\cite[Section 5.1]{brunerie2016spheres}. +%, and is used to compute $\pi_4(\S^3)$. + +The goal is to prove that the smash product defines a \emph{1-coherent symmetric +monoidal product on pointed types}~\cite[Definition 4.1.1]{brunerie2016spheres}, +which we repeat in \autoref{def:symmonprod}. Our proof strategy is to show that +the smash product is left adjoint to pointed maps and then use a Yoneda-style +argument to show that we get a 1-coherent symmetric monoidal product. + +This proof is known in 1-category theory~\cite[Chapter 2, Theorem +5.3]{eilenberg1966closedcategories}. Suppose given a closed category\footnote{A +\emph{closed category} is a category with internal hom-objects. We can view pointed +types as a higher closed category, where the internal hom-object is the type of pointed +maps, pointed by the constant map.} $\mc C$ with internal hom $[{-},{-}]:\mc C\op\times \mc C\to \mc C$. +Moreover suppose that for every $A,B:\mc C$ the functor $[A,[B,{-}]]:\mc C\to \mc C$ is +representable as a $\mc C$-enriched functor. This means that there is an object +$A\otimes B:\mc C$ and a $\mc C$-enriched natural transformation $[A\otimes +B,C]\cong [A,[B,C]]$. Then $\mc C$ is a monoidal closed category. We will spell +out the precise formulation for pointed types in \autoref{def:naturality}, where we will call +$\type^*$-enriched functors \emph{pointed functors} and $\type^*$-enriched +natural transformations \emph{pointed natural transformations}. + +In this section we will prove two main claims. +\begin{itemize} + \item We prove that $A\smsh B$ represents the functor $A\to^* B \to^* ({-})$ on +pointed types. In other words, that we have a natural equivalence + $$(A\smsh B \to^* C)\simeq^* (A\to^* B \to^* C).$$ +\item We prove that if we have a \emph{pointed} natural equivalence +$$(A\smsh B \to^* C)\simeq^* (A\to^* B \to^* C),$$ +then the smash product forms a 1-coherent symmetric +monoidal product on pointed types. +\end{itemize} +There is still a gap in this argument: we still need to show that the natural +equivalence above is a pointed natural equivalence. We did not manage to do +this, because of the high level of the path algebra involved, but we do not +expect theoretical difficulties. + +In this section, all types, maps, homotopies and equivalences are pointed, +unless mentioned otherwise. We will denote pointed homotopies using equalities +in diagrams. We will start with defining some categorical properties of pointed +types. We will use the notation established in \autoref{sec:pointed}. + +\subsection{The Category of Pointed Types}\label{sec:pointed-category} + +\begin{defn}\label{def:naturality} + Suppose we are given $F:\type^*\to\type^*$. We say that $F$ is a \emph{1-coherent functor} if + \begin{itemize} + \item $F$ acts on pointed maps: given $f : A \to A'$, there is a pointed map $Ff : F(A) \to F(A');$ + \item it respects identities: $F(\id_A)\sim \id_{FA};$ + \item it respects composition: $F(f' \o f) \sim Ff' \o Ff.$ + \end{itemize} + We will call a 1-coherent functor a \emph{functor} for short.\footnote{While + this is an abuse of terminology, it will not cause confusion in practice. Note that internally in the language of HoTT + it is an open problem whether we can even formulate the type of fully coherent functors.} + We say that a functor $F$ is a \emph{pointed functor} if moreover + $F\unit=\unit$, where $\unit$ is the unit type (which is the zero object in + pointed types). In this case we can show that + $F(\const_{A,B})=\const_{FA,FB}$, where $\const_{A,B}$ is the constant map. + + Let $F$, $G$ be functors of pointed types and suppose that $\theta$ is a + family of pointed maps $(X : \type^*) \to F(X) \to G(X)$. We say that $\theta$ + is a (1-coherent) \emph{natural transformation} or \emph{natural} if for every $f : A \to + B$ there is a diagram: + \begin{center} + \begin{tikzcd} + F(A) + \arrow[r, "F(f)"] + \arrow[d, swap, "\theta_A"] + & F(B) + \arrow[d, "\theta_B"] + \\ + G(A) + \arrow[r, swap, "G(f)"] + & G(B) + \end{tikzcd} + \end{center} + That is, a pointed homotopy + \[p_\theta(f) : \theta_B \o F(f) \sim G(f) \o \theta_A.\] + + We say that $\theta$ is \emph{pointed natural} if $\theta$ is natural and $p_\theta(\const) = (p_\theta)_0$, where + \[(p_\theta)_0 : G(\const) \o \theta_A \sim \const \o \theta_A \sim \const \sim \theta_B \o \const \sim \theta_B \o F(\const)\] + is the canonical proof of the pointed homotopy $G(\const) \o \theta_A \sim \theta_B \o F(\const)$. + + For $n$-ary functions $F:\type^*\to\cdots\to\type^*$ we define functoriality + similarly. We say that transformations between $n$-ary functors are natural if + they are natural in all arguments. +\end{defn} + +\begin{rmk} + We could define a notion of \emph{weak naturality}, which is like naturality, + but where the homotopy is not required to be pointed. However, this is + generally ill-behaved. For example, if $\theta$ is weakly natural, neither + $X\to\theta$ nor $\theta\to X$ needs to be weakly natural. +\end{rmk} + +\begin{defn}\label{def:symmonprod} + A 1-coherent symmetric monoidal product for pointed types is a binary operation $\otimes:\type^*\to\type^*\to\type^*$ that is functorial. Explicitly, this means that + \begin{itemize} + \item Given $f : A \to A'$ and $g : B \to B'$, there is a map $f\otimes g : A \otimes B \to A' \otimes B.$ + \item It respects identities: $\id_A\otimes\id_B\sim \id_{A\otimes B}.$ + \item It respects composition: $(f' \o f) \otimes (g' \o g) \sim (f' \otimes g') \o (f \otimes g).$ + \end{itemize} + Furthermore, there is a pointed type $I$ and natural equivalences + \begin{itemize} + \item $\alpha : (A \otimes B) \otimes C \simeq A \otimes (B \otimes C)$ (associativity of the smash product); + \item $\lambda : I \otimes B \simeq B$ (left unitor for the smash product); + \item $\gamma : A \otimes B \simeq B \otimes A$ (braiding for the smash product). + \end{itemize} + With pointed homotopies filling the following three diagrams. + %associativity pentagon; unitors triangle; braiding-unitors triangle; associativity-braiding hexagon; double braiding + %\[\alpha \o \alpha \sim (A \otimes \alpha) \o \alpha \o (\alpha \otimes D)\] + \begin{center} + \begin{tikzcd} + &((A \otimes B) \otimes (C \otimes D)) + \arrow[dr, "\alpha"] + \\ + (((A \otimes B) \otimes C) \otimes D) + \arrow[ru, "\alpha"] + \arrow[d, swap, "\alpha \otimes D"] + && (A \otimes (B \otimes (C \otimes D))) + \\ + ((A \otimes (B \otimes C)) \otimes D) + \arrow[rr, swap, "\alpha"] + && (A \otimes ((B \otimes C) \otimes D)) + \arrow[u, swap, "A \otimes \alpha"] + \end{tikzcd} + \end{center} + %\[(A \otimes \lambda) \o \alpha \sim (\rho \otimes B)\] % this is equivalent to below diagram, see picture on 2018-4-18 + \begin{center} + \begin{tikzcd} + ((I \otimes A) \otimes B) + \arrow[rr, "\alpha"] + \arrow[dr, swap, "\lambda \otimes B"] + && (I \otimes (A \otimes B)) + \arrow[dl, "\lambda"] + \\ + & (A \otimes B) + \end{tikzcd} + \end{center} + + %\[\lambda \o \gamma \sim \rho\] + % \begin{center} + % \begin{tikzcd} + % (A \otimes I) + % \arrow[rr, "\gamma"] + % \arrow[dr, swap, "\rho"] + % && (I \otimes A) + % \arrow[dl, "\lambda"] + % \\ + % & A + % \end{tikzcd} + % \end{center} + + %\[\alpha \o \gamma \o \alpha \sim (B \otimes \gamma) \o \alpha \o (\gamma \otimes C)\] + \begin{center} + \begin{tikzcd} + ((A \otimes B) \otimes C) + \arrow[r, "\alpha"] + \arrow[d, swap, "\gamma \otimes C"] + &(A \otimes (B \otimes C)) + \arrow[r, "\gamma"] + & ((B \otimes C) \otimes A) + \arrow[d, "\alpha"] + \\ + ((B \otimes A) \otimes C)) + \arrow[r, swap, "\alpha"] + & (B \otimes (A \otimes C)) + \arrow[r, swap, "B \otimes \gamma"] + & (B \otimes (C \otimes A)) + \end{tikzcd} + \end{center} +\end{defn} + +We have a version of the Yoneda Lemma for pointed types. +\begin{lem}[Yoneda]\label{lem:yoneda} + Let $A$, $B$ be pointed types, and assume, for all pointed types $X$, a pointed equivalence $\phi_X : (B \to X) \simeq (A \to X)$, natural in $X$, i.e. for all $f : X \to X'$ there is a homotopy \[ p_\phi(f) : (A \to f) \o \phi_X \sim \phi_X' \o (B \to f) \] +% making the following diagram commute for all $f : X \to X'$: +% \begin{center} +% \begin{tikzcd} +% (B \to X) +% \arrow[r, "\phi_X"] +% \arrow[d, swap, "f \o -"] +% & (A \to X) +% \arrow[d, "f \o -"] +% \\ +% (B \to X') +% \arrow[r, swap,"\phi_{X'}"] +% & (A \to X') +% \end{tikzcd} +% \end{center} + Then there exists a pointed equivalence $\psi_\phi : A \simeq B$. +\end{lem} +\begin{proof} + We define $\psi_\phi \defeq \phi_B(\idfunc[B]) : A \to B$ and $\psi_\phi\sy \defeq \phi_A\sy(\idfunc[A])$. The given naturality square for $X \defeq B$ and $g \defeq \psi_\phi\sy$ yields $\psi_\phi\sy \o \phi_B (\idfunc[B]) \judgeq \psi_\phi\sy \o \psi_\phi \sim \phi_A (\psi_\phi\sy \o \idfunc[B]) \judgeq \phi_A (\phi_A\sy (\idfunc[A])) \sim \idfunc[A]$, and similarly for the inverse composition. +\end{proof} + +\begin{lem}\label{lem:yoneda-pointed} + Assume $A$, $B$, $\phi_X$ and $p$ as in \autoref{lem:yoneda}, and assume moreover that $\phi_X$ is pointed natural. Then there is a pointed homotopy $(\psi_\phi \to X) \sim \phi_X$. +\end{lem} + +\begin{proof} + Let $f : B \to X$. The underlying homotopy is obtained by: + \begin{align*} + (\psi_\phi \to X)(f) &\judgeq f \o \psi_\phi\\ + &\sim \phi_X (f \o \idfunc) &&\text{(by $p_\phi(f)(\idfunc)$)}\\ + &\sim \phi_X (f) &&\text{(by $\mapfunc{\phi_X}(\oneh_f)$)} + \end{align*} + To show that this is a pointed homotopy, we need to prove that the following diagram commutes: + \begin{center} + \begin{tikzcd}[column sep=4em] + (\psi_\phi \to X)(\const) + \arrow[rr, equals, "p_\phi(\const)(\idfunc)\tr\mapfunc{\phi_X}(\oneh_\const)"] + \arrow[dr, equals, swap, "\zeroh_{\psi_\phi}"] + &&\phi_X(\const) + \arrow[dl, equals, "(\phi_X)_0"] + \\ + &\const + \end{tikzcd} + \end{center} + where the top-left expression is definitionally equal to $\const \o \phi_X(\idfunc)$, the horizontal path comes from the underlying homotopy and $(\phi_X)_0$ is the canonical path from $\phi_X(\const)$ to $\const$. Since $\phi_X$ is pointed natural, we have that + $p_{\phi_X}(\const)(\idfunc) = (p_{\phi_X})_0(\idfunc)$, which is the concatenation: + \begin{align*} + \const\o \phi_X(\idfunc) + &= \const &&\text{(by $\zeroh_{q_X(\idfunc)}$)}\\ + &= \phi_X(\const) &&\text{(by $(\phi_X)_0\sy$)}\\ + &= \phi_X(\const\o 1) &&\text{(by $(\mapfunc{\phi_X}(\zeroh_{\idfunc}))\sy$)} + \end{align*} + The diagram then commutes by cancellation of inverses and using that $\zeroh_{\idfunc} = \oneh_\const$. +\end{proof} + +\subsection{Basic Properties of the Smash Product}\label{sec:smash-basic} + +\begin{defn} + The smash of $A$ and $B$ is the HIT generated by the point constructor $(a,b)$ for $a:A$ and $b:B$ + and two auxiliary points $\auxl,\auxr:A\smsh B$ and path constructors $\gluel_a:(a,b_0)=\auxl$ + and $\gluer_b:(a_0,b)=\auxr$ (for $a:A$ and $b:B$). $A\smsh B$ is pointed with point $(a_0,b_0)$. +\end{defn} +\begin{rmk} + This definition of $A\smsh B$ is basically the pushout of + $\bool\leftarrow A+B\to A \times B$. A more traditional definition of $A\smsh B$ is the pushout + $\unit\leftarrow A\vee B\to A \times B$; here $\vee$ denotes the wedge product, which can be + equivalently described as either the pushout $A\leftarrow \unit\to B$ or + $\unit\leftarrow \bool\to A + B$. These two definitions of $A\smsh B$ are equivalent, because in + the following diagram the top-left square and the top rectangle are pushout squares, hence the + top-right square is a pushout square by applying the pushout lemma. Another application of the + pushout lemma then states that the two definitions of $A\smsh B$ are equivalent. +\begin{center} +\begin{tikzcd} +\bool \arrow[r]\arrow[d] & A+B \arrow[r]\arrow[d] & \bool \arrow[d] \\ +\unit \arrow[r] & A\vee B \arrow[r]\arrow[d] & \unit \arrow[d] \\ + & A\times B \arrow[r] & A\smsh B +\end{tikzcd} +\end{center} + +\end{rmk} +\begin{lem}\label{lem:smash-general} + The smash product is functorial: if $f:A\pmap A'$ and $g:B\pmap B'$, then + $f\smsh g:A\smsh B\pmap A'\smsh B'$. We write $A\smsh g$ or $f\smsh B$ if one of the + functions is the identity function. Moreover, if $p:f\sim f'$ and $q:g\sim g'$, then $p\smsh q:f\smsh g\sim f'\smsh g'$; this operation preserves reflexivities, symmetries and transitivies. We will write $p \smsh g$ or $f \smsh q$ if one of the homotopies is reflexivity. +% The smash product satisfies the following properties. +% \begin{itemize} +% \item The smash product is functorial: if $f:A\pmap A'$ and $g:B\pmap B'$ then +% $f\smsh g:A\smsh B\pmap A'\smsh B'$. We write $A\smsh g$ or $f\smsh B$ if one of the +% functions is the identity function. +% \item The smash product preserves composition, which gives rise to the interchange law: +% \[i:(f' \o f)\smsh (g' \o g) \sim f' \smsh g' \o f \smsh g\] +% \item If $p:f\sim f'$ and $q:g\sim g'$ then $p\smsh q:f\smsh g\sim f'\smsh g'$. This operation +% preserves reflexivities, symmetries and transitivies. +% \item There are homotopies $f\smsh0\sim0$ and $0\smsh g\sim 0$ such that the following diagrams +% commute for given homotopies $p : f\sim f'$ and $q : g\sim g'$. +% \begin{center} +%\begin{tikzcd} +%f\smsh 0 \arrow[rr, equals,"p\smsh1"]\arrow[dr,equals] & & +%f'\smsh 0\arrow[dl,equals] \\ +%& 0 & +%\end{tikzcd} +%\qquad +%\begin{tikzcd} +%0\smsh g\arrow[rr, equals,"1\smsh q"]\arrow[dr,equals] & & +%0\smsh g'\arrow[dl,equals] \\ +%& 0 & +%\end{tikzcd} +%\end{center} +%\end{itemize} +\end{lem} + +\begin{lem}\label{lem:interchange} + The smash product preserves composition, which gives rise to the interchange law: + \[i:(f_2 \o f_1)\smsh (g_2 \o g_1) \sim f_2 \smsh g_2 \o f_1 \smsh g_1\] + for maps $A_1\lpmap{f_1}A_2\lpmap{f_2}A_3$ and $B_1\lpmap{g_1}B_2\lpmap{g_2}B_3$. +\end{lem} +\begin{proof} + Let us denote the basepoints of $A_i$ and $B_i$ with $a_i$ and $b_i$ respectively. We first apply induction on the paths that all the maps in the statement respect the basepoint. We verify the underlying homotopy of $i$ by induction on terms $x$ of the domain $A_1 \smsh B_1$ of the two maps; this can be defined on point constructors $(a,b)$, $\auxl$ and $\auxr$ to be the identity path. If $x$ varies over $\gluel_a$, we need to fill the following square: + \begin{equation}\label{eq:i-gluel} + \begin{tikzcd} + (f_2(f_1(a)), b_3) + \arrow[r,equals,"1"] + \arrow[d,swap,equals,"\mapfunc{(f_2 \o f_1)\smsh (g_2 \o g_1)}(\gluel_a)"] + & (f_2(f_1(a)), b_3) + \arrow[d,equals,"\mapfunc{f_2 \smsh g_2 \o f_1 \smsh g_1}(\gluel_a)"] + \\ + \auxl + \arrow[r,swap,equals,"1"] + &\auxl + \end{tikzcd} + \end{equation} + This reduces to proving that + \[\mapfunc{(f_2(f_1(a)),-)}(g_2\o g_1)_0 \tr \gluel_{f_2(f_1(a))} = \mapfunc{(f_2(f_1(a)),-)}(\mapfunc{g_2}{(g_1)}_0 \tr {(g_2)}_0) \tr \gluel_{f_2(f_1(a))}\] + Since we assumed that ${(g_1)}_0$ and ${(g_2)}_0$ are the identity path, the claim is easily verified. The case for $x$ varying over $\gluer_b$ is entirely analogous, giving the square: + \begin{equation}\label{eq:i-gluer} + \begin{tikzcd} + (a_3, g_2(g_1(b)) + \arrow[r,equals,"1"] + \arrow[d,swap,equals,"\mapfunc{(f_2 \o f_1)\smsh (g_2 \o g_1)}(\gluer_b)"] + & (a_3, g_2(g_1(b)) + \arrow[d,equals,"\mapfunc{f_2 \smsh g_2 \o f_1 \smsh g_1}(\gluer_b)"] + \\ + \auxr + \arrow[r,swap,equals,"1"] + &\auxr + \end{tikzcd} + \end{equation} + The resulting homotopy is pointed, as $i(a_1,b_1) \judgeq 1$ and the proofs that the two maps respect the basepoint are assumed to be the identity path. +\end{proof} + +\begin{lem}\label{lem:smash-zero} + There are homotopies + \begin{align*} + t_g : \const\smsh g\sim \const && t'_f : f\smsh\const\sim\const + \end{align*} + such that the following diagrams + commute for given homotopies $p : g\sim g'$ and $q : f\sim f'$. + \begin{equation}\label{eq:t-triangles} + \begin{tikzcd} + \const\smsh g + \arrow[rr, equals,"1\smsh p"] + \arrow[dr,equals,swap,"t_g"] + && \const\smsh g'\arrow[dl,equals,"t_{g'}"] + &f\smsh \const + \arrow[rr, equals,"q\smsh 1"] + \arrow[dr,equals,swap, "t'_f"] + && f'\smsh \const\arrow[dl,equals,"t'_{f'}"] + \\ + & \const + &&& \const + \end{tikzcd} +% \qquad\qquad +% \begin{tikzcd} +% f\smsh 0 +% \arrow[rr, equals,"q\smsh 1"] +% \arrow[dr,equals,swap, "t'_f"] +% && f'\smsh 0\arrow[dl,equals,"t'_{f'}"] +% \\ +% & 0 +% \end{tikzcd} + \end{equation} +\end{lem} +\begin{proof} + We will define the homotopy $t_g : \const \smsh g$, with $\const : A_1 \to A_2$ and $g : B_1 \to B_2$ (with the notational convention for the basepoints as in \autoref{lem:interchange}); the definition for $t'_f$ is analogous. First, we apply induction on the path that $g$ respects the basepoint. The underlying homotopy of $t_g$ is given by induction on terms $x : A_1 \smsh B_1$. On point constructors, we define: + \begin{align*} + t_g (a,b) &\defeq \gluer_{g(b)} \tr \gluer_{b_2}\sy && : (a_2, g(b)) = (a_2, b_2)\\ + t_g (\auxl) &\defeq \gluel_{a_2}\sy && : \auxl = (a_2, b_2)\\ + t_g (\auxr) &\defeq \gluer_{b_2}\sy && : \auxr = (a_2, b_2) + \end{align*} + If $x$ varies over $\gluel_a$, after some reductions, we need to fill the following square: + \begin{equation}\label{eq:t-gluel} + \begin{tikzcd}[column sep=7em] + (a_2, g(b_1)) + \arrow[r,equals,"\gluer_{b_2} \tr \gluer_{b_2}\sy"] + \arrow[d,swap,equals, "\gluel_{a_2}"] + & (a_2, b_2) + \arrow[d,equals,"1"] + \\ + \auxl + \arrow[r,swap,equals, "\gluel_{a_2}\sy"] + & (a_2, b_2) + \end{tikzcd} + \end{equation} + Similarly, if $x$ varies over $\gluer_b$, we need to fill the following square: + \begin{equation}\label{eq:t-gluer} + \begin{tikzcd}[column sep=7em] + (a_2, g(b)) + \arrow[r,equals,"\gluer_{g(b)} \tr \gluer_{b_2}\sy"] + \arrow[d,swap,equals, "\gluer_{g(b)}"] + & (a_2, b_2) + \arrow[d,equals,"1"] + \\ + \auxr + \arrow[r,swap,equals, "\gluer_{b_2}\sy"] + & (a_2, b_2) + \end{tikzcd} + \end{equation} + The squares in (\ref{eq:t-gluel}) and (\ref{eq:t-gluer}) can both be filled by simple path algebra. The resulting homotopy is pointed, as $t_g(a_1,b_1)$ is equal to the identity path and the proof that $g$ respects the basepoint is also assumed to be the identity path. Finally, for $p : g \sim g'$, the diagram on the left in (\ref{eq:t-triangles}) commutes by induction on $p$. +\end{proof} + +\begin{lem}\label{lem:smash-coh} + Suppose that we have maps $A_1\lpmap{f_1}A_2\lpmap{f_2}A_3$ and $B_1\lpmap{g_1}B_2\lpmap{g_2}B_3$ + and suppose that either $f_1$ or $f_2$ is constant. Then there are two homotopies + $(f_2 \o f_1)\smsh (g_2 \o g_1)\sim \const$, one of which uses the interchange law and one that does not. These two homotopies are equal. Specifically, the following two diagrams commute: + \begin{center} + \begin{tikzcd} + (f_2 \o \const)\smsh (g_2 \o g_1) + \arrow[r, equals, "i"] + \arrow[dd, swap, equals, "\zeroh' \smsh (g_2 \o g_1)"] + &(f_2 \smsh g_2)\o (\const \smsh g_1) + \arrow[d, equals, "(f_2 \smsh g_2) \o t_{g_1}"] + \\ + & (f_2 \smsh g_2)\o \const + \arrow[d,equals, "\zeroh'"] + \\ + \const\smsh (g_2 \o g_1) + \arrow[r,equals, swap, "t_{g_2 \o g_1}"] + & \const + \end{tikzcd} + \qquad + \begin{tikzcd} + (\const \o f_1)\smsh (g_2 \o g_1) + \arrow[r, equals, "i"] + \arrow[dd, swap, equals, "\zeroh \smsh (g_2 \o g_1)"] + & (\const \smsh g_2)\o (f_1 \smsh g_1) + \arrow[d,equals, "t_{g_2} \o (f_1 \smsh g_1)"] + \\ + & \const\o (f_1 \smsh g_1) + \arrow[d,equals, "\zeroh"] + \\ + \const\smsh (g_2 \o g_1) + \arrow[r,swap, equals, "t_{g_2 \o g_1}"] + & \const + \end{tikzcd} + \end{center} + +\end{lem} +\begin{proof} +% We will only do the case where $f_1\jdeq \const$, i.e. fill the diagram on the left. The other case is similar (and slightly easier). + + We start by filling the diagram on the left. First apply induction on the paths that $f_2$, $g_1$ and $g_2$ + respect the basepoint. In this case $f_2\o\const$ is definitionally equal to $\const$, and the canonical + proof that $f_2\o \const\sim\const$ is (definitionally) equal to reflexivity. This means that the homotopy + $(f_2 \o \const)\smsh (g_2 \o g_1)\sim\const\smsh (g_2 \o g_1)$ is also equal to reflexivity, and also the + path that $f_2 \smsh g_2$ respects the basepoint is reflexivity, hence the homotopy + $(f_2 \smsh g_2)\o \const\sim\const$ is also reflexivity. This means we need to fill the following square: + \begin{center} + \begin{tikzcd} + (f_2 \o \const)\smsh (g_2 \o g_1) + \arrow[r, equals,"i"] + \arrow[d, swap, equals,"1"] + & (f_2 \smsh g_2)\o (\const \smsh g_1) + \arrow[d,equals,"(f_2\smsh g_2)\o t_{g_1}"] + \\ + \const \smsh (g_2 \o g_1) + \arrow[r, swap, equals,"t_{g_1 \o g_2}"] + & \const + \end{tikzcd} + \end{center} + For the underlying homotopy, take $x : A_1\smsh B_1$ and apply induction on $x$. Suppose + $x\equiv(a,b)$ for $a:A_1$ and $b:B_1$. With the notational convention for basepoints as in \autoref{lem:interchange}, we have to fill the square (we use that the paths that the maps respect the basepoints are reflexivity): + \begin{equation}\label{eq:pent-left-ab} + \begin{tikzcd}[column sep=5em] + (a_3,g_2(g_1(b))) + \arrow[r, equals,"1"] + \arrow[d,swap,equals,"1"] + %\arrow[d,equals,"\gluer_{g_2(g_1(b))}\tr\gluer_{g_2(g_1(b_1))}\sy"] + & (a_3,g_2(g_1(b))) + \arrow[d,equals,"\mapfunc{f_2\smsh g_2}(\gluer_{g_1(b)}\tr\gluer_{b_2}\sy)"] + \\ + (a_3,g_2(g_1(b))) + \arrow[r,swap,equals,"\gluer_{g_2(g_1(b))}\tr\gluer_{b_3}\sy"] + & (a_3,b_3) + \end{tikzcd} + \end{equation} + Now $\mapfunc{h\smsh k}(\gluer_z)=\gluer_{k(z)}$, so by general groupoid laws we see that the path on the bottom is equal to the path on the right, which means we can fill the square. For the other point constructors, the squares to fill are similar. If $x \judgeq \auxl$, we have: + \begin{equation}\label{eq:pent-left-auxl} + \begin{tikzcd}[column sep=5em] + \auxl \arrow[r, equals,"1"] + \arrow[d,swap,equals,"1"] & + \auxl \arrow[d,equals,"\mapfunc{f_2\smsh g_2}(\gluel_{a_2}\sy)"] \\ + \auxl \arrow[r,swap, equals,"\gluel_{a_3}\sy"] & + (a_3,b_3) + \end{tikzcd} + \end{equation} + We can fill this square, as the path on the bottom is definitionally equal to $\gluel_{a_3}\sy$ (as we applied path induction on the path that $f_2$ respects the basepoint) and the path on the right also reduces to $\gluel_{a_3}\sy$ using that $\mapfunc{h\smsh k}(\gluel_z)=\gluel_{h(z)}$. Similarly, we can fill the square for $x \judgeq \auxr$, which is: + \begin{equation}\label{eq:pent-left-auxr} + \begin{tikzcd}[column sep=5em] + \auxr \arrow[r, equals,"1"] + \arrow[d,swap,equals,"1"] & + \auxr \arrow[d,equals,"\mapfunc{f_2\smsh g_2}(\gluer_{b_2}\sy)"] \\ + \auxr \arrow[r,swap, equals,"\gluer_{b_3}\sy"] & + (a_3,b_3) + \end{tikzcd} + \end{equation} + If $x$ varies over $\gluel_a$, after some reductions, we need to fill the following cube, where the front and the back are the squares in (\ref{eq:pent-left-ab}) for $(a,b_1)$ and (\ref{eq:pent-left-auxl}) respectively; the left square is degenerate; the other three sides are the squares in the definition of $i$ and $t$ to show that they respect $\gluel_a$ (given in (\ref{eq:i-gluel}) and (\ref{eq:t-gluel}) respectively), where we also apply $f_2 \smsh g_2$ to the square on the right. We suppress in the diagram the arguments of $\gluer$ in $\gluer\tr\gluer\sy$ (which match, so the concatenation results equal to the identity path). + \begin{equation}\label{eq:pent-left-gluel} + \begin{tikzcd}[column sep=5em] + & \auxl + \arrow[rr, equals, "1"] + \arrow[dd, swap, equals, near end, "1"] + && \auxl + \arrow[dd, equals, "\mapfunc{f_2\smsh g_2} (\gluel_{a_2}\sy)"] + \\ + (a_3,b_3) + \arrow[ur, equals, "\gluel_{a_3}"] + \arrow[dd, swap, equals, "1"] + \arrow[rr, equals, crossing over, near end, "1"] + && (a_3,b_3) + \arrow[ur, equals, near start, "\mapfunc{f_2\smsh g_2}(\gluel_{a_2})"] + %\arrow[dd, equals, near start, "\mapfunc{f_2\smsh g_2}(\gluer\tr\gluer\sy)"] + \\ + & \auxl + \arrow[rr, swap, equals, near start, "\gluel_{a_3}\sy"] + && (a_3,b_3) + \\ + (a_3,b_3) + \arrow[ur, equals, "\gluel_{a_3}"] + \arrow[rr, swap, equals, "\gluer\tr\gluer\sy"] + && (a_3,b_3) + \arrow[ur, swap, equals, "1"] %1 + \arrow[from=uu, equals, crossing over, very near start, "\mapfunc{f_2 \smsh g_2}(\gluer\tr\gluer\sy)"] + \end{tikzcd} + \end{equation} + Similarly, if $x$ varies over $\gluer_b$, we need to fill the cube below: the front and the back are the squares in (\ref{eq:pent-left-ab}) for $(a_1,b)$ and (\ref{eq:pent-left-auxr}) respectively; the left square is again degenerate; the other three sides come from the fact that $i$ and $t$ respect $\gluer_b$ (given in (\ref{eq:i-gluer}) and (\ref{eq:t-gluer}) respectively). Again, we omit the arguments of $\gluer$ in $\gluer\tr\gluer\sy$ (in this case, not a priori judgmentally equal). + \begin{equation}\label{eq:pent-left-gluer} + \begin{tikzcd}[column sep=4em] + & \auxr + \arrow[rr, equals,"1"] + \arrow[dd, swap, equals, near end,"1"] + && \auxr + \arrow[dd,equals,"\mapfunc{f_2\smsh g_2}(\gluer_{b_2}\sy)"] + \\ + (a_3,g_2(g_1(b))) + \arrow[rr, equals, near end, crossing over, "1"] + \arrow[dd, swap, equals, "1"] + \arrow[ur, equals, "\gluer_{g_2(g_1(b))}"] + && (a_3,g_2(g_1(b))) + \arrow[ur, equals, near start, "\mapfunc{f_2\smsh g_2}(\gluer_{g_1(b)})"] + \\ + & \auxr + \arrow[rr, swap, equals, near start, "\gluer_{b_3}\sy"] + && (a_3,b_3) + \\ + (a_3,g_2(g_1(b))) + \arrow[rr, swap, equals,"\gluer\tr\gluer\sy"] + \arrow[ur, equals, near end, "\gluer_{g_2(g_1(b))}"] + && (a_3,b_3) + \arrow[from=uu, equals, crossing over, very near start, "\mapfunc{f_2\smsh g_2}(\gluer\tr\gluer\sy)"] + \arrow[ur, swap, equals, "1"] + \end{tikzcd} + \end{equation} + %After canceling applications of $\mapfunc{h\smsh k}(\gluer_z)=\gluer_{k(z)}$ on various sides of the squares (TODO). + In order to fill the cubes in (\ref{eq:pent-left-gluel}) and (\ref{eq:pent-left-gluer}), we generalize the paths and fill the cubes by path induction. The cube in (\ref{eq:pent-left-gluel}) can be generalized to a cube: + \begin{center} + \begin{tikzcd}[column sep=3em] + & h(y) + \arrow[rr, equals,"1"] + \arrow[dd, swap, equals, near end,"1"] + && h(y) + \arrow[dd,equals,"\mapfunc{h}(p_l\sy)"] + \\ + h(x) + \arrow[rr, equals, near end, crossing over, "1"] + \arrow[dd, swap, equals, "1"] + \arrow[ur, equals, "q_l"] + && h(x) + \arrow[ur, equals, near start, "\mapfunc{h}(p_l)"] + \\ + & h(y) + \arrow[rr, swap, equals, near start, "q_l\sy"] + && h(x) + \\ + h(x) + \arrow[rr, swap, equals,"q_r\tr q_r\sy"] + \arrow[ur, equals, "q_l"] + && h(x) + \arrow[from=uu, equals, crossing over, near start, "\mapfunc{h}(p_r\tr p_r\sy)"] + \arrow[ur, swap, equals, "1"] + \end{tikzcd} + \end{center} + for $X$ and $X'$ pointed types; a map $h : X \to X'$; terms $x$, $y$ $z : X$; paths $p_l : x = y$, $p_r : x = z$, $q_l : h(x) = h(y)$, $q_r : h(x) = h(z)$; and 2-paths $s_l : \mapfunc{h}(p_l) = q_l$ (for the back and the top) and $s_r : \mapfunc{h}(p_r) = q_r$ (for the right side). This cube is filled by path induction on $s_l$, $s_r$, $p_l$ and $p_r$. The cube in (\ref{eq:pent-left-gluer}) can be generalized to a similar cube: + \begin{center} + \begin{tikzcd}[column sep=3em] + & h(y) + \arrow[rr, equals,"1"] + \arrow[dd, swap, equals, near end,"1"] + && h(y) + \arrow[dd,equals,"\mapfunc{h}(p_b)"] + \\ + h(x) + \arrow[rr, equals, near end, crossing over, "1"] + \arrow[dd, swap, equals, "1"] + \arrow[ur, equals, "q_l"] + && h(x) + \arrow[ur, equals, near start, "\mapfunc{h}(p_l)"] + \\ + & h(y) + \arrow[rr, swap, equals, near start, "q_b"] + && h(z) + \\ + h(x) + \arrow[rr, swap, equals,"q_l\tr q_b"] + \arrow[ur, equals, "q_l"] + && h(z) + \arrow[from=uu, equals, crossing over, near start, "\mapfunc{h}(p_l\tr p_b)"] + \arrow[ur, swap, equals, "1"] + \end{tikzcd} + \end{center} + for paths $p_l : x = y$, $p_b : y = z$, $q_l : h(x) = h(y)$, $q_b : h(y) = h(z)$ and for 2-paths $s_l : \mapfunc{h}(p_l) = q_l$ (for the top) and $s_b : \mapfunc{h}(p_b) = q_b$ (for the back). + + The diagram on the right is similar to the previous case. It is not hard to show that these homotopies are pointed. + +\end{proof} + +\begin{thm}\label{thm:smash-functor-right} +Given pointed types $A$, $B$ and $C$, the functorial action of the smash product induces a map +$$({-})\smsh C:(A\pmap B)\pmap(A\smsh C\pmap B\smsh C)$$ +that is natural in $A$ and $B$ and dinatural in $C$. +\end{thm} +The naturality and dinaturality means that the following squares commute for $f : A' \to A$ $g:B\to B'$ and $h:C\to C'$. +\begin{center} +\begin{tikzcd}[column sep=5em] +(A\pmap B) \arrow[r,"({-})\smsh C"]\arrow[d,"f\pmap B"] & +(A\smsh C\pmap B\smsh C)\arrow[d,"f\smsh C\pmap B\smsh C"] \\ +(A'\pmap B) \arrow[r,"({-})\smsh C"] & +(A'\smsh C\pmap B\smsh C) +\end{tikzcd} +\qquad +\begin{tikzcd}[column sep=5em] +(A\pmap B) \arrow[r,"({-})\smsh C"]\arrow[d,"A\pmap g"] & +(A\smsh C\pmap B\smsh C)\arrow[d,"A\smsh C\pmap g\smsh C"] \\ +(A\pmap B') \arrow[r,"({-})\smsh C"] & +(A\smsh C\pmap B'\smsh C) +\end{tikzcd} +\begin{tikzcd}[column sep=5em] +(A\pmap B) \arrow[r,"({-})\smsh C"]\arrow[d,"({-})\smsh C'"] & +(A\smsh C\pmap B\smsh C)\arrow[d,"A\smsh C\pmap B\smsh h"] \\ +(A\smsh C'\pmap B\smsh C') \arrow[r,"A\smsh h\pmap B\smsh C'"] & +(A\smsh C\pmap B\smsh C') +\end{tikzcd} +\end{center} +\begin{proof} +First note that $\lam{f}f\smsh C$ preserves the basepoint so that the map is indeed pointed. + +Let $k:A\pmap B$. Then as homotopy the naturality in $A$ becomes +$(k\o f)\smsh C=k\smsh C\o f\smsh C$. To prove an equality between pointed maps, we need to give +a pointed homotopy, which is given by interchange. To show that this homotopy is pointed, we need to +fill the following square (after reducing out the applications of function extensionality), which follows from \autoref{lem:smash-coh}. +\begin{center} +\begin{tikzcd} +(\const \o f)\smsh C \arrow[r, equals]\arrow[dd,equals] & +(\const \smsh C)\o (f \smsh C)\arrow[d,equals] \\ +& \const \o (f \smsh C)\arrow[d,equals] \\ +\const\smsh C \arrow[r,equals] & +\const +\end{tikzcd} +\end{center} +The naturality in $B$ is almost the same: for the underlying homotopy we need to show +$i:(g \o k)\smsh C = g\smsh C \o k\smsh C$. For the pointedness we need to fill the following +square, which follows from the left pentagon in \autoref{lem:smash-coh}. +\begin{center} +\begin{tikzcd} +(g \o \const)\smsh C \arrow[r, equals]\arrow[dd,equals] & +(g \smsh C)\o (\const \smsh C)\arrow[d,equals] \\ +& (g\smsh C) \o \const\arrow[d,equals] \\ +\const\smsh C \arrow[r,equals] & +\const +\end{tikzcd} +\end{center} +% To show that this naturality is pointed, we need to show that if $g=\const$, then this homotopy is the same as the concatenation of the following pointed homotopies: +% $$q:({-})\smsh C \circ (A \to \const)\sim ({-})\smsh C \circ \const \sim \const \sim \const \circ ({-})\smsh C\sim \const\smsh B \circ ({-})\smsh C.$$ +% To show that the underlying homotopies are the same, we need to show that $i(0,f,\idfunc[C],\idfunc[C])$ is equal to the following concatenation of pointed homotopies +% $$q(f):(0\circ f)\smsh C\sim 0\smsh C \sim 0 \sim 0 \circ f\smsh C\sim 0\smsh B \circ f\smsh C,$$ +% which is the right pentagons in \autoref{lem:smash-coh}. +% To show that these pointed homotopies respect the basepoint in the same way, we need to show that (TODO) +% ``$R\mathrel\square(0\smsh C \circ t)\cdot q_0=L$ where $L$ and $R$ are the left and right pentagons applied to $0$ and $\square$ is whiskering.'' + +The dinaturality in $C$ is a bit harder. For the underlying homotopy we need to show +$B\smsh h\o k\smsh C=k\smsh C'\o A\smsh h$. This follows from applying interchange twice: +$$B\smsh h\o k\smsh C\sim(\idfunc[B]\o k)\smsh(h\o\idfunc[C])\sim(k\o\idfunc[A])\smsh(\idfunc[C']\o h)\sim k\smsh C'\o A\smsh h.$$ +To show that this homotopy is pointed, we need to fill the following square: +\begin{xcenter} + \begin{tikzcd} + B\smsh h\o \const\smsh C \arrow[r, equals]\arrow[d,equals] & + (\idfunc[B]\o \const)\smsh(h\o\idfunc[C]) \arrow[r, equals]\arrow[d,equals] & + (\const\o\idfunc[A])\smsh(\idfunc[C']\o h)\arrow[r, equals]\arrow[d,equals] & + \const\smsh C'\o A\smsh h\arrow[d,equals] \\ + B\smsh h\o \const \arrow[d,equals] & + \const\smsh(h\o\idfunc[C]) \arrow[r, equals]\arrow[d,equals] & + \const\smsh(\idfunc[C']\o h) \arrow[d,equals] & + \const\o A\smsh h\arrow[d,equals] \\ + B\smsh h\o \const \arrow[r, equals] & + \const \arrow[r, equals] & + \const \arrow[r, equals] & + \const + \end{tikzcd} +\end{xcenter} +The left and the right squares are filled by \autoref{lem:smash-coh}. The squares in the middle +are filled by (corollaries of) \autoref{lem:smash-general}. +\end{proof} + +\subsection{Adjunction}\label{sec:smash-adjunction} + +\begin{lem}\label{lem:unit-counit} + There is a unit $\eta_{A,B}\equiv\eta:A\pmap B\pmap A\smsh B$ natural in $A$ and counit + $\epsilon_{B,C}\equiv\epsilon : (B\pmap C)\smsh B \pmap C$ dinatural in $B$ and natural in $C$. + These maps satisfy the unit-counit laws: + $$(A\to\epsilon_{A,B})\o \eta_{A\to B,A}\sim \idfunc[A\to B]\qquad + \epsilon_{B,B\smsh C}\o \eta_{A,B}\smsh B\sim\idfunc[A\smsh B].$$ +\end{lem} +Note: $\eta$ is also dinatural in $B$, but we do not need this. +\begin{proof} + We define $\eta ab=(a,b)$. We define the path that $\eta a$ respects the basepoint as + $$(\eta a)_0\defeq\gluel_a\tr\gluel_{a_0}\sy:(a,b_0)=(a_0,b_0).$$ Also, $\eta$ itself respects the basepoint. To show this, we need to give $\eta_0:\eta (a_0)\sim \const$. The underlying maps are homotopic, by $$\eta_0b\defeq\gluer_b\cdot\gluer_{b_0}\sy:(a_0,b)=(a_0,b_0).$$ To show that + this homotopy is pointed, we need to show that the two given proofs of $(a_0,b_0)=(a_0,b_0)$ are + equal, but they are both equal to reflexivity: + $$\eta_{00}:\gluel_{a_0}\tr\gluel_{a_0}\sy=1=\gluer_{b_0}\tr\gluer_{b_0}\sy.$$ + This defines the unit. To show that it is natural in $A$, we need to give the following pointed homotopy $p_\eta(f)$ for $f:A\to A'$. + \begin{center} + \begin{tikzcd} + A \arrow[r,"\eta"]\arrow[d,"f"] & + (B\pmap A \smsh B)\arrow[d,"B\to f\smsh B"] \\ + A' \arrow[r,"\eta"] & + (B\pmap A'\smsh B) + \end{tikzcd} + \end{center} + We may assume that $f_0$ is reflexivity. For the underlying homotopy we need to define for $a:A$ that $p_\eta(f,a):\eta(fa)\sim f\smsh B \circ \eta a$, which is another pointed homotopy. For $b:B$ we have $\eta(fa,b)\equiv(fa,b)\equiv(f\smsh B)(\eta ab).$ + The homotopy $p_\eta(f,a)$ is pointed, since $$(f\smsh B \circ \eta a)_0=\apfunc{f\smsh B}(\gluel_a\cdot\gluel_{a_0}\sy)=\gluel_{fa}\cdot\gluel_{a_0'}\sy=(\eta(fa))_0.$$ + Now we need to show that $p_\eta(f)$ is pointed, for which we need to fill the following diagram. + \begin{center} + \begin{tikzcd} + \eta(fa_0) \arrow[equals,rr,"{p_\eta(f,a_0)}"]\arrow[dr,equals,"{\eta_0}"] & & + f\smsh B \circ \eta a_0\arrow[dl,equals,"{f\smsh B\circ\eta_0}"] \\ + & \const_{B,A'\smsh B} & + \end{tikzcd} + \end{center} + These pointed homotopies have equal underlying homotopies, since for $b:B$ we have + $$p_\eta(f,a_0,b)\cdot\apfunc{f\smsh B}(\eta_0 b)=1\cdot\apfunc{f\smsh B}(\gluer_b\cdot\gluer_{b_0}\sy)=\gluer_{b}\cdot\gluer_{b_0}\sy=\eta_0b.$$ + We will skip the proof that these homotopies respect the point in the same way. + + To define the counit, given $x:(B\pmap C)\smsh B$, we construct + $\epsilon (x):C$ by induction on $x$. If $x\jdeq(f,b)$, we set $\epsilon(f,b)\defeq f(b)$. If $x$ + is either $\auxl$ or $\auxr$, then we set $\epsilon (x)\defeq c_0:C$. If $x$ varies over $\gluel_f$, + then we need to show that $f(b_0)=c_0$, which is true by $f_0$. If $x$ varies over $\gluer_b$, we + need to show that $\const(b)=c_0$ which is true by reflexivity. Now $\epsilon_0\defeq 1:\epsilon(\const_{B,C},b_0)=c_0$ shows that $\epsilon$ is pointed. + + We will skip the proof that the counit is dinatural in $B$ and natural in $C$. + + Finally, we need to show the unit-counit laws. For the underlying homotopy of the first one, let + $f:A\to B$. We need to show that $p_f:\epsilon\o\eta f\sim f$. We define $p_f(a)=1:\epsilon(f,a)=f(a)$. To show that $p_f$ is a pointed homotopy, we need to show that + $p_f(a_0)\tr f_0=\mapfunc{\epsilon}(\eta f)_0\tr \epsilon_0$, which reduces to + $f_0=\mapfunc{\epsilon}(\gluel_f\tr\gluel_0\sy)$, but we can reduce the right hand side: (note: + $\const_0$ denotes the proof that $\const(a_0)=b_0$, which is reflexivity) + $$\mapfunc{\epsilon}(\gluel_f\tr\gluel_0\sy)=\mapfunc{\epsilon}(\gluel_f)\tr(\mapfunc{\epsilon}(\gluel_0))\sy=f_0\tr \const_0\sy=f_0.$$ + Now we need to show that $p$ itself respects the basepoint of $A\to B$, i.e. that the composite + $\epsilon\o\eta(\const)\sim\epsilon\o\const\sim\const$ is equal to $p_{\const_{A,B}}$. The underlying + homotopies are the same for $a : A$; on the one side we have + $\mapfunc{\epsilon}(\gluer_{a}\tr\gluer_{a_0}\sy)$ and on the other side we have reflexivity + (note: this type checks since $\const_{A,B}a\equiv\const_{A,B}a_0$). These paths are equal, since + $$\mapfunc{\epsilon}(\gluer_{a}\tr\gluer_{a_0}\sy)=\mapfunc{\epsilon}(\gluer_{a})\tr(\mapfunc\epsilon(\gluer_{a_0}))\sy=1\cdot1\sy\equiv1.$$ + Both pointed homotopies are pointed in the same way, which requires some path-algebra, and we skip the proof here. + + For the underlying homotopy of the second unit-counit law, we need to show for $x:A\smsh B$ that + $q(x):\epsilon((\eta\smsh B)x)=x$, which we prove by induction to $x$. If $x\equiv(a,b)$, then we can define $q(a,b)\defeq1_{(a,b)}$. + If $x$ is $\auxl$ or $\auxr$, then the left-hand side reduces to $(a_0,b_0)$, + so we can define $q(\auxl)\defeq\gluel_{a_0}$ and $q(\auxr)\defeq\gluer_{b_0}$. The following computation shows that $q$ respects $\gluel_a$: + \begin{align*} + \apfunc{\epsilon\circ\eta\smsh B}(\gluel_a)\cdot\gluel_{a_0}&= \apfunc{\epsilon}(\gluel_{\eta a})\cdot\gluel_{a_0}=(\eta a)_0\cdot\gluel_{a_0}=\gluel_a\cdot\gluel_{a_0}\sy\cdot\gluel_{a_0}\\ + &=\gluel_a. + \end{align*} + To show that it respects $\gluer_b$ we compute + \begin{align*} + \apfunc{\epsilon\circ\eta\smsh B}(\gluer_b)\cdot\gluer_{b_0}&= + \apfunc{\epsilon({-},b)}(\eta_0)\cdot\apfunc{\epsilon}(\gluer_b)\cdot\gluer_{b_0}= + \apfunc{\lam{f}fb}(\eta_0)\cdot\gluer_{b_0}\\ + &=\eta_0b\cdot\gluer_{b_0}= + %\gluer_b\cdot\gluer_{b_0}\sy\cdot\gluer_{b_0}= + \gluer_b. + \end{align*} + To show that $q$ is a pointed homotopy, we need to show that $(\epsilon\circ\eta\smsh B)_0=1$, For this we compute $$(\epsilon\circ\eta\smsh B)_0=\apfunc{\epsilon({-},b_0)}(\eta_0)=\eta_0b_0=\gluer_{b_0}\cdot\gluer_{b_0}\sy=1.$$ +\end{proof} + +\begin{defn} +The function $e\jdeq e_{A,B,C}:(A\pmap B\pmap C)\pmap(A\smsh B\pmap C)$ is defined as the composite +$$(A\pmap B\pmap C)\lpmap{({-})\smsh B}(A\smsh B\pmap (B\pmap C)\smsh B)\lpmap{A\smsh B \pmap\epsilon}(A\smsh B\pmap C).$$ +\end{defn} + +\begin{lem} + The function $e$ is invertible, hence gives a pointed equivalence $$(A\pmap B\pmap C)\simeq(A\smsh B\pmap C).$$ +\end{lem} +\begin{proof} + Define + $$e\sy_{A,B,C}:(A\smsh B\pmap C)\lpmap{B\pmap({-})}((B\pmap A\smsh B)\pmap (B\pmap + C))\lpmap{\eta\pmap(B\pmap C)}(A\pmap B\pmap C).$$ It is easy to show that $e$ and $e\sy$ are + inverses as unpointed maps from the unit-counit laws (\autoref{lem:unit-counit}) and naturality of $\eta$ and $\epsilon$. +% For $f : A\pmap B\pmap C$ we have +% \begin{align*} +% \inv{e}(e(f))&\equiv(\eta\pmap(B\pmap C))\o (B\pmap((A\smsh B\pmap\epsilon)\of\smsh B))\\ +% &= (\eta\pmap(B\pmap C))\o (B\pmap(A\smsh B\pmap\epsilon))\o(B\pmapf\smsh B)\\ +% % &= (\eta\pmap(B\pmap C))\o (B\pmap(A\smsh B\pmap\epsilon))\o(B\pmapf\smsh B)\\ +% \end{align*} +\end{proof} +\begin{lem}\label{lem:e-natural} + The function $e$ is natural in $A$, $B$ and $C$. +\end{lem} +\begin{proof} + \textbf{Naturality of $e$ in $A$}. Suppose that $f:A'\pmap A$. Then the following diagram commutes. The left square commutes by naturality of $({-})\smsh B$ in the first argument and the right square commutes because composition on the left commutes with composition on the right. + \begin{center} + \begin{tikzcd} + (A\pmap B\pmap C) \arrow[r,"({-})\smsh B"]\arrow[d,"f\pmap B\pmap C"] & + (A\smsh B\pmap (B\pmap C)\smsh B) \arrow[r,"A\smsh B\pmap\epsilon"]\arrow[d,"f\smsh B\pmap\cdots"] & + (A\smsh B\pmap C)\arrow[d,"f\smsh B\pmap C"] \\ + (A'\pmap B\pmap C) \arrow[r,"({-})\smsh B"] & + (A'\smsh B\pmap (B\pmap C)\smsh B) \arrow[r,"A\smsh B\pmap\epsilon"] & + (A'\smsh B\pmap C) + \end{tikzcd} + \end{center} + + \textbf{Naturality of $e$ in $C$}. Suppose that $f:C\pmap C'$. Then in the following diagram the left square commutes by naturality of $({-})\smsh B$ in the second argument (applied to $B\pmap f$) and the right square commutes by applying the functor $A\smsh B \pmap({-})$ to the naturality of $\epsilon$ in the second argument. + \begin{center} + \begin{tikzcd} + (A\pmap B\pmap C) \arrow[r]\arrow[d] & + (A\smsh B\pmap (B\pmap C)\smsh B) \arrow[r]\arrow[d] & + (A\smsh B\pmap C)\arrow[d] \\ + (A\pmap B\pmap C') \arrow[r] & + (A\smsh B\pmap (B\pmap C')\smsh B) \arrow[r] & + (A\smsh B\pmap C') + \end{tikzcd} + \end{center} +%Pointed naturality: TODO. + + \textbf{Naturality of $e$ in $B$}. Suppose that $f:B'\pmap B$. Here the diagram is a bit more +complicated, since $({-})\smsh B$ is dinatural (instead of natural) in $B$. Then we get the +following diagram. The front square commutes by naturality of $({-})\smsh B$ in the second argument + (applied to $f\pmap C$). The top square commutes by naturality of $({-})\smsh B$ in the third +argument, the back square commutes because composition on the left commutes with composition on the + right, and finally the right square commutes by applying the functor $A\smsh B' \pmap({-})$ to the + naturality of $\epsilon$ in the first argument. + \begin{center} + \begin{tikzcd}[row sep=scriptsize, column sep=-4em] + & (A\smsh B\pmap (B\pmap C)\smsh B) \arrow[rr] \arrow[dd] & & (A\smsh B'\pmap (B\pmap C)\smsh B)\arrow[dd] \\ + (A\pmap B\pmap C) \arrow[ur] \arrow[rr, crossing over] \arrow[dd] & & (A\smsh B'\pmap (B\pmap C)\smsh B') \arrow[ur] \\ + & (A\smsh B\pmap C)\arrow[rr] & & (A\smsh B'\pmap C) \\ + (A\pmap B'\pmap C) \arrow[rr] & & (A\smsh B'\pmap (B'\pmap C)\smsh B') \arrow[ur] \arrow[from=uu, crossing over] + \end{tikzcd} + \end{center} + +\end{proof} +\begin{rmk} + Instead of showing that $e$ is natural, we could show that $e^{-1}$ is natural. In + that case we need to show that the map $A\to({-}):(B\to C)\to(A\to B)\to(A\to C)$ is natural in + $A$, $B$ and $C$. This might actually be easier, since we do not need to work with any higher + inductive type to prove that. +\end{rmk} + +We have now obtained the following theorem +\begin{thm}\label{thm:smash-adjoint} + There is an equivalence + $$(A\pmap B\pmap C)\simeq(A\smsh B\pmap C)$$ + natural in $A$, $B$ and $C$. +\end{thm} + +\begin{rmk} + We can state \autoref{thm:smash-adjoint} as an adjunction $({-})\smsh B\dashv B\to({-})$ or by saying that $A\smsh B$ + represents the functor $A\pmap B\pmap ({-})$. + + In \autoref{sec:smash-monoidal} we show that the smash product forms a + 1-coherent symmetric monoidal product from the assumption that this adjunction + is pointed in $C$. Explicitly, this means that the naturality of $e$ in $C$ + applied to the map $\const_{C,C'}:C\to C'$ is equal to the composite + $$(A\smsh B \to \const_{C,C'})\o e_{A,B,C}\sim \const\o e_{A,B,C}\sim \const + \sim e_{A,B,C'}\o \const \sim e_{A,B,C'}\o (A\to B\to \const_{C,C'}).$$ + + To prove this, we need that the counit $\eps$ is pointed natural in $C$. To + prove that, we need to show that the map $({-})\smsh C$, defined in + \autoref{thm:smash-functor-right}, is pointed natural in $B$. In order to + prove that, we need to show that in the situation of \autoref{lem:smash-coh}, + if both $f_1$ and $f_2$ are (judgmentally) the constant map, then the two + pentagons stated in that lemma are equal (transported appropriately in order + to make this equality type check). This can be formulated as a 3-path in a + type of pointed maps, which is hard to fill. +\end{rmk} + +\subsection{Symmetric monoidal product}\label{sec:smash-monoidal} +In this section we will prove that the smash product is a 1-coherent symmetric +monoidal product~\autoref{def:symmonprod}, from the assumption that the +adjunction from \autoref{sec:smash-adjunction} is pointed natural in $C$. We +will need to following pointed equivalences. Without the proof that $e$ is +pointed natural, parts of this section are still true. In particular, the +natural equivalences defined in \autoref{def:smash-alrg} do not require pointed +naturality of $e$. + +\begin{defn}\label{def:b-and-tw} + We define the pointed equivalences: + \[\two : (\pbool \to X) \simeq X\] where $\pbool$ is the type of booleans (pointed in $\bfalse$) with underlying map defined with $\two(f) \defeq f(\btrue)$, and + \[\twist : (A \to B \to X) \simeq (B \to A \to X)\] + with underlying map defined with $\twist(f) \defeq \lam{b}\lam{a}f(a)(b)$. + \end{defn} + +Using \autoref{lem:yoneda} (Yoneda) we can prove associativity, left and right +unitality and braiding equivalences for the smash product, in the following way. + +\begin{defn}\label{def:equiv-precursors} + The following pointed equivalences are defined for $A$, $B$, $C$ and $X$ pointed types: + \begin{itemize} + \item $\alphabar_X : (A \smsh (B \smsh C) \to X) \simeq ((A \smsh B) \smsh C \to X)$ as the composition of the equivalences: + \begin{align*} + A \smsh (B \smsh C)\to X&\simeq A \to B\smsh C\to X && (e\sy)\\ + &\simeq A \to B\to C\to X && (A \to e\sy)\\ + &\simeq A \smsh B\to C\to X && (e)\\ + &\simeq (A \smsh B)\smsh C\to X. && (e) + \end{align*} + \item $\lambdabar_X : (B \to X) \simeq (\pbool \smsh B \to X)$ as the composition of the equivalences: + \begin{align*} + B \to X &\simeq \pbool \to B \to X && (\two\sy)\\ + &\simeq \pbool \smsh B \to X && (e) + \end{align*} + \item $\rhobar_X : (A \to X) \simeq (A \smsh \pbool \to X)$ as the composition of the equivalences: + \begin{align*} + A \to X &\simeq A \to \pbool \to X && (A \to \two\sy)\\ + &\simeq A \smsh \pbool \to X && (e) + \end{align*} + \item $\gammabar_X : (B \smsh A \to X) \simeq (A \smsh B \to X)$ as the composition of the equivalences: + \begin{align*} + B \smsh A \to X &\simeq B \to A \to X && (e\sy)\\ + &\simeq A \to B \to X && (\twist)\\ + &\simeq A \smsh B \to X && (e) + \end{align*} + \end{itemize} +\end{defn} + +\begin{rmk}\label{rmk:alrg-pointed-natural} + The equivalences in \autoref{def:equiv-precursors} are natural in all their + arguments and from the assumption that $e$ is pointed natural in $C$ we can + show that these maps are all pointed natural in $X$. + % In particular, we will + % use: + % \begin{align*} + % f \o \alphabar(g) &\sim \alphabar(f \o g) & f \o \lambdabar(g) &\sim \lambdabar(f \o g)\\ + % f \o \rhobar(g) &\sim \rhobar(f \o g) & f \o \gammabar(g) &\sim \gammabar(f \o g) + % \end{align*} +\end{rmk} + +\begin{defn}\label{def:smash-alrg} + We define the following equivalences, natural in all their arguments, with inverses provided as in \autoref{lem:yoneda}: + \begin{itemize} + \item $\alpha\defeq\alphabar_{A \smsh (B \smsh C)}(\idfunc) : (A \smsh B) \smsh C \simeq A \smsh (B \smsh C)$ (associativity of the smash product), with inverse $\alpha\sy\defeq\alphabar\sy_{(A \smsh B) \smsh C}(\idfunc)$; + \item $\lambda \defeq \lambdabar_B(\idfunc) : \pbool \smsh B \simeq B$ and $\rho \defeq \rhobar_A(\idfunc) : A \smsh \pbool \simeq A$ (left- and right unitors for the smash product), with inverses $\lambda\sy\defeq \lambdabar_{\pbool\smsh B}\sy(\idfunc)$ and $\rho\sy\defeq \rhobar_{A\smsh \pbool}\sy(\idfunc)$, respectively; + \item $\gamma \defeq \gammabar_{B\smsh A} (\idfunc) : A \smsh B \simeq B \smsh A$ (braiding for the smash product), with inverse $\gamma\sy \defeq \gammabar_{A \smsh B}\sy (\idfunc)$. + \end{itemize} + $\alpha$, $\lambda$, $\rho$ and $\gamma$ are natural in all their arguments, + as $\alphabar$, $\lambdabar$, $\rhobar$ and $\gammabar$ are. Note that these + definitions do \emph{not} require pointed naturality of $e$. +\end{defn} + +\begin{lem}\label{lem:bar-homotopy} + There are pointed homotopies + \begin{align*} + \alphabar_X &\sim \alpha \to X + & \lambdabar_X &\sim \lambda \to X + \\ + \rhobar_X &\sim \rho \to X + & \gammabar_X &\sim \gamma \to X + \end{align*} +\end{lem} + +\begin{proof} + This follows directly from \autoref{lem:yoneda-pointed} and \autoref{rmk:alrg-pointed-natural} (this does require pointed naturality of $e$). +\end{proof} + +\begin{thm}[Associativity pentagon]\label{thm:smash-associativity-pentagon} + For $A$, $B$, $C$ and $D$ pointed types, there is a homotopy + \[\alpha \o \alpha \sim (A \smsh \alpha) \o \alpha \o (\alpha \smsh D)\] + corresponding to the commutativity of the following diagram: + \begin{center} + \begin{tikzcd} + &((A \smsh B) \smsh (C \smsh D)) + \arrow[dr, "\alpha"] + \\ + (((A \smsh B) \smsh C) \smsh D) + \arrow[ru, "\alpha"] + \arrow[d, swap, "\alpha \smsh D"] + && (A \smsh (B \smsh (C \smsh D))) + \\ + ((A \smsh (B \smsh C)) \smsh D) + \arrow[rr, swap, "\alpha"] + && (A \smsh ((B \smsh C) \smsh D)) + \arrow[u, swap, "A \smsh \alpha"] + \end{tikzcd} +% \begin{tikzcd} +% (((A \smsh B) \smsh C) \smsh D) +% \arrow[rr, "\alpha"] +% \arrow[d, swap, "\alpha \smsh D"] +% && ((A \smsh B) \smsh (C \smsh D)) +% \arrow[d, "\alpha"] +% \\ +% ((A \smsh (B \smsh C)) \smsh D) +% \arrow[r, swap, "\alpha"] +% & (A \smsh ((B \smsh C) \smsh D)) +% \arrow[r, swap, "A \smsh \alpha"] +% & (A \smsh (B \smsh (C \smsh D))) +% \end{tikzcd} + \end{center} +\end{thm} +\begin{proof} + We articulate the proof in several steps. A map homotopic to both sides of the sought homotopy will be constructed via the equivalence + \begin{align*} + \alphabar^4 : (A \smsh (B \smsh (C \smsh D)) \to X) &\simeq (((A \smsh B) \smsh C) \smsh D \to X) + \intertext{(natural in all its arguments), defined as the composite:} + A \smsh (B \smsh (C \smsh D)) \to X + &\simeq A \to B \smsh (C \smsh D) \to X && \text{($e\sy$)}\\ + &\simeq A \to B \to C \smsh D \to X &&\text{($A \to e\sy$)}\\ + &\simeq A \to B \to C \to D \to X &&\text{($A \to B \to e\sy$)}\\ + &\simeq A \smsh B \to C \to D \to X &&\text{($e$)}\\ + &\simeq (A \smsh B) \smsh C \to D \to X &&\text{($e$)}\\ + &\simeq ((A \smsh B) \smsh C) \smsh D \to X && \text{($e$)} + \intertext{giving $\alphabar^4(\idfunc) : ((A \smsh B) \smsh C) \smsh D) \simeq A \smsh (B \smsh (C \smsh D))$. Moreover, in order to simplify the expressions of $\alpha \smsh D$ and $A \smsh \alpha$, we also define:} + \alphabar^R : ((A \smsh (B \smsh C)) \smsh D \to X) &\simeq (((A \smsh B) \smsh C) \smsh D \to X) + \intertext{as the composite:} + (A \smsh (B \smsh C)) \smsh D \to X + &\simeq A \smsh (B \smsh C) \to D \to X &&\text{($e\sy$)}\\ + &\simeq (A \smsh B) \smsh C \to D \to X &&\text{($\alphabar$)}\\ + &\simeq ((A \smsh B) \smsh C) \smsh D \to X &&\text{($e$)} + \intertext{and} + \alphabar^L : (A \smsh (B \smsh (C \smsh D)) \to X) &\simeq (A \smsh ((B \smsh C) \smsh D) \to X) + \intertext{as the composite:} + A \smsh (B \smsh (C \smsh D)) \to X + &\simeq A \to B \smsh (C \smsh D) \to X &&\text{($e\sy$)}\\ + &\simeq A \to (B \smsh C) \smsh D \to X &&\text{($A \to \alphabar$)}\\ + &\simeq A \smsh ((B \smsh C) \smsh D) \to X &&\text{($e$)} + \end{align*} + also natural in their arguments. Evaluating these equivalences to the identity function, we get new arrows that fit in the original diagram: + \begin{center} + \begin{tikzcd} + &((A \smsh B) \smsh (C \smsh D)) + \arrow[dr, "\alpha"] + \\ + (((A \smsh B) \smsh C) \smsh D) + \arrow[ru, "\alpha"] + \arrow[d, swap, "\alpha \smsh D"] + \arrow[d, bend left=40, "\alphabar^R(\idfunc)"] + \arrow[rr, "\alphabar^4(\idfunc)"] + && (A \smsh (B \smsh (C \smsh D))) + \\ + ((A \smsh (B \smsh C)) \smsh D) + \arrow[rr, swap, "\alpha"] + && (A \smsh ((B \smsh C) \smsh D)) + \arrow[u, swap, "A \smsh \alpha"] + \arrow[u, bend left=40, "\alphabar^L(\idfunc)"] + \end{tikzcd} + \end{center} + + The theorem is then proved once we show the chain of homotopies: + \begin{equation}\label{eq:alphafour} + \alpha \o \alpha + \sim \alphabar^4(\idfunc) + \sim \alphabar^L(\idfunc) \o \alpha \o \alphabar^R(\idfunc) + \sim (A \smsh \alpha) \o \alpha \o (\alpha \smsh D) + \end{equation} + + To verify the first homotopy in (\ref{eq:alphafour}), we see that: + \begin{align*} + \alpha \o \alpha + &\judgeq \alphabar(\idfunc) \o \alphabar(\idfunc)\\ + &\sim (\alphabar \o \alphabar) (\idfunc) &&\text{(naturality of $\alphabar$)}\\ + &\judgeq (e \o e \o (A \to e\sy) \o e\sy \o e \o e \o (A \to e\sy) \o e\sy)(\idfunc)\\ + &\sim (e \o e \o (A \to e\sy) \o e \o (A \to e\sy) \o e\sy)(\idfunc) &&\text{(cancelling)}\\ + &\sim (e \o e \o e \o (B \to A \to e\sy) \o (A \to e\sy) \o e\sy)(\idfunc) &&\text{(naturality of $e$)}\\ + &\judgeq \alphabar^4(\idfunc) + \end{align*} + + The second homotopy in (\ref{eq:alphafour}) is verified by (right-to-left): + \begin{align*} + \alphabar^L(\idfunc) \o \alpha \o \alphabar^R(\idfunc) + &\judgeq \alphabar^L(\idfunc) \o \alphabar(\idfunc) \o \alphabar^R(\idfunc)\\ + &\sim (\alphabar^R \o \alphabar \o \alphabar^L)(\idfunc) \\ &\mbox{}\qquad\text{(naturality of $\alphabar$ and $\alphabar^R$)}\\ + &\judgeq (e \o \alphabar \o e\sy \o e \o e \o (A \to e\sy) \o e\sy \o e \o (A \to \alphabar) \o e\sy)(\idfunc)\\ + &\sim (e \o \alphabar \o e \o (A \to e\sy) \o (A \to \alphabar) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(cancelling)}\\ + &\sim (e \o \alphabar \o e \o (A \to (e\sy \o \alphabar)) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(functoriality of $A\to -$)}\\ + &\judgeq (e \o e \o e \o (A \to e\sy) \o e\sy \o e \\ + &\hspace{3em}\o (A \to (e\sy \o e \o e \o (B \to e\sy) \o e\sy)) \o e\sy)(\idfunc)\\ + &\sim (e \o e \o e \o (A \to ((B \to e\sy) \o e\sy)) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(cancelling)}\\ + &\sim (e \o e \o e \o (B \to A \to e\sy) \o (A \to e\sy) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(funct. of $A \to -$)}\\ + &\judgeq \alphabar^4(\idfunc) + \end{align*} + + In order to prove the last homotopy in (\ref{eq:alphafour}), it is sufficient to show that $\alphabar^R(\idfunc) \sim \alpha \smsh D$ and that $\alphabar^L(\idfunc) \sim A \smsh \alpha$. We have: + \begin{align*} + \alphabar^R(\idfunc) + &\judgeq e(\alphabar (e\sy(\idfunc)))\\ + &\sim e (\alphabar (\eta))\\ + &\sim e (\eta \o \alphabar(\idfunc)) &&\text{(naturality of $\alphabar$)}\\ + &\judgeq \epsilon \o (\eta \o \alpha) \smsh D\\ + &\sim \epsilon \o (\eta \smsh D) \o (\alpha \smsh D) &&\text{(distrib. of $\smsh$)}\\ + &\sim \alpha \smsh D &&\text{(\autoref{lem:unit-counit})} + \end{align*} + and, lastly, + \begin{align*} + \alphabar^L(\idfunc) + &\judgeq e(\alphabar \o e\sy(\idfunc))\\ + &\sim e(\alphabar \o \eta)\\ + &\sim e((\alpha \to A \smsh (B \smsh (C \smsh D))) \o \eta) &&\text{(\autoref{lem:bar-homotopy})}\\ + &\sim e((B \smsh (C \smsh D) \to A \smsh \alpha) \o \eta) &&\text{(dinaturality of $\eta$)}\\ + &\sim (A \smsh \alpha) \o e(\eta) &&\text{(naturality of $e$)}\\ + &\sim A \smsh \alpha &&\text{(\autoref{lem:unit-counit})} + \end{align*} + thus proving the desired homotopy. +\end{proof} + +\begin{thm}[Unitors triangle]\label{thm:smash-unitors-triangle} + For $A$ and $B$ pointed types, there is a homotopy + \[(A \smsh \lambda) \o \alpha \sim (\rho \smsh B)\] + corresponding to the commutativity of the following diagram: + \begin{center} + \begin{tikzcd} + ((A \smsh \pbool) \smsh B) + \arrow[rr, "\alpha"] + \arrow[dr, swap, "\rho \smsh B"] + && (A \smsh (\pbool \smsh B)) + \arrow[dl, "A \smsh \lambda"] + \\ + & (A \smsh B) + \end{tikzcd} + \end{center} +\end{thm} +\begin{proof} + By an argument similar to the one for $\alphabar^L$ and $\alphabar^R$ in \autoref{thm:smash-associativity-pentagon}, one can verify the homotopies $A \smsh \lambda \sim (e \o (A \to \lambdabar) \o e\sy)(\idfunc)$ and $\rho \smsh B \sim (e \o \rhobar \o e)(\idfunc)$, simplifying the expressions in the sought homotopy. Then: + \begin{align*} + (A \smsh \lambda) \o \alpha + &\sim e(\lambdabar \o e\sy(\idfunc)) \o \alphabar(\idfunc) &&\text{(simplification)}\\ + &\sim \alphabar(e(\lambdabar \o e\sy(\idfunc)) &&\text{(naturality of $\alphabar$)}\\ + &\judgeq e(e(e\sy \o e\sy (e(\lambdabar \o e\sy(\idfunc)))))\\ + &\sim e(e(e\sy \o \lambdabar \o e\sy(\idfunc))) &&\text{(cancelling)}\\ + &\judgeq e(e(e\sy \o e \o \two\sy \o e\sy(\idfunc)))\\ + &\sim e(e(\two\sy \o e\sy(\idfunc))) &&\text{(cancelling)}\\ + &\judgeq (e \o \rhobar \o e\sy)(\idfunc)\\ + &\sim \rho \smsh B &&\text{(simplification)} + \end{align*} + gives the desired homotopy. +\end{proof} + +\begin{thm}[Braiding-unitors triangle]\label{thm:smash-braiding-unitors} + For a pointed type $A$, there is a homotopy + \[\lambda \o \gamma \sim \rho\] + corresponding to the commutativity of the following diagram: + \begin{center} + \begin{tikzcd} + (A \smsh \pbool) + \arrow[rr, "\gamma"] + \arrow[dr, swap, "\rho"] + && (\pbool \smsh A) + \arrow[dl, "\lambda"] + \\ + & A + \end{tikzcd} + \end{center} +\end{thm} +\begin{proof} + We have: + \begin{align*} + \lambda \o \gamma + &\judgeq \lambdabar(\idfunc) \o \gammabar(\idfunc)\\ + &\sim (\gammabar \o \lambdabar)(\idfunc) &&\text{(naturality of $\gammabar$)}\\ + &\judgeq (e \o \twist \o e\sy \o e \o \two\sy)(\idfunc)\\ + &\sim (e \o \twist \o \two\sy)(\idfunc) &&\text{(cancelling)}\\ + &\sim (e \o (A \to \two\sy))(\idfunc)\\ + &\judgeq \rhobar(\idfunc) \judgeq \rho + \end{align*} + where the last homotopy is given by $(A \to c) \o \two \sim \twist : (\pbool \to A \to X) \to (A \to X)$. +\end{proof} + +\begin{lem}\label{lem:pentagon-c} + The following diagram commutes, for $A$, $B$, $C$ and $X$ pointed types: + \begin{center} + \begin{tikzcd}[column sep=7em] + (B \smsh C \to A \to X) + \arrow[r, "e\sy"] + \arrow[dd, swap, "\twist"] + & (B \to C \to A \to X) + \arrow[d, "B \to \twist"] + \\ + & (B \to A \to C \to X) + \arrow[d, "\twist"] + \\ + (A \to B \smsh C \to X) + \arrow[r, swap, "A \to e\sy"] + & (A \to B \to C \to X) + \end{tikzcd} + \end{center} +\end{lem} +\begin{proof} + Unfolding the definition of $e\sy$, we get the diagram: + \begin{xcenter} + \begin{tikzcd}[column sep=6em,every node/.style={font=\sffamily\small}] + (B \smsh C \to A \to X) + \arrow[rr, bend left=10, "e\sy"] + \arrow[r, swap, "C\to -"] + \arrow[dd, swap, "\twist"] + & ((C \to B \smsh C) \to C \to A \to X) + \arrow[r, swap, "\eta \to C \to A \to X"] + \arrow[d, swap, "(C \to B \smsh C) \to \twist"] + & (B \to C \to A \to X) + \arrow[d, "B \to \twist"] + \\ + & ((C \to B \smsh C) \to A \to C \to X) + \arrow[r, swap, "\eta \to A \to C \to X"] + \arrow[d, swap, "\twist"] + & (B \to A \to C \to X) + \arrow[d, "\twist"] + \\ + (A \to B \smsh C \to X) + \arrow[rr, swap, bend right=10, "A \to e\sy"] + \arrow[r, "A \to (C \to -)"] + & (A \to (C \to B \smsh C) \to C \to X) + \arrow[r, "A \to (\eta \to C \to X)"] + & (A \to B \to C \to X) + \end{tikzcd} + \end{xcenter} + where the squares on the right are instances of naturality of $\twist$, while the commutativity of the pentagon on the left follows easily from the definition of $\twist$. +\end{proof} + + +\begin{thm}[Associativity-braiding hexagon]\label{thm:smash-associativity-braiding} + For pointed types $A$, $B$ and $C$, there is a homotopy + \[\alpha \o \gamma \o \alpha \sim (B \smsh \gamma) \o \alpha \o (\gamma \smsh C)\] + corresponding to the commutativity of the following diagram: + \begin{center} + \begin{tikzcd} + ((A \smsh B) \smsh C) + \arrow[r, "\alpha"] + \arrow[d, swap, "\gamma \smsh C"] + &(A \smsh (B \smsh C)) + \arrow[r, "\gamma"] + & ((B \smsh C) \smsh A) + \arrow[d, "\alpha"] + \\ + ((B \smsh A) \smsh C)) + \arrow[r, swap, "\alpha"] + & (B \smsh (A \smsh C)) + \arrow[r, swap, "B \smsh \gamma"] + & (B \smsh (C \smsh A)) + \end{tikzcd} + \end{center} +\end{thm} +\begin{proof} + The proof is structured similarly to the one for \autoref{thm:smash-associativity-pentagon}: the homotopies + \begin{align*} + B \smsh \gamma &\sim \gammabar^L(\idfunc) &\text{with\ \ } \gammabar^L &\defeq e \o (B \to \gammabar) \o e\sy\\ + \gamma \smsh C &\sim \gammabar^R(\idfunc) &\text{with\ \ } \gammabar^R &\defeq e \o \gammabar \o e\sy + \end{align*} + can be proven in exactly the same way and, using these simplifications, we will show that both sides of the sought homotopy are homotopic to the same equivalence. Indeed we have: + \begin{align*} + \alpha \o \gamma \o \alpha + &\judgeq \alphabar(\idfunc) \o \gammabar(\idfunc) \o \alphabar(\idfunc)\\ + &\sim (\alphabar \o \gammabar \o \alphabar)(\idfunc) \\ &\mbox{}\qquad\text{(naturality of $\gammabar$ and $\alphabar$)}\\ + &\judgeq (e \o e \o (A \to e\sy) \o e\sy \o e \o \twist \o e\sy \o e \o e \o (B \to e\sy) \o e\sy)(\idfunc)\\ + &\sim (e \o e \o (A \to e\sy) \o \twist \o e \o (B \to e\sy) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(cancelling)}\\ + &\sim (e \o e \o \twist \o (B \to \twist) \o e\sy \o e \o (B \to e\sy) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(\autoref{lem:pentagon-c})}\\ + &\sim (e \o e \o \twist \o (B \to \twist) \o (B \to e\sy) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(cancelling)} + \end{align*} + and + \begin{align*} + (B \smsh \gamma) \o \alpha \o (\gamma \smsh C) + &\sim \gammabar^L(\idfunc) \o \alphabar \o \gammabar^R(\idfunc) \\ &\mbox{}\qquad\text{(simplification)}\\ + &\sim (\gammabar^R \o \alphabar \o \gammabar^L)(\idfunc) \\ &\mbox{}\qquad\text{(naturality of $\alphabar$ and $\gammabar^R$)}\\ + &\judgeq (e \o \gammabar \o e\sy \o e \o e \o (B \to e\sy) \o e\sy \o e \o (B \to \gammabar) \o e\sy)(\idfunc)\\ + &\sim (e \o \gammabar \o e \o (B \to e\sy) \o (B \to \gammabar) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(cancelling)}\\ + &\sim (e \o \gammabar \o e \o (B \to (e\sy \o \gammabar)) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(functoriality of $B \to -$)}\\ + &\judgeq (e \o e \o \twist \o e\sy \o e \o (B \to (e\sy \o e \o \twist \o e\sy)) \o e\sy)(\idfunc)\\ + &\sim (e \o e \o \twist \o (B \to \twist) \o (B \to e\sy) \o e\sy)(\idfunc) \\ &\mbox{}\qquad\text{(cancelling)} + \end{align*} + proving the commutativity of the diagram. +\end{proof} + +\begin{thm}[Double braiding]\label{thm:smash-double-braiding} + For $A$ and $B$ pointed types, there is a homotopy + \[\gamma \o \gamma \sim \idfunc\] + corresponding to the commutativity of the following diagram: + \begin{center} + \begin{tikzcd} + (A \smsh B) + \arrow[r, "\gamma"] + \arrow[dr, equals] + & (B \smsh A) + \arrow[d, "\gamma"] + \\ + & (A \smsh B) + \end{tikzcd} + \end{center} +\end{thm} +\begin{proof} + Using that $\twist \o \twist \sim \idfunc$, we get: + \begin{align*} + \gamma \o \gamma + &\judgeq \gammabar(\idfunc) \o \gammabar(\idfunc)\\ + &\sim (\gammabar \o \gammabar)(\idfunc) &&\text{(naturality of $\gammabar$)}\\ + &\judgeq (e \o \twist \o e\sy \o e \o \twist \o e\sy)(\idfunc)\\ + &\sim \idfunc &&\text{(cancelling)} + \end{align*} + as desired. +\end{proof} + +Finally we get the result of this section. +\begin{thm} + $\smsh$ is a 1-coherent symmetric monoidal product, assuming that $e$ is pointed natural in $C$. +\end{thm} +\begin{proof} + This follows immediately from the theorems in this section. +\end{proof} + +% \subsection{Discussion} + +% Many results in \autoref{sec:smash-basic} about $({-}) \smsh B$ can be generalized to arbitrary pointed $\infty$-functors $\type^*\to\type^*$. Internally in HoTT we cannot talk about $\infty$-categories and $\infty$-functors (as far as we know), and therefore we cannot express this internally. But we only need finitely many coherences, which means that we can formulate these results internally for arbitrary functors. However, when we tried to do this, we ran into the problem that we needed all (or at least, some) coherences in the definition of a tricategory and functor for tricategories, which are harder to check than the result we wanted to prove. Here we sketch this argument. + +% $\type^*$ is a pointed $(\infty,1)$-category, meaning it has a zero object +% $\unit$, which is an object that is both (homotopy-)initial and +% (homotopy-)terminal. Given two pointed $(\infty,1)$-categories $\mc{A}$ and +% $\mc{B}$, we call a map $F:\mc{A}\to\mc{B}$ a 1-coherent functor if it satisfies +% all the coherences of a functor (i.e. an action on morphisms and a 2-cell +% stating that $F$ respects compositions and identities), a 2-coherent functor if +% it satisfies all the coherences of a bifunctor (a functor between bicategories) +% and a 3-coherent functor if it satisfies all the coherences of a trifunctor (a +% functor between tricategories). +% % Add reference: +% % @article{gurski2006algebraic, +% % title={An algebraic theory of tricategories}, +% % author={Gurski, Nick}, +% % year={2007}, +% % url={http://gauss.math.yale.edu/~mg622/tricats.pdf} +% % } + + +% If $F$ is a pointed 2-coherent functor (or more precisely a 1-coherent functor which a coherence for the associativity 2-cell), then we can show that \autoref{lem:smash-coh} holds for $F$. We can show that the two pentagons have the same filler if $F$ is a 3-coherent functor. + +% Now we can also show \autoref{thm:smash-functor-right} more generally, but we will only formulate that for functors between pointed types. If $F:\type^*\to\type^*$ is a 2-coherent pointed functor, then it induces a map +% $(A\to B)\to(FA\to FB)$ which is natural in $A$ and $B$. Moreover, if $F$ is 3-coherent, then this is a pointed natural transformation in $B$. + + +\chapter{The Serre Spectral Sequence}\label{cha:serre-spectr-sequ} + +Spectral sequences are important tools in algebraic topology.\footnote{The work in this chapter is joint work with Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman.} They give a +relationship between certain homotopy, homology and cohomology groups, in a way +that generalizes long exact sequences. This generalization comes at a cost of +being a lot more complicated than a long exact sequence. + +In this chapter we will start the study of spectral sequences in homotopy type +theory. We will introduce the notion of spectral sequences, and then construct +the Atiyah-Hirzebruch and Serre spectral sequences for cohomology. We follow the +construction due to Michael Shulman given in~\cite{shulman2013spectral}. We will also give a sketch on +how to construct the analogues for homology, and look at some of the +applications of these spectral sequences. + +There are a couple of notable differences between spectral sequences in homotopy +type theory compared to classical homotopy theory. +\begin{itemize} + \item As always, in HoTT all constructions have to be homotopy invariant, so + we cannot use classical constructions that are not homotopy invariant. For + example, the construction of the Serre spectral sequence for homology in~\cite{hatcher2004spectral} + uses CW-approximation of a space and the skeleton + of the obtained CW-complex to construct the spectral sequence. These + operations are not homotopy invariant, and therefore cannot be performed in + HoTT. + \item Another difference is that homology and cohomology are defined + differently in HoTT than in classical homotopy theory. In classical homotopy + theory (co)homology is defined using singular (co)homology. Since the intermediate + steps in the construction of singular (co)homology is not homotopy invariant, we use a different definition of + (co)homology (see \autoref{def:cohomology}), which impacts the definition of spectral sequences involving (co)homology. + \item %Spectral sequences consist of pages of abelian groups. + The first page of a spectral sequence is often not homotopy invariant, and + therefore cannot be constructed in HoTT. For this reason, we start counting + the pages of spectral sequences at 2. + \item HoTT offers a convenient language for formalizing proofs. Therefore, we + have formalized all constructed spectral sequences in this chapter. +\end{itemize} + +The spectral sequences we construct are not the most general version of these +spectral sequences. The spectral sequences we construct are still more general +than the formulation of the Serre spectral sequence in many textbooks (we give a +version of generalized and parametrized cohomology), but there exist more +general versions. There are two places where we compromised on generality for +the sake of making the formalization easier. The first compromise is that we +only formalized exact couples for graded $R$-modules for a ring $R$ (which is +not graded). More generally we could do this for any abelian category, which +would require building up the theory of abelian categories (this is done in +UniMath~\cite{unimath}). Furthermore, we did not look at convergence of spectral +sequences in the most general sense, since that can get quite complicated and +subtle. Instead, we only look at spectral sequences that are eventually constant +pointwise, so the $\infty$-page is just the eventual value. This restriction +adds the condition to the spectral sequences we construct that the coefficients +are only in truncated spectra. + +\section{Spectral Sequences}\label{sec:spectral-sequences} + +A spectral sequence consists of a sequence of pages, each of them containing a two-dimensional grid of abelian groups. There are maps between these groups, called \emph{differentials}. These differentials form (co)chain complexes, and the (co)homology of these complexes determine the groups on the next page. In \ref{fig:spectral-sequence-pages} we show an example of two pages of a spectral sequence, where each dot represents an abelian group. In this figure only the two first quadrants are shown, because in simple applications all other groups are trivial, though that need not be the case in general. + +\begin{figure}[ht] + \centering + \begin{subfigure}{.5\textwidth} + \centering + \begin{tikzpicture} + \draw [thick] (0.5,0.5) -- (0.5,5.5); + \draw [thick] (0.5,0.5) -- (5.5,0.5); + \node at (5.3,0.3) {$p$}; + \node at (0.3,5.3) {$q$}; + \node (x11) at (1,1) {$\bullet$}; + \node (x12) at (1,2) {$\bullet$}; + \node (x13) at (1,3) {$\bullet$}; + \node (x14) at (1,4) {$\bullet$}; + \node (x15) at (1,5) {$\bullet$}; + \node (x21) at (2,1) {$\bullet$}; + \node (x22) at (2,2) {$\bullet$}; + \node (x23) at (2,3) {$\bullet$}; + \node (x24) at (2,4) {$\bullet$}; + \node (x25) at (2,5) {$\bullet$}; + \node (x31) at (3,1) {$\bullet$}; + \node (x32) at (3,2) {$\bullet$}; + \node (x33) at (3,3) {$\bullet$}; + \node (x34) at (3,4) {$\bullet$}; + \node (x35) at (3,5) {$\bullet$}; + \node (x41) at (4,1) {$\bullet$}; + \node (x42) at (4,2) {$\bullet$}; + \node (x43) at (4,3) {$\bullet$}; + \node (x44) at (4,4) {$\bullet$}; + \node (x45) at (4,5) {$\bullet$}; + \node (x51) at (5,1) {$\bullet$}; + \node (x52) at (5,2) {$\bullet$}; + \node (x53) at (5,3) {$\bullet$}; + \node (x54) at (5,4) {$\bullet$}; + \node (x55) at (5,5) {$\bullet$}; + \path + (x12) edge[->] (x31) + (x13) edge[->] (x32) + (x14) edge[->] (x33) + (x15) edge[->] (x34) + (x22) edge[->] (x41) + (x23) edge[->] (x42) + (x24) edge[->] (x43) + (x25) edge[->] (x44) + (x32) edge[->] (x51) + (x33) edge[->] (x52) + (x34) edge[->] (x53) + (x35) edge[->] (x54) + (x42) edge (5.5,1.25) + (x43) edge (5.5,2.25) + (x44) edge (5.5,3.25) + (x45) edge (5.5,4.25) + (x52) edge (5.5,1.75) + (x53) edge (5.5,2.75) + (x54) edge (5.5,3.75) + (x55) edge (5.5,4.75) + (2,5.5) edge[->] (x35) + (3,5.5) edge[->] (x45) + (4,5.5) edge[->] (x55) + (5,5.5) edge (5.5,5.25) + ; + \end{tikzpicture} + \caption{The page $E_2^{p,q}$} + \label{fig:E2-page} + \end{subfigure}% + \begin{subfigure}{.5\textwidth} + \centering + \begin{tikzpicture} + \draw [thick] (0.5,0.5) -- (0.5,5.5); + \draw [thick] (0.5,0.5) -- (5.5,0.5); + \node at (5.3,0.3) {$p$}; + \node at (0.3,5.3) {$q$}; + \node (x11) at (1,1) {$\bullet$}; + \node (x12) at (1,2) {$\bullet$}; + \node (x13) at (1,3) {$\bullet$}; + \node (x14) at (1,4) {$\bullet$}; + \node (x15) at (1,5) {$\bullet$}; + \node (x21) at (2,1) {$\bullet$}; + \node (x22) at (2,2) {$\bullet$}; + \node (x23) at (2,3) {$\bullet$}; + \node (x24) at (2,4) {$\bullet$}; + \node (x25) at (2,5) {$\bullet$}; + \node (x31) at (3,1) {$\bullet$}; + \node (x32) at (3,2) {$\bullet$}; + \node (x33) at (3,3) {$\bullet$}; + \node (x34) at (3,4) {$\bullet$}; + \node (x35) at (3,5) {$\bullet$}; + \node (x41) at (4,1) {$\bullet$}; + \node (x42) at (4,2) {$\bullet$}; + \node (x43) at (4,3) {$\bullet$}; + \node (x44) at (4,4) {$\bullet$}; + \node (x45) at (4,5) {$\bullet$}; + \node (x51) at (5,1) {$\bullet$}; + \node (x52) at (5,2) {$\bullet$}; + \node (x53) at (5,3) {$\bullet$}; + \node (x54) at (5,4) {$\bullet$}; + \node (x55) at (5,5) {$\bullet$}; + \path + (x13) edge[->] (x41) + (x14) edge[->] (x42) + (x15) edge[->] (x43) + (x23) edge[->] (x51) + (x24) edge[->] (x52) + (x25) edge[->] (x53) + (x33) edge (5.5,1.3333) + (x34) edge (5.5,2.3333) + (x35) edge (5.5,3.3333) + (x43) edge (5.5,2) + (x44) edge (5.5,3) + (x45) edge (5.5,4) + (x53) edge (5.5,2.6667) + (x54) edge (5.5,3.6667) + (x55) edge (5.5,4.6667) + (1.75,5.5) edge[->] (x44) + (2.75,5.5) edge[->] (x54) + (3.25,5.5) edge[->] (x45) + (4.25,5.5) edge[->] (x55) + (3.75,5.5) edge (5.5,4.3333) + (4.75,5.5) edge (5.5,5) + (5.25,5.5) edge (5.5,5.3333) + ; + \end{tikzpicture} + \caption{$E_3^{p,q}$} + \label{fig:E3-page} + \end{subfigure} + \caption{Two pages of a spectral sequence.} + \label{fig:spectral-sequence-pages} + \end{figure} + + Before we start, we define the notion of a graded abelian group. We will give a + nonstandard definition that is equivalent to the standard one. + % Note that we do not assume that $R$ is a graded ring, which would be a more + % general notion. + \begin{defn}\label{def:graded} + For an abelian group $G$, an \emph{$G$-graded abelian group} is + a family of abelian groups indexed over $G$. If $M$ and $M'$ are $G$-graded + abelian groups, the type of \emph{graded abelian group homomorphism from $M$ to + $M'$} is a triple consisting of a \emph{degree} $e : G \simeq G$ (this is an equivalence of types, not a group isomorphism), a proof of + $(g : G) \to e(g)=g+e(0)$ and a term of type + $$\{x\ y : I\} \to (p : e(x)=y) \to M_x \to M'_y.$$ + We will denote the type of homomorphisms as $M\to M'$. + For $\phi:M\to M'$ we write $\deg_\phi$ for the first projection. We will + often call $\deg_\phi(0)$ the degree of $\phi$. For $x : I$ we will write + $$\phi_x\defeq\phi_{\refl_x}:M_x\to M'_{\deg_\phi(x)}$$ + and + $$\phi_{[x]}\defeq\phi_{p_x}:M_{\deg_\phi^{-1}(x)}\to M'_x$$ + where $p_x:\deg_\phi(\deg_\phi^{-1}(x))=x$ is the proof obtained from the equivalence $\deg_\phi$. + \end{defn} + \begin{rmk} + This definition looks a bit cumbersome, since the condition on $e$ forces + $e$ to be homotopic to the function $\lam{g}g+h$ for some group element $h$. + Furthermore, the type of $\phi$ is equivalently $(x : I) \to M_x \to + M'_{x+h}$. We will now discuss why we made these choices. + + % Traditionally, an abelian group is graded over a group $G$, and the degree of a graded homomorphism + % is just a group element, and a map of degree $h$ has type $(g : G) \to M_g\to M'_{g+h}$. The + % definition given here is more general, because we can define such a map as having degree + % $\lam{g}g+h:G\simeq G$. + To see why this is more convenient, we consider the composition of two graded homomorphisms. Suppose we + have two graded homomorphisms $\phi:M\to M'$ and $\psi:M'\to M''$ of degrees $h:G$ and $k:G$, + respectively. Then the pointwise composition $\lam{g:G}{m:M_g}\psi_{g+h}(\phi_g(m))$ has type + $(g : G) \to M_g \to M''_{(g+h)+k}$. So to get a graded homomorphism of degree $h+k$, with the more straightforward representation, we would need to + transport along the equality $(g+h)+k=g+(h+k)$. Since compositions are ubiquitous, this would happen all over the place. + However, in our setting, the composite of two graded homomorphisms of degree $e$ and + $e'$ will have degree $e' \o e$, without using any transports. + + We eliminated a transport to define composition, but there are other places where we cannot get rid + of them so easily. For example, given morphisms $\phi:M\to M'$ and $\psi:M'\to M''$ with + $\psi\o\phi=0$ (the graded map that is constantly 0), we are interested in the + homology of $\phi$ and $\psi$. This is the kernel of $\psi$ quotiented by the image of $\phi$ in + $M'_x$. However, if $\phi_x$ has type $M_x \to M'_{\deg_\phi(x)}$, there is no map that (without + transports) lands in $M'_x$. We would need to transport along the path + $p_x:\deg_\phi\sy(\deg_\phi(x))=x$ and take the image of this composite: + $$M_{\deg_\phi\sy(x)} \xrightarrow{\phi_{\deg_\phi\sy(x)}} M'_{\deg_\phi(\deg_\phi\sy(x))} \xrightarrow{\sim} M'_x.$$ + For this reason, we allow graded homomorphisms to be applied to paths, so that we have a ``built-in'' + transport. Then we can define the homology as $H_x\defeq \ker(\psi_x)/\im(\phi_{[x]})$, or diagramatically + $$M_{\deg_\phi\sy(x)}\xrightarrow{\phi_{[x]}}M'_x\xrightarrow{\psi_x}M''_{\deg_\psi(x)}.$$ + + For the construction of spectral sequences, we do not actually need the + second component of a graded homomorphism: all constructions also work if + the degrees are arbitrary equivalences of type $I\simeq I$, where $I$ is an + arbitrary set. This is the definition used in the formalization. In this + document we add this condition, so that our definition is equivalent to the + usual definition of graded morphism. + \end{rmk} + +\begin{defn} + A \emph{spectral sequence} consists of the following data. + \begin{itemize} + \item A sequence $E_r$ of abelian groups graded over $\Z\times\Z$ for $r\geq2$. $E_r$ is called the \emph{$r$-page} of the spectral sequence; + \item \emph{differentials}, which are graded morphisms $d_r:E_r\to E_r$ such that $d_r\circ d_r=0$; + \item isomorphisms $\alpha_r^{p,q}:H^{p,q}(E_r)\simeq E_{r+1}^{p,q}$ where $H^{p,q}(E_r)=\ker(d_r^{p,q})/\im(d_r^{[p,q]})$ is the cohomology of the cochain complex determined by $d_r$. + \end{itemize} +\end{defn} + We use the notation for cohomologically indexed spectral sequences, since we + will construct spectral sequences in cohomology in this chapter. For the + spectral sequences in this chapter, the degree of $d_r$ will be $(r,1-r)$, which + signifies a cohomologically indexed spectral sequence. + + As mentioned before, we start counting the pages at 2, since the first page of + the spectral sequences we construct will not be homotopy invariant. In the + formalization we start counting at 0 for convenience. Also, in the + formalization, we assume that the grading of $E_r$ is over some set $I$ + instead of fixing it to $\Z\times \Z$. It is not clear whether this extra + generality is useful. Instead of abelian groups, we could take objects of an + arbitrary abelian category, but for concreteness and to simplify things, we + choose to develop the theory only for abelian groups. In the formalization we + developed the theory for graded $R$-module for a (non-graded) ring $R$, but we + have only applied it to abelian groups so far. + + Note that $(E_r,d_r)$ determines $E_{r+1}$ but \emph{not} $d_{r+1}$. + Furthermore, $E_r$ is a subquotient (subgroup of a quotient) of $E_2$, so if + $E_2^{p,q}$ is trivial, then $E_r^{p,q}$ is trivial for all $r$. + + In many cases, the spectral sequence will \emph{converge}. That means that for + a fixed $(p,q):\Z\times \Z$ the sequence $E_r^{p,q}$ will be constant for $r$ + large enough. For example, suppose that the degree of $d_r$ is $(r,r-1)$, and + $E_2$ is limited to the first quadrant. Now for any $(p,q)$ all differentials + in or out of $E_r^{p,q}$ will go out the first quadrant for sufficiently large + $r$. This means that the image of $d_r^{[p,q]}$ is trivial, and the kernel of + $d_r^{p,q}$ is the full group. This implies that $E_{r+1}^{p,q}\simeq + E_r^{p,q}$, so the spectral sequence converges. + + Whenever a spectral sequence converges, we write $E_\infty^{p,q}$ for the + eventual value of $E_r^{p,q}$ for $r$ large enough. Now the power of spectral + sequences is that there is often a relation between $E_2^{p,q}$ and + $E_\infty^{p,q}$. This relation does not specify $E_\infty^{p,q}$ exactly, but + specifies that $E_\infty^{p,q}$ build up some group $D^n$ for the diagonals + where $p+q=n$. + + \begin{defn} + Suppose given an abelian group $D$ and a finite sequence of abelian groups + $(E^n)_n$. We say that \emph{$D$ is built from $(E^n)_n$} if there is a + sequence of abelian groups $(D^n)_n$ and short exact sequences + \begin{align*} + E^0\to &D \to D^1\\ + &\vdots\\ + E^k\to &D^k \to D^{k+1}\\ + E^{k+1}\to &D^{k+1} \to D^{k+2}\\ + &\vdots\\ + E^m\to &D^m \to 0\\ + \end{align*} + The sequence $(D^n)_n$ is called a \emph{cofiltration} of $D$, they are successive quotients of $D$. + \end{defn} + \begin{defn} + Given a graded abelian group $D^n$ and a bigraded abelian group $C^{p,q}$, we write + $$E_2^{p,q}=C^{p,q}\Rightarrow D^{p+q}$$ + if there is a spectral sequence $E$ such that + \begin{itemize} + \item $E_2^{p,q}=C^{p,q}$; + \item $E$ converges to $E_\infty$; + \item $D^n$ is built from $E_\infty^{p,q}$ where $p+q=n$. + \end{itemize} + \end{defn} + \begin{rmk} + This definition implicitly requires that for $p+q=n$ only finitely many $E_\infty^{p,q}$ are nontrivial. + This is sufficient for the spectral sequences we consider in this chapter, but this condition can be relaxed in more general constructions of spectral sequences. + \end{rmk} + + % We can define the ordinary (unreduced) cohomology of a type $X$ with coefficients in an abelian group $A$ to be + % $$H^n(X;A):=\|X\to K(A,n)\|_0$$ + % Here $K(A,n)$ is the Eilenberg-MacLane space (see \autoref{sec:eilenb-macl-spac}). We will see a more general definition in \autoref{sec:spectra}. + +\section{Exact Couples}\label{sec:exact-couples} + +As we said before, the pair $(E_r,d_r)$ in a spectral sequence specifies +$E_{r+1}$, but not $d_{r+1}$. If we have some more information about page $r$, +then we can construct page $r+1$ and the extra information for page $r+1$. Now +we can iterate this construction and obtain a spectral sequence by forgetting +about the extra information. + +An \emph{exact couple} exactly gives this extra information~\cite{massey1952exactcouple}. From it, we can compute the \emph{derived exact +couple}, which gives us the information next page of the spectral sequence. +\begin{defn} +An \emph{exact couple} is a pair $(D,E)$ of $\Z\times\Z$-graded abelian groups with graded homomorphisms +\begin{center} + \begin{tikzpicture}[thick,node distance=1cm] + \node (tl) at (0,0) {$D$}; + \node[below right = of tl] (b) {$E$}; + \node[above right = of b] (tr) {$D$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above] {$i$} (tr) + (tr) edge[->] node [below right] {$j$} (b) + (b) edge[->] node [below left] {$k$} (tl); + \end{tikzpicture} +\end{center} +that is exact in all three vertices. This means that for all $p:\deg_j(x)=_Iy$ and +$q:\deg_k(y)=z$ that $\ker(k_q)=\im(j_p)$, and similarly for the other two pairs of maps. + +For an exact couple we will write $\iota\defeq\deg_i$ and $\eta\defeq\deg_j$ and $\kappa\defeq\deg_k$ for the degrees. +\end{defn} +\begin{lem} +Given an exact couple $(D,E,i,j,k)$, we can define a \emph{derived exact couple} $(D',E',i',j',k')$ +where $E'$ is the homology of $d\defeq j\o k:E\to E$. The degrees of the derived maps are +$\deg_{i'}\equiv\iota$, $\deg_{k'}\equiv\kappa$ and $\deg_{j'}\equiv\eta\o\iota\sy$. +\begin{center} + \begin{tikzpicture}[thick,node distance=1cm] + \node (tl) at (0,0) {$D'$}; + \node[below right = of tl] (b) {$E'$}; + \node[above right = of b] (tr) {$D'$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above] {$i'$} (tr) + (tr) edge[->] node [below right] {$j'$} (b) + (b) edge[->] node [below left] {$k'$} (tl); + \end{tikzpicture} +\end{center} +\end{lem} +\begin{proof} + In this proof we will be explicit about the grading of $D$ and $E$, which is a lot trickier (at least in intensional type theory) than a proof without the grading. For a proof that does not take the grading into account, see for example~\cite[Lemma 1.1]{hatcher2004spectral}. + We define for $x:\Z\times \Z$ the graded abelian groups $D'$ and $E'$ by $D_x'=\im i_{[x]}$ and $E_x'=\ker d_x/\im d_{[x]}$. Now $i_x':D_x'\to D_{\iota x}'$ is defined as the composite + $$D_x' \hookrightarrow D_x \xrightarrow{i} D_{\iota x}'.$$ + This is sufficient to define $i'$ on all paths $\iota x=y$ as a function $D_x \to D_y$.\\ + We first define $j_{px}':D_{\iota x}'\to E_{\eta x}'$ for the canonical path $px:\eta(\iota\sy(\iota x))=\eta x$, which is sufficient to define $j'$ in general. + Note that $D_{\iota x}'\equiv \im i_{[\iota x]}\simeq\im i_x$, so to define $j_{px}'$ it is sufficient to define $\tilde\jmath : D_x \to E_{\eta x}'$ such that $(a : D_x) \to i_x(a)=0 \to \tilde\jmath(a)=0$. We define $\tilde\jmath(a)\defeq [j_xa]$. This is well-defined, since $j_xa\in\ker d_{\eta x}$,\footnote{We use the set-theoretical notation $g\in H$ to say that a group element $g:G$ is in subgroup $H$. Formally, a subgroup $H$ is an element of $G\to\prop$ (containing 0, and closed under addition and negation) and $g\in H$ is defined as $H(g)$. Note that $(g:G)\times H(g)$ can be endowed with a group structure, which is $H$ viewed as a group.} because + $$d_{\eta x}(j_xa)=j_{\kappa(\eta x)}(k_{\eta x}(j_xa))=j_{\kappa(\eta x)}\const=\const.$$ + Now suppose that $i_x(a)=0$. Without loss of generality we may assume that $x\equiv \kappa y$. By exactness, this means that $a\in\im k_y$, so there is $b:E_y$ such that $k_y(b)=a$. + Now $j_xa=j_x(k_yb)\equiv d_yb$, so + $$j_xa\in\im d_y\simeq \im d_{[\deg_d y]}\equiv \im d_{[\eta x]}.$$ + This shows that $\tilde\jmath(a)=0$, completing the definition of $j'$. Note that + $j_{px}'(i_xa)=[j_xa]$. + To define $k_x':E_x'\to D_{\kappa x}'$, first note that if $a\in\ker d_x$, then $k_xa\in\ker j_{\kappa x}=\im i_{[\kappa x]}$ by exactness. Now we need to show that if $a\in\im d_{[x]}$, then $k_xa=0$. By assumption, we have $b:E_{\eta\sy\kappa\sy x}$ such that $d_{[x]}(b)=a$. Now we compute (using $k_xj_{[x]}=\const$) + $$k_xa=k_x(d_{[x]}b)=k_x(j_{[x]}(k_{[\eta\sy x]}b))=0.$$ + This defines $k$. + + Showing exactness of the derived couple involves some diagram chasing. + To show that $j'i'=\const$ it is sufficient to show that for all $a:D_{\iota x}'$ we have + $j'_{p(\iota x)}(i'_{\iota x}a)=0$. Since $a\in\im i_{[\iota x]}\simeq i_x$ we know that $a=i_x b$ for some $b: D_x$. We compute + $$j'_{p(\iota x)}(i'_{\iota x}a)=j'_{p(\iota x)}(i_{\iota x}a)=[j_{\iota x}a]=[j_{\iota x}(i_x b)]=[0]=0.$$ + To show that $\ker j'\subseteq \im i'$, it is sufficient to show that + $\ker j'_{p(\kappa x)}\subseteq \im i'_{\kappa x}$. Suppose $a : D'_{\iota(\kappa x)}$ such that $j'_{p(\kappa x)}(a)=0$, we know that $a=i_{\kappa x}(b)$ for some $b$. Now + $$0=j'_{p(\kappa x)}(a)=j'_{p(\kappa x)}(i_{\kappa x}(b))=[j_{\kappa x}(b)],$$ + which means that $j_{\kappa x}(b)\in\im d_{[{\kappa (\eta x)}]}\simeq \im d_x$. This means that for some $c: E_x$ we have $j_{\kappa x}(b)=d_x(c)=j_{\kappa x}(k_x c)$. This means that + $j_{\kappa x}(b-k_x c)=0$, hence $b-k_xc\in \ker j_{\kappa x}=\im i_{[\kappa x]}$. This means that we can define $b-k_xc: D'_{\kappa x}$. Now we compute + $$i'_{\kappa x}(b-k_xc)=i_{\kappa x}b-i_{\kappa x}(k_xc)=a-0=a,$$ + which means $a\in\im i'_{\kappa x}$, as desired. + + We will omit the other cases, which are similar but easier. + %%possibly : clarify notation x \in \ker \phi ?? +\end{proof} +Repeating the process of deriving exact couples, we get a sequence of exact couples $(D_r,E_r,i_r,j_r,k_r)$.\footnote{We will now put the grading of $D$, $E$ and the maps as superscript, so that we can put the page as subscript.} We get a spectral sequence $(E_r,d_r)$ where $d_r\defeq j_r\o k_r$. Note that +$$\deg_{d_r}=\deg_{j_r}\o\deg_{k_r}=\eta\o\iota^r\o\kappa%\defeqr f_r +$$ + +Given some extra conditions on the exact couple, we can show that this spectral sequence converges. +% Spectral sequences converge under some boundedness assumptions of the initial exact couple. In~\cite{shulman2013spectral} the boundedness conditions is formulated in terms of the iterated +% fibration sequence from which we construct an exact couple. However, it should be possible to +% formulate it directly in terms of an exact couple. The following is a first attempt, which might not +% work exactly. The conditions and/or theorem statements might need to be changed. +\begin{defn}\label{def:bounded} +We call an exact couple \emph{bounded} if for every $x:\Z\times\Z$ there is are bounds $B_x : \N$ such that for all $s \geq B_x$ we have +$$E^{\iota^{-s}(x)}=0 \qquad\text{and}\qquad D^{\iota^s(x)}=0$$ +\end{defn} +\begin{rmk}\label{rmk:bounded} + The condition on $D$ also shows that if you go sufficiently far in the $\iota$-direction, then $E$ is trivial, since $D \xrightarrow{j} E \xrightarrow{k} D$ is exact and the occurrences of $D$ will be trivial. Converse, the condition on $E$ shows that if you go sufficiently far in the + $\iota^{-1}$ direction, $i$ will be an equivalence, by the following exact sequence. + $$E \xrightarrow{k} D \xrightarrow{i} D \xrightarrow{j} E$$ + We call $x:\Z\times\Z$ a \emph{stable index} whenever $i^{[\iota^{-s}x]}$ is surjective for \emph{all} $s\geq0$. +\end{rmk} +Given a bounded exact couple, the pages stabilize pointwise, which is the content of the next lemma. + +\begin{lem}\label{lem:exact-couple-stabilize} + For a bounded exact couple $(D,E,i,j,k)$ we have for all sufficiently large $r$ that $D_{r+1}^x=D_r^x$ and $E_{r+1}^x=E_r^x$. +\end{lem} +\begin{proof} + Note that $E_{r+1}^x=\ker d_r^{x}/\im d_r^{[x]}$. Since $d_r$ has degree + $\eta\o\iota^r\o\kappa$, and because $\Z\times \Z$ is an abelian group, the + degrees commute.\footnote{In the formalization, we do not assume that the + degrees are shifts by a group element, and we explicitly assume that + $\kappa\iota=\iota\kappa$ and $\iota\eta=\eta\iota$.} The codomain of $d_r^x$ + is $E_r^{\eta(\iota^r(\kappa x))}=E_r^{\iota^r(\eta(\kappa x))}$, which is + trivial for sufficiently large $r$ by \autoref{rmk:bounded}. Also, the domain + of $d_r^{[x]}$ is $E_r^{\kappa\sy(\iota^{-r}(\eta\sy + x))}=E_r^{\iota^{-r}(\kappa\sy(\eta\sy x))}$, which is trivial for sufficiently large $r$ by the definition of boundedness. + + To show that $D$ stabilizes, first note that if $i_r^{[\iota\sy x]}$ is surjective, then $i_{r+1}^{[x]}$ is surjective. The reason is that + $$i_{r+1}: D_{r+1}^{\iota\sy x}\xrightarrow{\sim} D_r^{\iota\sy x} \to D_{r+1}^{x}$$ + is now a composite of two surjective maps. This means that if the maps $i_{r_0}^{[\iota^{-s}x]}$ are surjections for all $s\geq B+1$, then the maps $i_{r_0+1}^{[\iota^{-s}x]}$ will be surjections for all $s\geq B$. In this case, for $r\geq r_0+B$ we have that $i_r^{[x]}$ is a surjection, hence that $D_{r+1}^x=D_r^x$. Since $i_0^{[\iota^{-s}x]}$ are surjections for sufficiently large $s$ by \autoref{rmk:bounded}, we finish the proof. +\end{proof} + +By the proof of \autoref{lem:exact-couple-stabilize} we get explicit bounds $B_x^D$ and $B_x^E$ such that $D_r^x=D_{B_x^D}^x$ and $E_{r'}^x=E_{B_x^E}^x$ for all $r\geq B_x^D$ and $r'\geq B_x^E$. We define $D_\infty^x\defeq D_{B_x^D}^x$ and $E_\infty^x\defeq E_{B_x^E}^x$. Both $B_x^D$ and $B_x^E$ will be the maximum of $B_y$ for some sequence of indices $y$. + +\begin{thm}[Convergence Theorem]\label{thm:exact-couple-convergence} +Let $(D,E,i,j,k)$ be a bounded exact couple and let $x$ be a stable index. +Then $D^{\kappa x}$ is built from $(E_\infty^{\iota^n(x)})_{0\le n < B_{\kappa x}}$. +\end{thm} +\begin{proof} + Define + $C^n = D_\infty^{\kappa(\iota^nx)}$. Let $n:\N$ be arbitrary, then for sufficiently large $r$ the following is a short exact sequence + $$0 \xrightarrow{j_r} E_r^{\iota^nx} \xrightarrow{k_r} D_r^{\kappa(\iota^nx)} \xrightarrow{i_r} + D_r^{\iota(\kappa(\iota^nx))} \xrightarrow{j_r} 0.$$ + This is the case, because for sufficiently large $r$ the domain of $j_r^{[\iota^nx]}$ and the codomain of $j_r{\iota(\kappa(\iota^nx))}$ are contractible. Now (possibly by increasing $r$) these groups are in the stable range, so we get a short exact sequence + $$0 \to E_\infty^{\iota^nx} \to C^n \to C^{n+1} \to 0.$$ Moreover, we have + $C^0\equiv D_\infty^{\kappa x}=D^{\kappa x}$ because $x$ is a stable index. + Lastly, for $s\geq B_{\kappa x}$ we know that $C^s$ is trivial, because + $D^{\kappa(\iota^n)}$ is trivial by the condition of being bounded. This shows + that $D^{\kappa x}$ is built from $(E_\infty^{\iota^n(x)})_{0\le n < B_{\kappa + x}}$. +\end{proof} + +\section{Spectra}\label{sec:spectra} + +We have not yet discussed how to get an exact couple in the first place. Recall that from a pointed +map we get a long exact sequence of homotopy groups. For a \emph{sequence} of pointed maps we get a sequence of long exact sequences. However, we do not want to do this for pointed maps, but for maps between \emph{spectra}. + +You can think of a spectrum as a generalized space with negative dimensions. Suppose we are given a pointed type $X$ and a chosen delooping $Y$ of $X$. That is, $Y$ is a pointed type such that $\Omega Y\simeq^* X$. Now the $(n+1)$-th homotopy group of $Y$ is equal to the $n$-th homotopy group of $X$. The $0$-th homotopy group of $Y$ is new information, and we can think of it as the $(-1)$-th homotopy group of $X$. Spectra go further on this idea: it is a pointed type with infinitely many deloopings. + +\begin{defn} + A \emph{prespectrum} is a pair consisting of a sequence of pointed types $Y:\Z\to\type^*$ and a sequence of pointed maps $e:(n:\Z) \to Y_n \to^* \Omega Y_{n+1}$. An \emph{$\Omega$-spectrum} or \emph{spectrum} is a prespectrum $(Y,e)$ where $e_n$ is a pointed equivalence for all $n$. We will often just write $Y$ for the pair $(Y,e)$, and we denote the type of (pre)spectra by $\prespectrum$ and $\spectrum$.\\ + A map between (pre)spectra $(Y,e)\to (Y',e')$ is a pair consisting of $f:(n : \Z) \to Y_n \to Y_n'$ and $p : (n : \N) \to e_n' \o f_n \sim^* \Omega f_{n+1} \o e_n$. +\end{defn} + +\begin{rmk} + Usually a (pre)spectrum is indexed over $\N$ and not over $\Z$. We index it over $\Z$ so that we do not have to do a case split in --- for example --- the definition of homotopy group of a spectrum, see \autoref{def:spectrum-homotopy-group}. +\end{rmk} + +\begin{ex}\mbox{} + \begin{itemize} + \item If $A$ is an abelian group, we have $HA:\spectrum$ where $(HA)_n=K(A,n)$ for $n\geq0$ and $(HA)_n=\unit$ for $n<0$. + \item Given $Y:\spectrum$ and $k:\Z$, we can define two new spectra $\Omega^kY$ and $\susp^k Y:\spectrum$ with + \begin{align*} + (\Omega^kY)_n&\defeq Y_{n-k}&(\susp^kY)_n&\defeq Y_{n+k} + \end{align*} + \item Given a spectrum map $f: X \to Y$, we have a spectrum $\fib_f:\spectrum$ with $(\fib_f)_n\defeq \fib_{f_n}$. Furthermore we have a spectrum map $p_1:\fib_f\to X$. This follows from the following two facts about fibers (which we will not prove here). + \begin{enumerate} + \item Given a pointed map $g : A \to B$, there is a pointed equivalence $e_1:\Omega\fib_g\simeq^*$ with a pointed homotopy + \begin{center} + \begin{tikzpicture}[thick,node distance=1cm] + \node (tl) at (0,0) {$\Omega\fib_g$}; + \node[below right = of tl] (r) {$A$}; + \node[below left = of r] (bl) {$\fib_{\Omega g}$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above right] {$\Omega p_1$} (r) + (bl) edge[->] node [below right] {$p_1$} (r) + (tl) edge[->] node [left] {$e_1$} (bl); + \end{tikzpicture} + \end{center} + \item $\fib$ is a functor from pointed maps to pointed types and $p_1$ is a natural transformation. This means the following. Suppose we are given a square of pointed maps and a homotopy filling the following square. + \begin{center} + \begin{tikzpicture}[thick,node distance=1cm] + \node (tl) at (0,0) {$A$}; + \node[right = of tl] (tr) {$A'$}; + \node[below = of tl] (bl) {$B$}; + \node (br) at (tr |- bl) {$B'$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above] {$f$} (tr) + edge[->] node [right] {$h$} (bl) + (bl) edge[->] node [above] {$g$} (br) + (tr) edge[->] node [right] {$h'$} (br); + \end{tikzpicture} + \end{center} + Then there is a pointed map $e_2:\fib_f\to\fib_g$, functorial in $(h,h')$. In particular this means that if $h$ and $h'$ are equivalences, then $e_2$ is. The naturality of $p_1$ means that we have the following pointed homotopy. + \begin{center} + \begin{tikzpicture}[thick,node distance=1cm] + \node (tl) at (0,0) {$\fib_f$}; + \node[right = of tl] (tr) {$A$}; + \node[below = of tl] (bl) {$\fib_g$}; + \node (br) at (tr |- bl) {$B$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node [above] {$p_1$} (tr) + edge[->] node [right] {$e_2$} (bl) + (bl) edge[->] node [above] {$p_1$} (br) + (tr) edge[->] node [right] {$h$} (br); + \end{tikzpicture} + \end{center} + \end{enumerate} + \end{itemize} +\end{ex} + +Given an $\Omega$-spectrum $Y$ and $n : \Z$, we define can the $n$-th homotopy group of $Y$ to be +$$\pi_n(Y)\defeqp\pi_{n+k}(Y_k):\abgroup$$ +for any $k$ such that $n+k\geq 0$. This is independent of $k$, because +$$\pi_{n+(k+1)}(Y_{k+1})\simeq \pi_{n+k}(\Omega Y_{k+1})\simeq \pi_{n+k}(Y_k).$$ +For concreteness, in the following definition we pick $k=2-n$. We make this choice so that $\pi_n(Y)$ directly carries the structure of an abelian group. + +The homotopy group of a prespectrum $Y$ is a bit different, since $\pi_{n+k}(Y_k)$ is not independent of $k$. In this case, it is the colimit as $k\to\infty$. We make the substitution +$\ell=n+k-2$ to make the index of the homotopy group always positive. + +\begin{defn}\label{def:spectrum-homotopy-group} + Given an $\Omega$-spectrum $Y$ and $n : \Z$, we define the \emph{$n$-th homotopy group of $Y$} as + $$\pi_n(Y)\defeq \pi_2(Y_{2-n}).$$ + For a prespectrum $Y$ we define + $$\pi_n(Y)\defeq\colim_{\ell\to\infty}(\pi_{\ell+2}(Y_{\ell+2-n})).$$ +\end{defn} + +Note that the homotopy group of a prespectrum is a set by \autoref{cor:trunc_colim}\ref{part:colim_is_trunc}, +and the colimit can be equipped with a group structure, making $\pi_n(Y)$ an abelian group for a prespectrum $Y$. + +The long exact sequence of homotopy groups for pointed types, constructed in \autoref{sec:les-homotopy}, induces one on spectra. + +\begin{thm}\label{thm:spectrum-LES} + Given a spectrum map $f:X\to Y$ with fiber $F$, we get the following long exact sequence of homotopy groups indexed over $\Z\times \fin_3$. + \begin{center}\begin{tikzpicture}[thick, node distance=18mm] + \node (Y) at (0,0) {$\pi_k(Y)$}; + \node[left = of Y] (X) {$\pi_k(X)$}; + \node[left = of X] (F) {$\pi_k(F)$}; + \node[above = of Y] (OY) {$\pi_{k+1}(Y)$}; + \node[above = of X] (OX) {$\pi_{k+1}(X)$}; + \node[above = of F] (OF) {$\pi_{k+1}(F)$}; + \node[above = of OY] (O2Y) {$\pi_{k+2}(Y)$}; + \node[above = of OX] (O2X) {$\pi_{k+2}(X)$}; + \node[above = of OF] (O2F) {$\pi_{k+2}(F)$}; + \node[above = 5mm of O2X] {$\vdots$}; + \node[below = 5mm of X] {$\vdots$}; + % \node[above = 4mm of O2X] (O3X) {$\vdots$}; + \path[every node/.style={font=\sffamily\small}] + (X) edge[->] node [above] (f){$\pi_k(f)$} (Y) + (F) edge[->] node [below] (f){$\pi_k(p_1)$} (X) + (OY) edge[->] (F) + (OX) edge[->] node [above] (f){$\pi_{k+1}(f)$} (OY) + (OF) edge[->] node [below] (f){$\pi_{k+1}(p_1)$} (OX) + (O2Y) edge[->] (OF) + (O2X) edge[->] node [above] (f){$\pi_{k+2}(f)$} (O2Y) + (O2F) edge[->] node [below] (f){$\pi_{k+2}(p_1)$} (O2X); +\end{tikzpicture} +\end{center} +\end{thm} +We will use the following lemma. Recall the definition of successor structure from \autoref{def:chain-complex}. +\begin{lem}\label{lem:splice} + Suppose given two successor structures $N$ and $M$, and for each $n:N$ let $G^n$ be a long exact sequence index by $M$. Let $m:M$ and $k\geq 2$. Suppose that + \begin{itemize} + \item for all $n:N$, $G_m^{n+1} \simeq G_{m+k}^n$ and $G_{m+1}^{n+1} \simeq G_{m+k+1}^n$ + \item for all $n:N$ the following diagram commutes. + \begin{center}\begin{tikzpicture}[thick, node distance=10mm] + \node (Y) at (0,0) {$G_{m+k}^n$}; + \node[left = of Y] (X) {$G_{m+k+1}^n$}; + \node[above = 12mm of Y] (OY) {$G_m^{n+1}$}; + \node[above = 12mm of X] (OX) {$G_{m+1}^{n+1}$}; + \path[every node/.style={font=\sffamily\small}] + (X) edge[->] (Y) + (OY) edge[->] node [sloped, above] {$\sim$} (Y) + (OX) edge[->] node [sloped, above] {$\sim$} (X) + (OX) edge[->] (OY); + \end{tikzpicture}\end{center} + \end{itemize} + Then there is a long exact sequence $H:N\times\fin_{k-1}\to\set^*$ with $H_{(n,\ell)}\defeq G_{m+\ell}^n$ +\end{lem} +For $k=3$ the hypotheses can be represented in the diagram below. +\begin{center}\begin{tikzpicture}[thick, node distance=10mm] +\node (Y) at (0,0) {$G_m^n$}; +\node[left = of Y] (X) {$G_{m+1}^n$}; +\node[left = of X] (F) {$G_{m+2}^n$}; +\node[left = of F] (G) {$G_{m+3}^n$}; +\node[left = of G] (H) {$G_{m+4}^n$}; +\node[left = of H] (I) {$\cdots$}; +\node[above = 12mm of Y] (OY) {$G_m^{n+1}$}; +\node[above = 12mm of X] (OX) {$G_{m+1}^{n+1}$}; +\node[above = 12mm of F] (OF) {$G_{m+2}^{n+1}$}; +\node[above = 12mm of G] (OG) {$G_{m+3}^{n+1}$}; +\node[above = 12mm of H] (OH) {$G_{m+4}^{n+1}$}; +\node (OI) at (I |- OH) {$\cdots$}; +\node[above = 12mm of OY] (O2Y) {$G_{m}^{n+2}$}; +\node[above = 12mm of OX] (O2X) {$G_{m+1}^{n+2}$}; +\node[above = 12mm of OF] (O2F) {$G_{m+2}^{n+2}$}; +\node[above = 12mm of OG] (O2G) {$G_{m+3}^{n+2}$}; +\node[above = 12mm of OH] (O2H) {$G_{m+4}^{n+2}$}; +\node (O2I) at (OI |- O2H) {$\cdots$}; +\node[above = 5mm of O2F] {$\vdots$}; +\node[below = 5mm of F] {$\vdots$}; +\path[every node/.style={font=\sffamily\small}] +(X) edge[->] (Y) +(F) edge[->] (X) +(G) edge[->] (F) +(H) edge[->] (G) +(I) edge[->] (H) +(OY) edge[->] node [sloped, below, pos = 0.4] {$\sim$} (G) +(OX) edge[->] node [sloped, above, pos = 0.6] {$\sim$} (H) +(OX) edge[->] (OY) +(OF) edge[->] (OX) +(OG) edge[->] (OF) +(OH) edge[->] (OG) +(OI) edge[->] (OH) +(O2Y) edge[->] node [sloped, below, pos = 0.4] {$\sim$} (OG) +(O2X) edge[->] node [sloped, above, pos = 0.6] {$\sim$} (OH) +(O2X) edge[->] (O2Y) +(O2F) edge[->] (O2X) +(O2G) edge[->] (O2F) +(O2H) edge[->] (O2G) +(O2I) edge[->] (O2H); +\end{tikzpicture}\end{center} +\begin{proof}[Proof (\autoref{lem:splice})] + The map $H_{(n,\ell+1)}\to H_{(n,\ell)}$ is defined to be the given map $G_{m+\ell+1}^n\to G_{m+\ell}^n$. The map $H_{(n+1,0)}\to H_{(n,k-1)}$ is defined to be the composite + $$G_m^{n+1}\xrightarrow{\sim}G_{m+k}^n\to G_{m+k-1}^n.$$ + It is easy to check that this is a long exact sequence from the conditions. +\end{proof} +\begin{proof}[Proof (\autoref{thm:spectrum-LES})] +For each $n:\Z$ we get a long exact sequence of homotopy groups for $f_{2-n}$ by +\autoref{thm:les-homotopy}. We splice them together using \autoref{lem:splice} with $N=(\Z,\lam{n}n+1)$ and $M=(\N,\lam{n}n+1)$ and with $k=3$ and $m=(2,0)$. This means that the resulting sequence is +\begin{center}\begin{tikzpicture}[thick, node distance=18mm] + \node (Y) at (0,0) {$\pi_2(Y_{2-n})$}; + \node[left = of Y] (X) {$\pi_2(X_{2-n})$}; + \node[left = of X] (F) {$\pi_2(F_{2-n})$}; + \node[above = of Y] (OY) {$\pi_2(Y_{2-(n+1)})$}; + \node[above = of X] (OX) {$\pi_2(X_{2-(n+1)})$}; + \node[above = of F] (OF) {$\pi_2(F_{2-(n+1)})$}; + \node[above = of OY] (O2Y) {$\pi_2(Y_{2-(n+2)})$}; + \node[above = of OX] (O2X) {$\pi_2(X_{2-(n+2)})$}; + \node[above = of OF] (O2F) {$\pi_2(F_{2-(n+2)})$}; + \node[above = 5mm of O2X] {$\vdots$}; + \node[below = 5mm of X] {$\vdots$}; + % \node[above = 4mm of O2X] (O3X) {$\vdots$}; + \path[every node/.style={font=\sffamily\small}] + (X) edge[->] node [above] (f){$\pi_2(f)$} (Y) + (F) edge[->] node [below] (f){$\pi_2(p_1)$} (X) + (OY) edge[->] (F) + (OX) edge[->] node [above] (f){$\pi_2(f)$} (OY) + (OF) edge[->] node [below] (f){$\pi_2(p_1)$} (OX) + (O2Y) edge[->] node [left] (f){$\pi_2(\delta)\quad\mbox{}$} (OF) + (O2X) edge[->] node [above] (f){$\pi_2(f)$} (O2Y) + (O2F) edge[->] node [below] (f){$\pi_2(p_1)$} (O2X); +\end{tikzpicture} +\end{center} +We still need to check the conditions for the Lemma. The first isomorphism is given by the following composition +$$\pi_2(Y_{2-(n+1)})\simeq\pi_2(\Omega Y_{2-(n+1)+1})\simeq\pi_2(\Omega Y_{2-n})\equiv \pi_3(Y_{2-n}),$$ +The second isomorphism is the same, replacing $Y$ by $X$. The square commutes because the two isomorphisms are both natural in $Y$. +\end{proof} + +Suppose given a sequence of spectra $A$ and a sequence of spectrum maps +$$\cdots \to A_{s} \xrightarrow{f_{s}} A_{s-1} \xrightarrow{f_{s-1}} A_{s-2} \to \cdots $$ +Let $B_s\defeq\fib_{f_s}$. Then $D^{n,s}\defeq\pi_n(A_s)$ and $E^{n,s}\defeq\pi_n(B_s)$ are +graded abelian groups and the maps of the long exact sequences become graded +homomorphisms. This gives exactly the data of an exact couple. + +For cohomology, it is customary to reindex the pages of the spectral sequence +with the base change $(p,q)=(s-n,-s)$, or equivalently $(n,s)=(-(p+q),-q)$. + +\begin{thm}\label{thm:spectral-sequence-spectrum} + Given a sequence of spectra + $$\cdots \to A_{s} \xrightarrow{f_{s}} A_{s-1} \xrightarrow{f_{s-1}} A_{s-2} + \to \cdots $$ with fibers $B_s\defeq\fib_{f_s}$, suppose for all $n$ there is + a $\beta_n$ such that for all $s\le \beta_n$ we have $\pi_n(A_s)=0$ and suppose that + for all $n$ there is a $\gamma_n$ such that for all $s>\gamma_n$ the map $\pi_n(f_s)$ is + an isomorphism. Then the exact couple constructed from this sequence is + bounded. This spectral sequence gives + %%$$E_2^{n,s}=\pi_{n}(B_s)\Rightarrow \pi_{n}(A_{s_0}).$$ + $$E_2^{p,q}=\pi_{-(p+q)}(B_{-q})\Rightarrow \pi_{-(p+q)}(A_{\gamma_{-(p+q)}}).$$ +\end{thm} +\begin{proof} + Note that for this spectral sequence we have + $\iota(n,s)\equiv\deg_i(n,s)\equiv(n,s-1)$ and $\kappa=\id$. This means that + we need to show that for all $(n,s):\Z\times\Z$ there is a bound $\beta'_{n,s}$ + such that for all $t\geq \beta'_{n,s}$ we have + $$E^{n,s+t}\equiv \pi_n(B_{s+t})=0\quad\text{ and }\quad D^{n,s-t}\equiv\pi_n(A_{s-t})=0.$$ + Note that the right equation holds if $s-t\le \beta_n$, i.e. if $t\ge s-\beta_n$. By + the long exact sequence of homotopy groups we know that if $f_s:A_s\to + A_{s-1}$ induces an equivalence on both $\pi_n$ and $\pi_{n+1}$, then + $\pi_n(B_{s+t})=0$. So if we define + $$\beta'_{n,s}\defeq\max(s-\beta_n,\gamma_n-s,\gamma_{n+1}-s),$$ + we know that the exact couple is bounded with bound $\beta'$. + + Now note that $x=(n,\gamma_n)$ is a stable index, because $\pi_n(f_{\gamma_n+t})$ is + surjective for all $t\geq0$. Therefore, by + \autoref{thm:exact-couple-convergence} we know that $D^{n,\gamma_n}$ is built from + $(E_\infty^{n,\gamma_n-s})_{0\le s\le \beta'_{n,\gamma_n}}$. If we apply the reindexing + $(p,q)=(s-n,-s)$, we get the desired relation + $$E_2^{p,q}=\pi_{-(p+q)}(B_{-q})\Rightarrow \pi_{-(p+q)}(A_{\gamma_{-(p+q)}}).$$ + % the second page becomes + % $$E_2^{n,s} = E_2^{-(p+q),-q} = \pi_{-(p+q)}(B_{-q}),$$ + % and that $\pi_m(A_{\gamma_m})\equiv D^{m,\gamma_m}$ is built from $(E_\infty^{m,s})_s$, which is the subspace where $n=-(p+q)$ is constant. +\end{proof} + +\section{Spectral Sequences for Cohomology}\label{sec:spectral-sequence-cohomology} + +Cohomology groups are algebraic invariants of types. They are often easier to +compute than homotopy groups, but they can also be used to compute certain +homotopy groups, often via the universal coefficient theorem and the Hurewicz +theorem (neither of which have been proven in HoTT yet). + +The intermediate steps of most classical constructions of the singular cohomology are not homotopy invariant. +\emph{Cellular cohomology} is only defined for cell complexes and not for arbitrary +spaces, but it can be defined in HoTT~\cite{buchholtz2018cellular}. \emph{Singular +cohomology} is defined as a quotient of a large abelian group that is not +homotopy invariant, which makes this definition impossible in HoTT. However, +classically, Eilenberg-MacLane spaces represent cohomology, and we can use this +fact as the \emph{definition} of cohomology in HoTT~\cite{cavallo2015cohomology}. + +Normally cohomology groups have coefficients in an abelian group, but more +generally they can have coefficients in a spectrum, or even a family of spectra. +In this section we will define cohomology groups and construct the +Atiyah-Hirzebruch spectral sequence for cohomology. This is a generalization of +the spectral sequence defined in~\cite{atiyah1961spectral} in the special case +of topological K-theory. From the Atiyah-Hirzebruch spectral sequence we can +construct the Serre spectral sequence, sometimes also called the Leray-Serre spectral sequence. +\begin{defn} + Suppose given $X:\type^*$ and $Y:X\to\spectrum$. We define $(x:X)\to^* Yx:\spectrum$ such that + $((x:X)\to^* Yx)_n\defeq (x:X)\to^*(Yx)_n$. If $Y$ does not depend on $X$, we write $X\to^* Y$. + + For an unpointed type $X:\type$ and $Y:X\to\spectrum$, we similarly define $(x:X)\to Yx:\spectrum$ such that + $((x:X)\to Yx)_n\defeq (x:X)\to(Yx)_n$ (this has as basepoint the constant map into the basepoint of $(Yx)_n$), and abbreviate this to $X\to Y$ if $Y$ does not depend on $X$. +\end{defn} +These spectra are well-defined, since we have +$$\Omega((a:A)\to^*(Ba))\simeq (a:A)\to^*\Omega(Ba)$$ +and +$$\Omega((a:A)\to(Ba))\simeq (a:A)\to\Omega(Ba).$$ +Moreover, they satisfy the expected properties of dependent product. In particular, if $X:\type^*$ and $Y,Z:X\to\spectrum$ and moreover if we have a fiberwise spectrum map $f:(x:X)\to Yx \to Zx$, this induces a map on the dependent products +$$\Pi_f:((x:X)\to Yx)\to ((x:X)\to Zx).$$ +\begin{defn}\label{def:cohomology} + Suppose given $X:\type^*$, $Y:X\to\spectrum$ and $n:\Z$. We define the \emph{geneneralized, parametrized, reduced cohomology} of $X$ with coefficients in $Y$ as\footnote{We will write $\lam{x}Yx$ in $\eta$-expanded form to remember that this is parametrized cohomology.} + $$\tilde H^n(X;\lam{x}Yx)\defeq \pi_{-n}((x:X)\to^* Yx)\simeq \|(x:X)\to^* (Yx)_n\|_0.$$ + If $Y$ does not depend on $X$, we have the \emph{unparametrized cohomology} as + $$\tilde H^n(X;Y)\defeq \pi_{-n}(X\to^* Yx)\simeq \|X\to^* Y_n\|_0.$$ + If $X:\type$ is an arbitrary type, we define the \emph{unreduced cohomology} as + $$H^n(X;\lam{x}Yx)\defeq \pi_{-n}((x:X)\to Yx)\simeq \|(x:X)\to (Yx)_n\|_0\simeq \tilde H^n(X_+;\lam{x}Y_+x).$$ + Here $X_+\defeq X+1:\type^*$ and $Y_+:X_+\to\spectrum$ is defined as $Y_+(\inl(x))\defeq Yx$ and $Y_+(\inr(\star))\defeq 1$. + If $X:\type^*$ and $A:X\to\abgroup$, we define the \emph{ordinary cohomology} as + $$\tilde H^n(X;\lam{x}Ax)\defeq \tilde H^n(X;\lam{x}H(Ax).$$ + We can combine the attributes ordinary/generalized, parametrized/unparametrized and reduced/unreduced for cohomology however we want, leading to eight different notions. + + We define + $$\tilde H^n(X)\defeq \tilde H^n(X;\Z)$$ + and similarly for unreduced cohomology. +\end{defn} +Unparametrized cohomology satisfies the Eilenberg-Steenrod axioms for cohomology. Although we will not use this fact in this chapter, for completeness we will state it here. + +To give the definition we need to introduce one more concept. +\begin{defn}\label{def:choice} +A type $X$ has $n$-choice for $n\geq-2$ if for all $P:X\to\type$ the canonical map +$$\|(x:X)\to Px\|_n\to ((x:X) \to \|Px\|_n)$$ +is an equivalence. +\end{defn} +Note that in particular $\fin_k$ has $n$-choice for all $k,n$. +\begin{defn} + A \emph{unparametrized reduced cohomology theory} is a contravariant functor + $\tilde E^n:\type^*\to\abgroup$ for every $n:\Z$ satisfying the + \emph{Eilenberg-Steenrod axioms}. Functoriality means that for a pointed map + $f:X\to^* Y$ there is a map $\tilde E^n(f):\tilde E^n(Y)\to \tilde E^n(X)$ such that $\tilde E(\id)\sim^*\id$ and $\tilde E(g\o + f)\sim^*\tilde E(f)\o\tilde E g$. The Eilenberg-Steenrod axioms are + \begin{itemize} + \item \emph{(Suspension axiom)} There is a natural transformation $\tilde E^{n+1}(\susp X)\simeq \tilde E^n(X)$. + \item \emph{(Exactness)} Given a cofiber sequence $X\xrightarrow{f}Y\xrightarrow{g}Z$, the sequence + $$\tilde E^n(Z)\xrightarrow{\tilde E^n(g)}\tilde E^nY\xrightarrow{\tilde E^n(f)}\tilde E^n(X)$$ + is exact at $\tilde E^n(Y)$. + \item \emph{(Additivity)} Suppose given a type $I$ satisfying 0-choice and $X:I\to\type^*$. Then the canonical homomorphism + $$\tilde E^n\big(\bigvee_i Xi\big)\to ((i:I)\to \tilde E^n(Xi))$$ + is an isomorphism. + \end{itemize} + A cohomology theory is called \emph{ordinary} if it also satisfies the following axiom. + \begin{itemize} + \item \emph{(Dimension)} If $n\neq 0$, then $\tilde E^n(\pbool)$ is trivial. + \end{itemize} +\end{defn} +The following theorem has been proven in~\cite{cavallo2015cohomology}. We will not repeat the proof here. +\begin{thm}\label{thm:cohomology-theory} + Unparametrized generalized reduced cohomology is a cohomology theory. Ordinary cohomology also satisfies the dimension axiom. +\end{thm} +% Conversely, we can ask whether all cohomology theories come from a spectrum. +% Classically, the answer is true, which is known as Brown's representability +% theorem. This has not been proven in HoTT yet, and it is likely that it will depend on whether +% \begin{conj}\label{con:brown-representability} +% For every unparametrized reduced cohomology theory there exists a spectrum representing that theory. +% \end{conj} +We will not use \autoref{thm:cohomology-theory} in the remainder of this chapter. + +To construct the Atiyah-Hirzebruch spectral sequence, we need the Postnikov tower of a spectrum. + +\begin{defn} + We say that for $k:\Z$ a spectrum $Y$ is \emph{$k$-truncated} if $Y_n$ is $(k+n)$-truncated for all $n:\Z$ (using the convention that any type is $\ell$-truncated for $\ell\leq-2$). + + The \emph{$k$-truncation} of a spectrum $Y$, written $\|Y\|_k$, is defined as $(\|Y\|_k)_n\defeq \|Y_n\|_{k+n}$ where we define $\|A\|_\ell=\unit$ for $\ell\leq-2$. +\end{defn} +\begin{lem}\label{lem:spectrum-trunc-properties} + The usual properties of truncations also hold for spectra. In particular we will use that there is a spectrum map $|{-}|_k:Y\to\|Y\|_k$ and that if $Z$ is $k$-truncated, then a spectrum map $f:Y\to Z$ induces a spectrum map $\|Y\|_k\to Z$. +\end{lem} +\begin{proof} + The underlying maps are the corresponding facts for pointed maps. The fact that these maps are spectrum maps comes from the fact that these operations commute with taking loop spaces. We omit the details here. +\end{proof} +\begin{lem}[Postnikov Tower for spectra]\label{lem:postnikov-tower-spectra} + For $s:\Z$ and $Y:\spectrum$ there is a spectrum map $f^s:\|Y\|_s\to\|Y\|_{s-1}$ that levelwise has fiber $\susp^n(H\pi_s(Y))$. That is, + $$(\fib_{f^s})_k\simeq^*(\susp^s(H\pi_s(Y)))_k.$$ +\end{lem} +We should be able to extend this equivalence to a spectrum equivalence, but we do not need this strengthening for the remainder of the proof. +\begin{proof} + Note that $\|Y\|_{s-1}$ is $(s-1)$-truncated, and therefore $s$-truncated. By + the elimination of spectrum truncation in + \autoref{lem:spectrum-trunc-properties} we get a spectrum map + $f^s:\|Y\|_s\to\|Y\|_{s-1}$. For the levelwise pointed equivalence, we need to + show that + $$\fib_{f^s_k}\simeq^*K(\pi_s(Y),s+k)$$ + To show this, by \autoref{thm:em-unique} we need to show that $\fib_{f^s_k}$ + is $(s+k)$-truncated, $(s+k-1)$-connected and $\pi_{s+k}(\fib_{f^s_k})\simeq + \pi_s(Y)$. + + Note that $f^s_k:\|Y_k\|_{s+k}\to\|Y_k\|_{s+k-1}$, so the truncatedness + follows because the domain and codomain of $f^s_k$ are both $(s+k)$-truncated. + For the connectedness, we know that $|{-}|_{s+k-1}:Y_k\to \|Y_k\|_{s+k-1}$ is + $(s+k-1)$-connected, and the elimination principle for truncations preserve + connectedness, therefore $f^s_k$ is $(s+k-1)$-connected. To compute the + homotopy group, we look at a piece of the long exact sequence for homotopy + groups for $f^s_k$ at level $s+k$ and $s+k+1$. + \begin{center}\begin{tikzpicture}[node distance=10mm] + \node (Y) at (0,0) {$0$}; + \node[left = of Y] (X) {$\pi_{s+k}(Y)$}; + \node[left = of X] (F) {$\pi_{s+k}(\fib_{f^s_k})$}; + \node[above = of Y] (OY) {$0$}; + \node[above = of X] (OX) {$0$}; + \node[above = of F] (OF) {$\bullet$}; + % \node[above = 5mm of OX] {$\vdots$}; + % \node[below = 5mm of X] {$\vdots$}; + \path[every node/.style={font=\sffamily\small}] + (X) edge[->] (Y) + (F) edge[->] (X) + (OY) edge[->] (F) + (OX) edge[->] (OY) + (OF) edge[->] (OX); + \end{tikzpicture} + \end{center} + Since we have the exact sequence + $0\to\pi_{s+k}(\fib_{f^s_k})\to\pi_{s+k}(Y)\to0$, the middle map must be an + equivalence, which finishes the proof. +\end{proof} +For a spectrum $Y$, we get the Postnikov tower +$$\cdot\to\|Y\|_s\to\|Y\|_{s-1}\to\|Y\|_{s-2}\to\cdots$$ +This satisfies the conditions of \autoref{thm:spectral-sequence-spectrum}, but +unfortunately the spectral sequence constructed from this is trivial. We need another ingredient to get an interesting spectral sequence. + +\begin{lem}\label{lem:spi-functor} + Suppose given $X:\type^*$ and two family of spectra $Y,Z:X\to\spectrum$. A family of spectrum maps + $$f:(x:X)\to Yx \to Zx$$ + induces a spectrum map between the spectra of sections for $Y$ and $Z$: + $$f\o({-}):((x:X)\to Yx)\to((x:X)\to Zx).$$ + Moreover, the fiber of this spectrum map is levelwise $(x:X)\to \fib_{fx}$, that is + $$(\fib_{f\o({-})})_n \simeq^* ((x:X)\to \fib_{fx})_n.$$ +\end{lem} +The levelwise equivalence should be extendable to a spectrum equivalence, but we do not need that in this chapter. +\begin{proof} + We define (see \autoref{lem:pointed-types-basic}.\ref{item:fiber-composition}) + $$(f\o({-}))_n\defeq f_n \o ({-}):((x:X)\to^* (Yx)_n)\to^*((x:X)\to^* (Zx)_n).$$ + This is a spectrum map because of the pointed function extensionality mentioned in \autoref{lem:pointed-types-basic}.\ref{item:pointed-function-extensionality}. + + By \autoref{lem:pointed-types-basic}.\ref{item:fiber-composition} the fiber of this map is levelwise $(x:X)\to \fib_{fx}$. +\end{proof} + +We now have all the ingredients of the Atiyah-Hirzebruch spectral sequence. + +\begin{thm}[Atiyah-Hirzebruch spectral sequence for reduced cohomology]\label{thm:atiyah-hirzebruch-reduced} + If $X:\type^*$ is a pointed type and $Y:X\to k\operatorname{-\spectrum}$ is a family of + $k$-truncated spectra over $X$, then we get a spectral sequence with + $$E_2^{p,q}=\tilde H^p(X;\lam{x}\pi_{-q}(Yx))\Rightarrow \tilde H^{p+q}(X;\lam{x}Yx).$$ +\end{thm} +\begin{proof} + Define $A_s\defeq((x:X)\to^* \|Yx\|_s)$ and consider the sequence of spectra + $$\cdots \to A_s\xrightarrow{f_s} A_{s-1} \xrightarrow{f_{s-1}} A_{s-2} \to \cdots$$ + where $f_s$ is the map induced by the Postnikov tower. By + \autoref{lem:spi-functor} and \autoref{lem:postnikov-tower-spectra} $f_s$ + levelwise has fiber $B_s\defeq(x:X)\to^*\susp^sH\pi_s(Yx)$. We want to apply + \autoref{thm:spectral-sequence-spectrum}, so we need to check the conditions + of that theorem. For $n:\Z$ we define $\beta_n\defeq n-1$. Notice that $A_s$ + is $s$-truncated, and thus for $s\le B_n$ we have + $$\pi_n(A_s)\defeq \pi_n((x:X)\to^* \|Yx\|_s)=0.$$ + For $n:\Z$ define $\gamma_n\defeq k$. Then for $s\geq \gamma_n$ the spectrum + $A_s$ is levelwise equivalent to $(x:X)\to^* Yx$, so for $s>\gamma_n$ the map + $A_s\to A_{s-1}$ becomes levelwise the identity map under that equivalence. + This means that $f_s$ is an equivalence, so in particular $\pi_n(f_s)$ is an + isomorphism. By \autoref{thm:spectral-sequence-spectrum} we now get the + spectral sequence + $$E_2^{p,q}=\pi_{-(p+q)}(B_{-q})\Rightarrow \pi_{-(p+q)}(A_k).$$ + We now compute + \begin{align*} + \pi_{-(p+q)}(B_{-q})&\simeq\pi_{-(p+q)}((x:X) \to^* \susp^{-q}H\pi_{-q}(Yx))\\ + &\simeq\tilde H^{p+q}(X;\lam{x}\susp^{-q}H\pi_{-q}(Yx))\\ + &\simeq\tilde H^{p}(X;\lam{x}\pi_{-q}(Yx)) + \intertext{and} + \pi_{-(p+q)}(A_k)&\simeq \pi_{-(p+q)}((x:X)\to^*\|Yx\|_k)\\ + &\simeq\pi_{-(p+q)}((x:X)\to^* Yx)\\ + &\simeq\tilde H^{p+q}(X;\lam{x}Yx). + \end{align*} +\end{proof} + +We also have the corresponding spectral sequence for unreduced cohomology. + +\begin{cor}[Atiyah-Hirzebruch spectral sequence for unreduced cohomology]\label{cor:atiyah-hirzebruch-unreduced} + If $X:\type$ is any type and $Y:X\to k\operatorname{-\spectrum}$ is a family + of $k$-truncated spectra over $X$, then + $$E_2^{p,q}=H^p(X;\lam{x}\pi_{-q}(Yx))\Rightarrow H^{p+q}(X;\lam{x}Yx).$$ +\end{cor} +\begin{proof} +Apply \autoref{thm:atiyah-hirzebruch-reduced} to $X_+$ and $Y_+$ (defined in \autoref{def:cohomology}). +\end{proof} +From the Atiyah-Hirzebruch spectral sequence we can construct the Serre spectral sequence. +\begin{thm}[Serre spectral sequence for cohomology]\label{thm:serre-spectral-sequence} + Suppose given $B:\type$, a family of types $F:B\to\type$ and a spectrum + $Y:\spectrum$ that is $k$-truncated. Then + $$E_2^{p,q}=H^p(B;\lam{b}H^q(Fb;Y))\Rightarrow H^{p+q}((b:B)\times Fb;Y).$$ +\end{thm} +\begin{proof} +Apply \autoref{cor:atiyah-hirzebruch-unreduced} to the type $B$ and and the family of spectra $\lam{b}Fb\to Y$, which is $k$-truncated. Then we get +$$E_2^{p,q}=H^p(B;\lam{b}\pi_{-q}(Fb\to Y))\Rightarrow H^{p+q}(B;\lam{b}Fb\to Y).$$ +Note that $\pi_{-q}(Fb\to Y)\simeq H^q(Fb;Y)$, so the second page is the desired group, and for the $\infty$-page we compute +\begin{align*} + H^{p+q}(B;\lam{b}Fb\to Y) &= \pi_{-(p+q)}((b:B) \to Fb \to Y)\\ + &= \pi_{-(p+q)}(((b:B) \times Fb)\to Y)\\ + &= H^{p+q}((b:B)\times Fb;Y). +\end{align*} +\end{proof} +Equivalent to the data given in \autoref{thm:serre-spectral-sequence} is a map +$X\to B$ and a $k$-truncated spectrum $Y$. In that case we get the spectral +sequence +$$E_2^{p,q}=H^p(B;\lam{b}H^q(\fib_f(b);Y))\Rightarrow H^{p+q}(X;Y).$$ Analogous +to the proof of \autoref{thm:serre-spectral-sequence} we also have a version +when $Y$ is parametrized over $B$. In that case we get +$$E_2^{p,q}=H^p(B;\lam{b}H^q(\fib_f(b);Y))\Rightarrow H^{p+q}(X;\lam{x}Y(fx)).$$ + +We get a useful special case of the Serre spectral sequence when the family +$\lam{b}H^q(Fb;Y)$ is constant. This happens in particular when $B$ is simply +connected. +\begin{cor}\label{cor:serre-spectral-sequence-conn} + Suppose given a simply connected pointed type $B:\type^*$, a family of types $F:B\to\type$ and a spectrum + $Y:\spectrum$ that is $k$-truncated. Then + $$E_2^{p,q}=H^p(B;H^q(Fb_0;Y))\Rightarrow H^{p+q}((b:B)\times Fb;Y).$$ +\end{cor} +\begin{proof} + Apply \autoref{thm:serre-spectral-sequence}. The family + $\lam{b}H^q(Fb;Y):B\to\abgroup$ is a family of sets. Since $B$ is simply + connected, every such family is constant, so all fibers are equal to $H^q(Fb_0;Y)$. +\end{proof} +% Note that if in \autoref{cor:serre-spectral-sequence-conn} instead of a family of types $F:B\to\type$ we have a map $f:X\to B$, we get the spectral sequence +% $$E_2^{p,q}=H^p(B;H^q(\fib_f;Y))\Rightarrow H^{p+q}(X;Y)$$ +% by applying \autoref{cor:serre-spectral-sequence-conn} to the family $\lam{b}\fib_f(b)$. + +In the spectral sequences constructed we assumed that the spectra were truncated. +The reason we need this assumption is that the notion of convergence we used for spectral sequence is the eventual value of the sequence. +If we had a stronger notion of convergence, we might be able to relax the truncatedness condition. +However, there is another reason why the spectral sequence can become (pointwise) eventually constant. +Instead of assuming that the spectra are truncated, we can pose a restriction on the base space. +\begin{defn}\label{def:weak-pointed-choice} + We say that $X:\type^*$ satisfies \emph{weak pointed choice} if there is a natural number $n$ such that for all families $Y:X\to\type^*$ of $n$-connected types the type of dependent pointed maps $(x:X)\to^* Y(x)$ is 0-connected. +\end{defn} +\begin{ex}\mbox{} + \begin{itemize} + \item The spheres $\S^n$ satisfy weak pointed choice. The proof is easy for $n=0$, which we will skip. + For $\S^{n+1}$, note that + $$((x:S^{n+1})\to^* Y(x))\simeq \star =_{\surf}^{\Omega^nY({-})} \star$$ + where $\surf:\Omega^{n+1}\S^{n+1}$ is the surface of $\S^{n+1}$ and + $\star:\Omega^nY(\base)$ is the basepoint. Now if $Y$ is a family of $(n+1)$-connected types, then $\Omega^nY({-})$ is a family of 1-connected types, and a pathover in that family is 0-connected, as desired. + \item Suppose $I$ is a type that satisfies 0-choice (see \autoref{def:choice}). Then the collection of types that satisfy weak pointed choice are closed under $I$-indexed wedges. This follows from the dependent universal property of the wedge. + $$\big(\bigvee_{i:I}X_i\to^* P(x)\big)\simeq (i:I) \to (x : X_i) \to^* P(\inm_i(x)).$$ + \end{itemize} +\end{ex} + +\begin{thm} + If $X:\type^*$ satisfies weak pointed choice and $Y:X\to\spectrum$ is any family of spectra, we get the spectral sequence in \autoref{thm:atiyah-hirzebruch-reduced}: + $$E_2^{p,q}=\tilde H^p(X;\lam{x}\pi_{-q}(Yx))\Rightarrow \tilde H^{p+q}(X;\lam{x}Yx).$$ +\end{thm} +\begin{proof} + The proof is the mostly the same as for \autoref{thm:atiyah-hirzebruch-reduced}. The only difference is in showing that the sequence stabilizes on homotopy groups when $s$ is large. Suppose $X$ satisfies choice with respect to $k$-connected families. + For $n:\Z$ define $\gamma_n\defeq n+k$. Then for $s>\gamma_n$ we know that the fiber of $f_s$ has as $k$-th homotopy group + $$\pi_\ell(B_s)=\pi_0(\Omega^\ell B_s)=\|(x:X)\to^* K(\pi_s(Yx),s-\ell)\|_0.$$ + This is a product into a family of $(s-\ell-1)$-connected types, which for $\ell=n$ and $\ell=n-1$ is a family of at least $k$-connected types. By the weak choice principle on $X$ this type is 0-connected, so these homotopy groups are trivial. Now by the long exact sequence of homotopy groups for $f_s$ the map $\pi_n(f_s)$ is an isomorphism, as required. +\end{proof} + +\section{Spectral Sequences for Homology}\label{sec:spectral-sequence-homology} + +Homology theory has not been developed as much as cohomology theory in HoTT. It is known that the homology given by a prespectrum forms a homology theory~\cite{graham2017homology}. Lemma 18 in that paper was not proven carefully, but it follows from the results in \autoref{sec:smash-product}. + +In this section, we sketch the construction of the Atiyah-Hirzebruch and Serre spectral sequences for homology~\cite{serre1951homology}. The results in the section are not proven in HoTT, and are therefore stated as remarks without proof. + +If $X:\type^*$ and $Y:\prespectrum$, we can define $X\wedge Y:\prespectrum$ with +$$(X\wedge Y)_n\defeq X\wedge Y_n.$$ +To show that it is a prespectrum, recall the adjunction between the suspension and the loop. For pointed types $X$ and $Y$ we have a natural equivalence +$$(\susp X \to^* Y)\simeq (X \to^* \susp Y).$$ +Therefore, to characterize a prespectrum, it is sufficient to give a map $f_n:\susp Y_n \to Y_{n+1}$. This is given for the smash prespectrum as the composite +$$\susp(X\wedge Y_n)\xrightarrow{\sim} X \wedge \susp Y_n \xrightarrow{X\wedge f_n} X \wedge Y_{n+1}.$$ +We can define reduced homology +$$\tilde H_n(X;Y)\defeq \pi_n(X\wedge Y).$$ +For the construction of parametrized homology we need to generalize the smash product. + +\begin{defn} + Given $A : \type^*$ and $B : A \to \type^*$, we define the parametrized smash + $$(x : A) \wedge B(x)$$ + to be the pushout + \begin{center} + \begin{tikzpicture}[node distance=10mm,baseline={([yshift={-\ht\strutbox}]current bounding box.north)}] + \node (tl) at (0,0) {$A+B$}; + \node[right = 25mm of tl] (tr) {$2$}; + \node[below = of tl] (bl) {$(x:A)\times B(x)$}; + \node (br) at (tr |- bl) {$(x : A) \wedge B(x)$}; + \path[every node/.style={font=\sffamily\small}] + (tl) edge[->] node {} (tr) + edge[->] node {} (bl) + (tr) edge[->] node {} (br) + (bl) edge[->] node {} (br); + \end{tikzpicture}\quad + \begin{tikzpicture}[baseline={([yshift={-\ht\strutbox}]current bounding box.north)}] + \draw[fill=black!5, thin] (0,0) ellipse (20mm and 8mm); + \node[label=below:{$a_0$}] at (0,-1.98) {$\bullet$}; + \node at (2.3,-2) {$A$}; + \node at (2.3,0) {$B$}; + \path[every node/.style={font=\sffamily\scriptsize}] + (-2,-2) edge[thick] (2,-2) + (0,-1.1) edge[->] (0,-1.8) + (0,0) edge[bend left = 10, thick] (-2,0) + edge[bend left = 10, thick] node[below] {$b_0$} (2,0) + edge[bend left = 5, thick] node[left] {$B(a_0)$} (0,0.8) + edge[bend left = 5, thick] (0,-0.8); + \end{tikzpicture} + \end{center} +\end{defn} +\begin{rmk}The strategy for constructing the spectral sequences for homology is as follows. +\begin{itemize} +\item The parametrized smash is (should be) left adjoint to pointed dependent maps. +That means that there is a natural equivalence +$$(((x:A) \wedge Bx) \to^* C) \simeq (x:A) \to^* Bx \to^* C.$$ +\item From this we get (natural) equivalences +$$\susp((x:A)\wedge Bx) \simeq ((x:A) \wedge \susp(Bx));$$ +$$((x:A)\wedge Bx) \wedge C \simeq (x:A) \wedge (Bx \wedge C);$$ +$$(x:A_+)\wedge B_+x) \wedge C \simeq (x:A) \times Bx.$$ +The proofs of these properties should be similar to the proofs in \autoref{sec:smash-monoidal}. +\item Therefore, for $X:\type^*$ and $Y:X\to\prespectrum$ we have a prespectrum $(x : X)\wedge Yx$. The maps are given by the above equivalence. +\item We can now define parametrized (reduced, generalized) homology as +$$H_n(X;\lam{x}Yx) \defeq \pi_n((x : X) \wedge Yx).$$ +We can define unreduced homology by adding a point to $X$, in the same way as for cohomology. +\item As before, given $X:\type^*$ and $Y:X\to\spectrum$, we can again form the Postnikov tower of $Yx$ for any $x:X$. We now want to take the parametrized smash over $X$, but there is no hope to compute the fiber of this spectrum. +\item However, we should be able to do it when we work in spectra. The forgetful +functor $\spectrum \to \prespectrum$ has a left adjoint, called +\emph{spectrification}. The spectrification $LY$ of a prespectrum $Y$ can be +constructed either as a higher inductive family of types~\cite{shulman2011spectrification} or as the colimit +$$(LY)_n\defeq\colim_{k\to\infty}\Omega^k Y_{n+k}.$$ +For neither definition a careful proof of the adjunction has been given. +\item We can now define the parametrized smash of spectra as the spectrification of the parametrized smash for prespectra. This should preserve cofiber sequences of spectra, in the sense that if +$$Ax \to Bx \to Cx$$ +is a family of cofiber sequences of spectra indexed by $x:\type^*$, the following sequence is also a cofiber sequence of spectra +$$((x:X)\wedge Ax)\to ((x:X)\wedge Bx)\to ((x:X)\wedge Cx)$$ +\item A sequence of spectra should be a fiber sequence of spectra if and only if it is a cofiber sequence of spectra. This is true classically, and should also hold in HoTT. +\item Assuming that all the above properties have been proven, we can get the Atiyah-Hirzebruch spectral sequence for reduced homology. Suppose given a pointed type $X$ and $Y:X\to\spectrum$ a family of spectra. We can apply +\autoref{thm:spectral-sequence-spectrum} to the iterated fiber sequence +$$((x:X)\wedge \susp^nH)\to((x:X)\wedge \|Yx\|_s)\to ((x:X)\wedge +\|Yx\|_{s-1}).$$ To satisfy the conditions for that theorem we need to assume +some conditions on $X$ and/or $Y$. In particular it is sufficient if $Y$ is a family of +truncated and connected spectra, but weaker conditions might also suffice. +Using homological indexing (where $p$ and $q$ have their sign reversed) we get +$$E^2_{p,q}=\pi_{p+q}(B_q)\Rightarrow \pi_{p+q}(A_{\gamma_{p+q}}).$$ +Now we compute +\begin{align*} + \pi_{p+q}(B_q)&\simeq\pi_{p+q}((x:X) \wedge \susp^qH\pi_q(Yx))\\ + &\simeq\tilde H_{p+q}(X;\lam{x}\susp^qH\pi_q(Yx))\\ + &\simeq\tilde H_p(X;\lam{x}\pi_q(Yx)) + \intertext{and} + \pi_{p+q}(A_k)&\simeq \pi_{p+q}((x:X)\wedge\|Yx\|_k)\\ + &\simeq\pi_{p+q}((x:X)\to Yx)\\ + &\simeq\tilde H_{p+q}(X;\lam{x}Yx). +\end{align*} +This gives the desired spectral sequence: +$$E^2_{p,q}=\tilde H_p(X;\lam{x}\pi_q(Yx))\Rightarrow \tilde H_{p+q}(X;\lam{x}Yx).$$ +\item We get the Atiyah-Hirzebruch spectral sequence for \emph{unreduced} homology in the same way as for cohomology, by applying the version for reduced homology to $X_+$ and $Y_+$. +\item We get the Serre spectral sequence for homology also in the same way. Suppose given +$B:\type$ and $F:B\to\type$ and a truncated spectrum $Y$. Applying the Atiyah-Hirzebruch spectral sequence for unreduced homology to the type $B$ and the spectrum $\lam{b}Fb\wedge Y$ we get +$$E^2_{p,q}=H_p(B;\lam{b}\pi_q(Fb\wedge Y))\Rightarrow H_{p+q}(B;\lam{b}Fb\wedge Y).$$ +The second page is what we want. For the $\infty$-page we compute +\begin{align*} + H_{p+q}(B;\lam{b}Fb\to Y) &= \pi_{p+q}((b:B_+) \wedge (F_+b \wedge Y))\\ + &= \pi_{p+q}(((b:B_+) \wedge F_+b)\wedge Y)\\ + &= \pi_{p+q}(((b:B) \times Fb)\wedge Y)\\ + &= H_{p+q}((b:B)\times Fb;Y). +\end{align*} +This gives the Serre spectral sequence for homology: +$$E^2_{p,q}=H_p(B;\lam{b}H_q(Fb;Y))\Rightarrow H_{p+q}((b:B)\times Fb;Y).$$ +\end{itemize} +\end{rmk} +\begin{rmk} + We can also use the parametrized smash to get a spectral sequence for reduced + homology and reduced cohomology. Suppose given $B:\type^*$ and a family of types $F:B\to\type^*$ and a spectrum $Y:\spectrum$ that is $k$-truncated. Then we get the following two spectral sequences + $$E_2^{p,q}=\tilde H^p(B;\lam{b}\tilde H^q(Fb;Y))\Rightarrow \tilde H^{p+q}((b:B)\wedge Fb;Y);$$ + $$E^2_{p,q}=\tilde H_p(B;\lam{b}\tilde H_q(Fb;Y))\Rightarrow \tilde H_{p+q}((b:B)\wedge Fb;Y).$$ + For homology, the proof is the same as above. For cohomology, we apply the Atiyah-Hirzebruch spectral sequence for reduced cohomology to the pointed type $B$ and the family of spectra $\lam{b}Fb\to^* Y$. We get the desired spectral sequence by the adjunction between parametrized smash and dependent pointed maps. + + These spectral sequences generalize \autoref{thm:serre-spectral-sequence} and the corresponding version for homology: we get those versions back when we add a point to $B$ and $F$. Whether this extra generality is useful is unknown. +\end{rmk} + +\section{Applications of Spectral Sequences}\label{sec:applications-spectral-sequences} + +Classically, there are many applications of the Serre and Atiyah-Hirzebruch spectral sequences. +Here we will list some of these applications, and give thoughts on how to translate these results in HoTT. +The results in this section have not been formalized. Before we start, we compute the cohomology of spheres. + +\begin{lem}\label{lem:cohomology-spheres} + If $n\geq 1$, then +\begin{equation}H^k(\S^n;A)=\begin{cases}A&\text{if $k\in\{0,n\}$}\\0&\text{otherwise.}\end{cases}\label{eq:cohomology-spheres}\end{equation} +\end{lem} +This is a special case of the universal coefficient theorem, which we do not have yet in HoTT. +However, we can prove these equalities directly from the definition of cohomology. +\begin{proof} +For $k=0$ we have +$$H^0(\S^n;A)=\|\S^n\to A\|_0=(\S^n\to A) = A,$$ +where we use that $\S^n$ is 0-connected. For $k\neq 0$ we have +$$H^k(\S^n;A)=\tilde H^k(\S^n+1;A)=\tilde H^k(\S^n;A)=\|\S^n\to^* K(A,k)\|_0=\|\Omega^n K(A,k)\|_0.$$ +Now for $nk$ the type $\Omega^n K(A,k)$ itself is contractible. +\end{proof} + +The first application is the path fibration. Suppose given a simply connected pointed type $B$ we have a map $\unit\to B$ that has fiber $\Omega B$.\footnote{It is called the \emph{path fibration} because classically to get a Serre fibration we need to take the path space $PB$ instead of $\unit$.} In other words, we have the fiber sequence +$$\Omega B \to \unit \to B.$$ +Now the Serre spectral sequence for cohomology gives (say, with integer coefficients) +$$E_2^{p,q}=H^p(B;H^q(\Omega B))\Rightarrow H^{p+q}(\unit).$$ +Note that the $\infty$-page vanishes, except when $p+q=0$, when the coefficient is $\Z$. +For ordinary cohomology $H^n$ is trivial for $n<0$, which means that the second page is only nontrivial in the first quadrant of the plane, +hence this is true for all pages, including the $\infty$-page. +Therefore, the $\infty$-page has one group $\Z$ at the origin, and trivial groups everywhere else, as shown in \autoref{fig:infty-path-fibration}. +\begin{figure}[ht] + \begin{center} + \begin{tikzpicture} + \draw [thick] (0.5,0.5) -- (0.5,3.5); + \draw [thick] (0.5,0.5) -- (5.5,0.5); + \node[font=\normalsize] at (5.4,0.3) {$p$}; + \node[font=\normalsize] at (0.3,3.4) {$q$}; + \node at (0.1,1) {$0$}; + \node at (0.1,2) {$1$}; + \node at (0.1,3) {$2$}; + \node at (1,0.1) {$0$}; + \node at (2,0.1) {$1$}; + \node at (3,0.1) {$2$}; + \node at (4,0.1) {$3$}; + \node at (5,0.1) {$4$}; + \node (x11) at (1,1) {$\Z$}; + \node (x12) at (1,2) {$0$}; + \node (x13) at (1,3) {$0$}; + \node (x21) at (2,1) {$0$}; + \node (x22) at (2,2) {$0$}; + \node (x23) at (2,3) {$0$}; + \node (x31) at (3,1) {$0$}; + \node (x32) at (3,2) {$0$}; + \node (x33) at (3,3) {$0$}; + \node (x41) at (4,1) {$0$}; + \node (x42) at (4,2) {$0$}; + \node (x43) at (4,3) {$0$}; + \node (x51) at (5,1) {$0$}; + \node (x52) at (5,2) {$0$}; + \node (x53) at (5,3) {$0$}; + \end{tikzpicture} +\end{center} + \caption{$E_\infty^{p,q}$ for the path fibration.} + \label{fig:infty-path-fibration} +\end{figure} + +This gives a relation between the cohomology of $B$ and the cohomology of $\Omega B$. +If we know the cohomology for one of the spaces one of them, +then we can sometimes compute the cohomology from the other using this. +Using the Serre spectral sequence for homology, we have the same relationship between the homology of $B$ and the homology of $\Omega B$. +The computations in the next example will work exactly the same for homology. + +\begin{ex}\label{ex:cohomology-KZ2} +As an example, we can compute the cohomology groups of $B = K(\Z,2)$ +(which is the complex projective space $\mathbf{CP}^\infty$). +Its loop space is $\Omega K(\Z,2) = K(\Z,1) = \S^1$, and by \autoref{lem:cohomology-spheres} we have +$$H^n(\S^1)=\begin{cases}\Z&\text{if $n=0,1$}\\ 0&\text{otherwise.}\end{cases}$$ +The resulting second page of the spectral sequence is shown in \autoref{fig:two-page-KZ2-path-fibration}, all other groups on the second page are trivial. +\begin{figure}[ht] + \begin{center} + \begin{tikzpicture}[every node/.style={font=\small},x=15mm,y=15mm] + \draw [thick] (0.5,0.5) -- (0.5,2.5); + \draw [thick] (0.5,0.5) -- (5.5,0.5); + \node[font=\normalsize] at (5.4,0.3) {$p$}; + \node[font=\normalsize] at (0.3,2.4) {$q$}; + \node at (0.1,1) {$0$}; + \node at (0.1,2) {$1$}; + \node at (1,0.1) {$0$}; + \node at (2,0.1) {$1$}; + \node at (3,0.1) {$2$}; + \node at (4,0.1) {$3$}; + \node at (5,0.1) {$4$}; + \node (x11) at (1,1) {$H^0\!(B)$}; + \node (x12) at (1,2) {$H^0\!(B)$}; + \node (x21) at (2,1) {$H^1\!(B)$}; + \node (x22) at (2,2) {$H^1\!(B)$}; + \node (x31) at (3,1) {$H^2\!(B)$}; + \node (x32) at (3,2) {$H^2\!(B)$}; + \node (x41) at (4,1) {$H^3\!(B)$}; + \node (x42) at (4,2) {$H^3\!(B)$}; + \node (x51) at (5,1) {$H^4\!(B)$}; + \node (x52) at (5,2) {$H^4\!(B)$}; + \path + (x12) edge[->] (x31) + (x22) edge[->] (x41) + (x32) edge[->] (x51) + (x42) edge (5.5,1.25) + (x52) edge (5.5,1.75); + \end{tikzpicture} + \end{center} + \caption{$E_2^{p,q}$ for the path fibration of $K(\Z,2)$.} + \label{fig:two-page-KZ2-path-fibration} +\end{figure} +Note that the shown differentials are the only nontrivial differentials on the second page, and all differentials on all later pages are also trivial. This means that $E_3=E_\infty$, depicted in \autoref{fig:infty-path-fibration}. Note that there are no nontrivial differentials going in or out of the $H^0(B)$ and $H^1(B)$ in the bottom line. This means that +$$H^0(B)=E_\infty^{0,0}=\Z$$ +and +$$H^1(B)=E_\infty^{1,0}.$$ +All other groups displayed on the second page vanish on the $\infty$-page. Therefore, all shown differentials must be isomorphisms. +This means that $H^{n+2}(B)=H^n(B)$, which shows that $H^n(B)$ is $\Z$ for even $n$ and $0$ for odd $n$. +\end{ex} +Another simple application of the Serre spectral sequence is to compute the homology and cohomology groups of $\Omega \S^n$, given in~\cite[Example 1.5]{hatcher2004spectral}. +In this case, we know the (co)homology of the base space $\S^n$, and from it we can deduce the (co)homology of the loop space $\Omega \S^n$. We will do the computation here for cohomology. + +\begin{ex}\label{ex:cohomology-OSn} + If we take the Serre spectral sequence for the path fibration of $B=S^n$ for $n\geq 2$, then the second page has entries + $$E_2^{p,q}=H^p(S^n;H^q(\Omega S^n))=\begin{cases}H^q(\Omega S^n)&\text{if $p=0,n$}\\ 0&\text{otherwise.}\end{cases}$$ + using \autoref{lem:cohomology-spheres}. Therefore, the only nontrivial groups are in the columns $p=0$ and $p=n$. + This means that by looking at the degree of the differentials, the only nonzero differentials can occur in page $n$, as shown in \autoref{fig:n-page-path-fibration-sphere}. +\begin{figure}[ht] + \begin{center} + \begin{tikzpicture}[every node/.style={font=\small},x=25mm,y=12mm] + \draw [thick] (0.5,0.5) -- (2.5,0.5); + \draw [thick] (0.5,0.5) -- (0.5,4.5); + \node[font=\normalsize] at (2.4,0.3) {$p$}; + \node[font=\normalsize] at (0.4,4.4) {$q$}; + \node at (1,0.1) {$0$}; + \node at (2,0.1) {$n$}; + \node at (0.1,1) {$0$}; + \node at (0.1,2) {$n-1$}; + \node at (0.1,3) {$2(n-1)$}; + \node at (0.1,4) {$3(n-1)$}; + \node (x11) at (1,1) {$H_0(\Omega S^n)$}; + \node (x12) at (1,2) {$H_{n-1}(\Omega S^n)$}; + \node (x13) at (1,3) {$H_{2(n-1)}(\Omega S^n)$}; + \node (x14) at (1,4) {$H_{3(n-1)}(\Omega S^n)$}; + \node (x21) at (2,1) {$H_0(\Omega S^n)$}; + \node (x22) at (2,2) {$H_{n-1}(\Omega S^n)$}; + \node (x23) at (2,3) {$H_{2(n-1)}(\Omega S^n)$}; + \node (x24) at (2,4) {$H_{3(n-1)}(\Omega S^n)$}; + \path + (x12) edge[->] (x21) + (x13) edge[->] (x22) + (x14) edge[->] (x23) + (1.5,4.5) edge[->] (x24); + \end{tikzpicture} + \end{center} + \caption{$E_n^{p,q}$ for the path fibration of $S^n$.} + \label{fig:n-page-path-fibration-sphere} +\end{figure} + Because all later differentials are trivial, $E_{n+1}=E_\infty$, which is depicted in \autoref{fig:infty-path-fibration}. This means that all differentials on page $n$ from the $p=0$ column to the $p=n$ column must be isomorphisms, except for the differential from $(0,0)$ to $(n,-(n-1))$. Hence we can conclude by induction that +\begin{equation}H^k(\Omega\S^n)\begin{cases}\Z&\text{if $n-1\mid k$}\\0&\text{otherwise.}\end{cases}\label{eq:cohomology-loop-spheres}\end{equation} +\end{ex} + +As a generalization of \autoref{ex:cohomology-KZ2}, we can construct the \emph{Gysin sequence} from the Serre spectral sequence~\cite[Theorem 3.3.3]{holmbergperoux2014models}. +The Gysin sequence for homology states that if $f:E\to B$ is a pointed map with fiber $\S^{n-1}$ for $n\geq 2$ and if $B$ is simply connected, then there exists a long exact sequence +$$\cdots \to H_i(E)\to H_i(B)\to H_{i-n}(B)\to H_{i-1}(E) \to \cdots.$$ +There is also an analogue for cohomology, which states that under the same assumptions there exists a long exact sequence of cohomology groups +$$\cdots \to H^{i-1}(E)\to H^{i-n}(B)\to H^i(B)\to H^i(E) \to \cdots.$$ +The proof given in~\cite[Theorem 3.3.3]{holmbergperoux2014models} works the same in HoTT. +An alternative construction of the Gysin sequence in HoTT is given in~\cite[Section 6.1]{brunerie2016spheres}, which was used as a main ingredient to compute $\pi_4(\S^3)$. + +We can also generalize \autoref{ex:cohomology-OSn} to get the \emph{Wang sequence}. +For homology this states that if $E\to \S^n$ is a pointed map for $n\geq 2$ with fiber $F$, then there exists a long exact sequence +$$\cdots \to H_i(F)\to H_i(E)\to H_{i-n}(F)\to H_{i-1}(F) \to \cdots.$$ +Again, a similar long exact sequence holds for cohomology, and the proof given in~\cite[Theorem 3.3.6]{holmbergperoux2014models} works the same in HoTT. + +As another application, we can prove the Hurewicz theorem from the Serre spectral sequence~\cite{holmbergperoux2014models}. +The Hurewicz theorem only holds for homology, and requires the Serre spectral sequence for homology. +The theorem states that if $X$ is a simply connected pointed type, $n\geq 2$ and $\pi_q(X)$ is trivial for $qn$, except for $\pi_{4k-1}(\S^{2k})$, which are the direct sum of $\Z$ and a finite group~\cite[Theorem 1.21]{hatcher2004spectral}. +\item For a prime $p$ the $p$-torsion subgroup of $\pi_i(\S^3)$ is 0 for $i<2p$ and $\Z_p$ for $i=2p$~\cite[Example 1.18]{hatcher2004spectral}. +\item From the two above results we can immediately conclude that $\pi_4(\S^3)=\Z_2$. +\item Using additionally the localization of a space at a prime, we can show that for $p$ a prime, the $p$-torsion subgroup of $\pi_i(\S^{n+3})$ is 0 for $i A -> Type and eq' a b = eq a b. +% - functors between (pointed) types and natural transformations would be useful + +\chapter*{Acknowledgements} +\addcontentsline{toc}{chapter}{Acknowledgements} +First and foremost I would like to thank my advisor Jeremy Avigad, who was always ready to give useful feedback, proofread drafts of all my written work and provide support. +Futhermore, I would like to thank Steve Awodey for always being ready to answer any questions I have about HoTT or category theory. +I would like to thank Mike Shulman for many helpful remarks and insights whenever I show my work. +I also want to thank Ulrik Buchholtz, Egbert Rijke, Jakob von Raumer, Stefano Piceghello and Kristina Sojakova for the collaborations and discussions. +I am grateful towards Leonardo de Moura for all his help with getting me up to speed with Lean, and answering all my stupid questions and ideas I brought up early in the development of Lean. +I would like to thank Marc Bezem and Dan Christensen to invite me for academic visits. +More generally, I would like to thank everyone in the HoTT community for maintaining such a good research community. +It is very nice to be part of such a friendly and collaborative research community, +where it is normal to have unfinished projects on Github or discuss half-baked ideas on a mailing list. + +For moral support, I would like to thank my parents, Peter van Doorn and Judith van Wakeren, for supporting me during times when I was struggling. Dank jullie wel! +Lastly I would like to thank Cecilia Hornberger for the moral support over the last months. + +I gratefully acknowledge the support of the Air Force Office of Scientific Research through MURI +grant FA9550-15-1-0053. Any opinions, findings and conclusions or recommendations expressed in this +material are those of the authors and do not necessarily reflect the views of the AFOSR. % Air force grant + +The author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, +for support and hospitality during the programme Big Proof where work on this paper +was undertaken. This work was supported by EPSRC grant no EP/K032208/1. % Big Proof + +%This material is based upon work supported by the National Science Foundation under Grant Number DMS 1321794. % MRC +This material is based upon work supported by the National Science Foundation under Grant Number DMS 1641020. % MRC (2nd)? + + + +\clearpage +\phantomsection +\addcontentsline{toc}{chapter}{Bibliography} +\Urlmuskip=0mu plus 1mu %adds glue to urls +\bibliographystyle{amsalphaurl} +\bibliography{references} + + +\end{document} diff --git a/resources/VanDoornDissertation/lstlean.tex b/resources/VanDoornDissertation/lstlean.tex new file mode 100644 index 0000000..c7600a2 --- /dev/null +++ b/resources/VanDoornDissertation/lstlean.tex @@ -0,0 +1,293 @@ +% Listing style definition for the Lean Theorem Prover. +% Defined by Jeremy Avigad, 2015, by modifying Assia Mahboubi's SSR style. +% Unicode replacements taken from Olivier Verdier's unixode.sty + +\lstdefinelanguage{lean} { + +% Anything betweeen $ 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metaclasses, raw, migrate, replacing, +calc, have, obtains, show, suffices, by, by+, in, at, let, forall, Pi, fun, +exists, if, dif, then, else, assume, assert, take, +obtain, from, aliases +}, + +% Sorts +morekeywords=[2]{Type, Prop}, + +% tactics, list taken from lean-syntax.el +morekeywords=[3]{ +Cond, or_else, then, try, when, assumption, eassumption, rapply, +apply, fapply, eapply, rename, intro, intros, all_goals, fold, focus, focus_at, +generalize, generalizes, clear, clears, revert, reverts, back, beta, done, exact, rexact, +refine, repeat, whnf, rotate, rotate_left, rotate_right, inversion, cases, rewrite, +xrewrite, krewrite, blast, simp, esimp, unfold, change, check_expr, contradiction, +exfalso, split, existsi, constructor, fconstructor, left, right, injection, congruence, reflexivity, +symmetry, transitivity, state, induction, induction_using, fail, append, +substvars, now, with_options, with_attributes, with_attrs, note +}, + +% modifiers, taken from lean-syntax.el +% 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+sensitive=true, + +% Automatic breaking of long lines +breaklines=true, + +% Default style fors listingsred +basicstyle=\ttfamily, + +% Position of captions is bottom +captionpos=b, + +% Full flexible columns +columns=[l]fullflexible, + + +% Style for (listings') identifiers +identifierstyle={\ttfamily\color{black}}, +% Note : highlighting of Coq identifiers is done through a new +% delimiter definition through an lstset at the begining of the +% document. Don't know how to do better. + +% Style for declaration keywords +keywordstyle=[1]{\ttfamily\color{keywordcolor}}, + +% Style for sorts +keywordstyle=[2]{\ttfamily\color{sortcolor}}, + +% Style for tactics keywords +keywordstyle=[3]{\ttfamily\color{tacticcolor}}, + +% Style for attributes +keywordstyle=[4]{\ttfamily\color{attributecolor}}, + +% Style for strings +stringstyle=\ttfamily, + +% Style for comments +% commentstyle={\ttfamily\footnotesize }, + +} diff --git a/resources/VanDoornDissertation/macros.tex b/resources/VanDoornDissertation/macros.tex new file mode 100644 index 0000000..1e2e2a8 --- /dev/null +++ b/resources/VanDoornDissertation/macros.tex @@ -0,0 +1,783 @@ +%%%% MACROS FOR NOTATION %%%% +% Use these for any notation where there are multiple options. +% Slightly modified by Floris van Doorn + +%%% Notes and exercise sections +\makeatletter +\newcommand{\sectionNotes}{\phantomsection\section*{Notes}\addcontentsline{toc}{section}{Notes}\markright{\textsc{\@chapapp{} \thechapter{} Notes}}} +\newcommand{\sectionExercises}[1]{\phantomsection\section*{Exercises}\addcontentsline{toc}{section}{Exercises}\markright{\textsc{\@chapapp{} \thechapter{} Exercises}}} +\makeatother + +%%% Definitional equality (used infix) %%% +\newcommand{\jdeq}{\equiv} % An equality judgment +\let\judgeq\jdeq +%\newcommand{\defeq}{\coloneqq} % An equality currently being defined +\newcommand{\defeq}{\vcentcolon\equiv} % A judgmental equality currently being defined + +%%% Term being defined +\newcommand{\define}[1]{\textbf{#1}} + +%%% Vec (for example) + +\newcommand{\Vect}{\ensuremath{\mathsf{Vec}}} +\newcommand{\Fin}{\ensuremath{\mathsf{Fin}}} +\newcommand{\fmax}{\ensuremath{\mathsf{fmax}}} +\newcommand{\seq}[1]{\langle #1\rangle} + +%%% Dependent products %%% +\def\prdsym{\textstyle\prod} +%% Call the macro like \prd{x,y:A}{p:x=y} with any number of +%% arguments. Make sure that whatever comes *after* the call doesn't +%% begin with an open-brace, or it will be parsed as another argument. +\makeatletter +% Currently the macro is configured to produce +% {\textstyle\prod}(x:A) \; {\textstyle\prod}(y:B),\ +% in display-math mode, and +% \prod_{(x:A)} \prod_{y:B} +% in text-math mode. +\def\prd#1{\@ifnextchar\bgroup{\prd@parens{#1}}{\@ifnextchar\sm{\prd@parens{#1}\@eatsm}{\prd@noparens{#1}}}} +\def\prd@parens#1{\@ifnextchar\bgroup% + {\mathchoice{\@dprd{#1}}{\@tprd{#1}}{\@tprd{#1}}{\@tprd{#1}}\prd@parens}% + {\@ifnextchar\sm% + {\mathchoice{\@dprd{#1}}{\@tprd{#1}}{\@tprd{#1}}{\@tprd{#1}}\@eatsm}% + {\mathchoice{\@dprd{#1}}{\@tprd{#1}}{\@tprd{#1}}{\@tprd{#1}}}}} +\def\@eatsm\sm{\sm@parens} +\def\prd@noparens#1{\mathchoice{\@dprd@noparens{#1}}{\@tprd{#1}}{\@tprd{#1}}{\@tprd{#1}}} +% Helper macros for three styles +\def\lprd#1{\@ifnextchar\bgroup{\@lprd{#1}\lprd}{\@@lprd{#1}}} +\def\@lprd#1{\mathchoice{{\textstyle\prod}}{\prod}{\prod}{\prod}({\textstyle #1})\;} +\def\@@lprd#1{\mathchoice{{\textstyle\prod}}{\prod}{\prod}{\prod}({\textstyle #1}),\ } +\def\tprd#1{\@tprd{#1}\@ifnextchar\bgroup{\tprd}{}} +\def\@tprd#1{\mathchoice{{\textstyle\prod_{(#1)}}}{\prod_{(#1)}}{\prod_{(#1)}}{\prod_{(#1)}}} +\def\dprd#1{\@dprd{#1}\@ifnextchar\bgroup{\dprd}{}} +\def\@dprd#1{\prod_{(#1)}\,} +\def\@dprd@noparens#1{\prod_{#1}\,} + +%%% Lambda abstractions. +% Each variable being abstracted over is a separate argument. If +% there is more than one such argument, they *must* be enclosed in +% braces. Arguments can be untyped, as in \lam{x}{y}, or typed with a +% colon, as in \lam{x:A}{y:B}. In the latter case, the colons are +% automatically noticed and (with current implementation) the space +% around the colon is reduced. You can even give more than one variable +% the same type, as in \lam{x,y:A}. +\def\lam#1{{\lambda}\@lamarg#1:\@endlamarg\@ifnextchar\bgroup{.\,\lam}{.\,}} +\def\@lamarg#1:#2\@endlamarg{\if\relax\detokenize{#2}\relax #1\else\@lamvar{\@lameatcolon#2},#1\@endlamvar\fi} +\def\@lamvar#1,#2\@endlamvar{(#2\,{:}\,#1)} +% \def\@lamvar#1,#2{{#2}^{#1}\@ifnextchar,{.\,{\lambda}\@lamvar{#1}}{\let\@endlamvar\relax}} +\def\@lameatcolon#1:{#1} +\let\lamt\lam +% This version silently eats any typing annotation. +\def\lamu#1{{\lambda}\@lamuarg#1:\@endlamuarg\@ifnextchar\bgroup{.\,\lamu}{.\,}} +\def\@lamuarg#1:#2\@endlamuarg{#1} + +%%% Dependent products written with \forall, in the same style +\def\fall#1{\forall (#1)\@ifnextchar\bgroup{.\,\fall}{.\,}} + +%%% Existential quantifier %%% +\def\exis#1{\exists (#1)\@ifnextchar\bgroup{.\,\exis}{.\,}} + +%%% Dependent sums %%% +\def\smsym{\textstyle\sum} +% Use in the same way as \prd +\def\sm#1{\@ifnextchar\bgroup{\sm@parens{#1}}{\@ifnextchar\prd{\sm@parens{#1}\@eatprd}{\sm@noparens{#1}}}} +\def\sm@parens#1{\@ifnextchar\bgroup% + {\mathchoice{\@dsm{#1}}{\@tsm{#1}}{\@tsm{#1}}{\@tsm{#1}}\sm@parens}% + {\@ifnextchar\prd% + {\mathchoice{\@dsm{#1}}{\@tsm{#1}}{\@tsm{#1}}{\@tsm{#1}}\@eatprd}% + {\mathchoice{\@dsm{#1}}{\@tsm{#1}}{\@tsm{#1}}{\@tsm{#1}}}}} +\def\@eatprd\prd{\prd@parens} +\def\sm@noparens#1{\mathchoice{\@dsm@noparens{#1}}{\@tsm{#1}}{\@tsm{#1}}{\@tsm{#1}}} +\def\lsm#1{\@ifnextchar\bgroup{\@lsm{#1}\lsm}{\@@lsm{#1}}} +\def\@lsm#1{\mathchoice{{\textstyle\sum}}{\sum}{\sum}{\sum}({\textstyle #1})\;} +\def\@@lsm#1{\mathchoice{{\textstyle\sum}}{\sum}{\sum}{\sum}({\textstyle #1}),\ } +\def\tsm#1{\@tsm{#1}\@ifnextchar\bgroup{\tsm}{}} +\def\@tsm#1{\mathchoice{{\textstyle\sum_{(#1)}}}{\sum_{(#1)}}{\sum_{(#1)}}{\sum_{(#1)}}} +\def\dsm#1{\@dsm{#1}\@ifnextchar\bgroup{\dsm}{}} +\def\@dsm#1{\sum_{(#1)}\,} +\def\@dsm@noparens#1{\sum_{#1}\,} + +%%% W-types +\def\wtypesym{{\mathsf{W}}} +\def\wtype#1{\@ifnextchar\bgroup% + {\mathchoice{\@twtype{#1}}{\@twtype{#1}}{\@twtype{#1}}{\@twtype{#1}}\wtype}% + {\mathchoice{\@twtype{#1}}{\@twtype{#1}}{\@twtype{#1}}{\@twtype{#1}}}} +\def\lwtype#1{\@ifnextchar\bgroup{\@lwtype{#1}\lwtype}{\@@lwtype{#1}}} +\def\@lwtype#1{\mathchoice{{\textstyle\mathsf{W}}}{\mathsf{W}}{\mathsf{W}}{\mathsf{W}}({\textstyle #1})\;} +\def\@@lwtype#1{\mathchoice{{\textstyle\mathsf{W}}}{\mathsf{W}}{\mathsf{W}}{\mathsf{W}}({\textstyle #1}),\ } +\def\twtype#1{\@twtype{#1}\@ifnextchar\bgroup{\twtype}{}} +\def\@twtype#1{\mathchoice{{\textstyle\mathsf{W}_{(#1)}}}{\mathsf{W}_{(#1)}}{\mathsf{W}_{(#1)}}{\mathsf{W}_{(#1)}}} +\def\dwtype#1{\@dwtype{#1}\@ifnextchar\bgroup{\dwtype}{}} +\def\@dwtype#1{\mathsf{W}_{(#1)}\,} + +\newcommand{\suppsym}{{\mathsf{sup}}} +\newcommand{\supp}{\ensuremath\suppsym\xspace} + +\def\wtypeh#1{\@ifnextchar\bgroup% + {\mathchoice{\@lwtypeh{#1}}{\@twtypeh{#1}}{\@twtypeh{#1}}{\@twtypeh{#1}}\wtypeh}% + {\mathchoice{\@@lwtypeh{#1}}{\@twtypeh{#1}}{\@twtypeh{#1}}{\@twtypeh{#1}}}} +\def\lwtypeh#1{\@ifnextchar\bgroup{\@lwtypeh{#1}\lwtypeh}{\@@lwtypeh{#1}}} +\def\@lwtypeh#1{\mathchoice{{\textstyle\mathsf{W}^h}}{\mathsf{W}^h}{\mathsf{W}^h}{\mathsf{W}^h}({\textstyle #1})\;} +\def\@@lwtypeh#1{\mathchoice{{\textstyle\mathsf{W}^h}}{\mathsf{W}^h}{\mathsf{W}^h}{\mathsf{W}^h}({\textstyle #1}),\ } +\def\twtypeh#1{\@twtypeh{#1}\@ifnextchar\bgroup{\twtypeh}{}} +\def\@twtypeh#1{\mathchoice{{\textstyle\mathsf{W}^h_{(#1)}}}{\mathsf{W}^h_{(#1)}}{\mathsf{W}^h_{(#1)}}{\mathsf{W}^h_{(#1)}}} +\def\dwtypeh#1{\@dwtypeh{#1}\@ifnextchar\bgroup{\dwtypeh}{}} +\def\@dwtypeh#1{\mathsf{W}^h_{(#1)}\,} + + +\makeatother + +% Other notations related to dependent sums +\let\setof\Set % from package 'braket', write \setof{ x:A | P(x) }. +% \newcommand{\pair}{\ensuremath{\mathsf{pair}}\xspace} +\newcommand{\tup}[2]{(#1,#2)} +\newcommand{\proj}[1]{\ensuremath{\mathsf{pr}_{#1}}\xspace} +\newcommand{\fst}{\ensuremath{\proj1}\xspace} +\newcommand{\snd}{\ensuremath{\proj2}\xspace} +\newcommand{\ac}{\ensuremath{\mathsf{ac}}\xspace} % not needed in symbol index +\newcommand{\un}{\ensuremath{\mathsf{upun}}\xspace} % not needed in symbol index, uniqueness principle for unit type + +%%% recursor and induction +\newcommand{\rec}[1]{\mathsf{rec}_{#1}} +\newcommand{\ind}[1]{\mathsf{ind}_{#1}} +\newcommand{\indid}[1]{\ind{=_{#1}}} % (Martin-Lof) path induction principle for identity types +\newcommand{\indidb}[1]{\ind{=_{#1}}'} % (Paulin-Mohring) based path induction principle for identity types + +%%% the uniqueness principle for product types, formerly called surjective pairing and named \spr: +\newcommand{\uppt}{\ensuremath{\mathsf{uppt}}\xspace} + +% Paths in pairs +\newcommand{\pairpath}{\ensuremath{\mathsf{pair}^{\mathord{=}}}\xspace} +% \newcommand{\projpath}[1]{\proj{#1}^{\mathord{=}}} +\newcommand{\projpath}[1]{\ensuremath{\apfunc{\proj{#1}}}\xspace} + +%%% For quotients %%% +%\newcommand{\pairr}[1]{{\langle #1\rangle}} +\newcommand{\pairr}[1]{{\mathopen{}(#1)\mathclose{}}} +\newcommand{\Pairr}[1]{{\mathopen{}\left(#1\right)\mathclose{}}} + +% \newcommand{\type}{\ensuremath{\mathsf{Type}}} % this command is overridden below, so it's commented out +\newcommand{\im}{\ensuremath{\mathsf{im}}} % the image + +%%% 2D path operations +\newcommand{\leftwhisker}{\mathbin{{\ct}_{\ell}}} +\newcommand{\rightwhisker}{\mathbin{{\ct}_{r}}} +\newcommand{\hct}{\star} + +%%% modalities %%% +\newcommand{\modal}{\ensuremath{\ocircle}} +\let\reflect\modal +\newcommand{\modaltype}{\ensuremath{\type_\modal}} +% \newcommand{\ism}[1]{\ensuremath{\mathsf{is}_{#1}}} +% \newcommand{\ismodal}{\ism{\modal}} +% \newcommand{\existsmodal}{\ensuremath{{\exists}_{\modal}}} +% \newcommand{\existsmodalunique}{\ensuremath{{\exists!}_{\modal}}} +% \newcommand{\modalfunc}{\textsf{\modal-fun}} +% \newcommand{\Ecirc}{\ensuremath{\mathsf{E}_\modal}} +% \newcommand{\Mcirc}{\ensuremath{\mathsf{M}_\modal}} +\newcommand{\mreturn}{\ensuremath{\eta}} +\let\project\mreturn +%\newcommand{\mbind}[1]{\ensuremath{\hat{#1}}} +\newcommand{\ext}{\mathsf{ext}} +%\newcommand{\mmap}[1]{\ensuremath{\bar{#1}}} +%\newcommand{\mjoin}{\ensuremath{\mreturn^{-1}}} +% Subuniverse +\renewcommand{\P}{\ensuremath{\type_{P}}\xspace} + +%%% Localizations +% \newcommand{\islocal}[1]{\ensuremath{\mathsf{islocal}_{#1}}\xspace} +% \newcommand{\loc}[1]{\ensuremath{\mathcal{L}_{#1}}\xspace} + +%%% Identity types %%% +\newcommand{\idsym}{{=}} +% \newcommand{\id}[3][]{\ensuremath{#2 =_{#1} #3}\xspace} +\newcommand{\idtype}[3][]{\ensuremath{\mathsf{Id}_{#1}(#2,#3)}\xspace} +\newcommand{\idtypevar}[1]{\ensuremath{\mathsf{Id}_{#1}}\xspace} +% A propositional equality currently being defined +\newcommand{\defid}{\coloneqq} + +%%% Dependent paths +\newcommand{\dpath}[4]{#3 =^{#1}_{#2} #4} + +%%% singleton +% \newcommand{\sgl}{\ensuremath{\mathsf{sgl}}\xspace} +% \newcommand{\sctr}{\ensuremath{\mathsf{sctr}}\xspace} + +%%% Reflexivity terms %%% +% \newcommand{\reflsym}{{\mathsf{refl}}} +% \newcommand{\refl}[1]{\ensuremath{\mathsf{refl}_{#1}}\xspace} + +%%% Path concatenation (used infix, in diagrammatic order) %%% +\newcommand{\ct}{% + \mathchoice{\mathbin{\raisebox{0.5ex}{$\displaystyle\centerdot$}}}% + {\mathbin{\raisebox{0.5ex}{$\centerdot$}}}% + {\mathbin{\raisebox{0.25ex}{$\scriptstyle\,\centerdot\,$}}}% + {\mathbin{\raisebox{0.1ex}{$\scriptscriptstyle\,\centerdot\,$}}} +} + +%%% Path reversal %%% +\newcommand{\opp}[1]{\mathord{{#1}^{-1}}} +\let\rev\opp + +%%% Transport (covariant) %%% +\newcommand{\trans}[2]{\ensuremath{{#1}_{*}\mathopen{}\left({#2}\right)\mathclose{}}\xspace} +\let\Trans\trans +%\newcommand{\Trans}[2]{\ensuremath{{#1}_{*}\left({#2}\right)}\xspace} +\newcommand{\transf}[1]{\ensuremath{{#1}_{*}}\xspace} % Without argument +%\newcommand{\transport}[2]{\ensuremath{\mathsf{transport}_{*} \: {#2}\xspace}} +\newcommand{\transfib}[3]{\ensuremath{\mathsf{transport}^{#1}(#2,#3)\xspace}} +\newcommand{\Transfib}[3]{\ensuremath{\mathsf{transport}^{#1}\Big(#2,\, #3\Big)\xspace}} +\newcommand{\transfibf}[1]{\ensuremath{\mathsf{transport}^{#1}\xspace}} + +%%% 2D transport +\newcommand{\transtwo}[2]{\ensuremath{\mathsf{transport}^2\mathopen{}\left({#1},{#2}\right)\mathclose{}}\xspace} + +%%% Constant transport +\newcommand{\transconst}[3]{\ensuremath{\mathsf{transportconst}}^{#1}_{#2}(#3)\xspace} +\newcommand{\transconstf}{\ensuremath{\mathsf{transportconst}}\xspace} + +%%% Map on paths %%% +\newcommand{\mapfunc}[1]{\ensuremath{\mathsf{ap}_{#1}}\xspace} % Without argument +\newcommand{\map}[2]{\ensuremath{{#1}\mathopen{}\left({#2}\right)\mathclose{}}\xspace} +\let\Ap\map +%\newcommand{\Ap}[2]{\ensuremath{{#1}\left({#2}\right)}\xspace} +\newcommand{\mapdepfunc}[1]{\ensuremath{\mathsf{apd}_{#1}}\xspace} % Without argument +% \newcommand{\mapdep}[2]{\ensuremath{{#1}\llparenthesis{#2}\rrparenthesis}\xspace} +\newcommand{\mapdep}[2]{\ensuremath{\mapdepfunc{#1}\mathopen{}\left(#2\right)\mathclose{}}\xspace} +\let\apfunc\mapfunc +\let\ap\map +\let\apdfunc\mapdepfunc +\let\apd\mapdep + +%%% 2D map on paths +\newcommand{\aptwofunc}[1]{\ensuremath{\mathsf{ap}^2_{#1}}\xspace} +\newcommand{\aptwo}[2]{\ensuremath{\aptwofunc{#1}\mathopen{}\left({#2}\right)\mathclose{}}\xspace} +\newcommand{\apdtwofunc}[1]{\ensuremath{\mathsf{apd}^2_{#1}}\xspace} +\newcommand{\apdtwo}[2]{\ensuremath{\apdtwofunc{#1}\mathopen{}\left(#2\right)\mathclose{}}\xspace} + +%%% Identity functions %%% +\newcommand{\idfunc}[1][]{\ensuremath{\mathsf{id}_{#1}}\xspace} + +%%% Homotopies (written infix) %%% +\newcommand{\htpy}{\sim} + +%%% Other meanings of \sim +\newcommand{\bisim}{\sim} % bisimulation +\newcommand{\eqr}{\sim} % an equivalence relation + +%%% Equivalence types %%% +\newcommand{\eqv}[2]{\ensuremath{#1 \simeq #2}\xspace} +\newcommand{\eqvspaced}[2]{\ensuremath{#1 \;\simeq\; #2}\xspace} +\newcommand{\eqvsym}{\simeq} % infix symbol +\newcommand{\texteqv}[2]{\ensuremath{\mathsf{Equiv}(#1,#2)}\xspace} +\newcommand{\isequiv}{\ensuremath{\mathsf{isequiv}}} +\newcommand{\qinv}{\ensuremath{\mathsf{qinv}}} +\newcommand{\ishae}{\ensuremath{\mathsf{ishae}}} +\newcommand{\linv}{\ensuremath{\mathsf{linv}}} +\newcommand{\rinv}{\ensuremath{\mathsf{rinv}}} +\newcommand{\biinv}{\ensuremath{\mathsf{biinv}}} +\newcommand{\lcoh}[3]{\mathsf{lcoh}_{#1}(#2,#3)} +\newcommand{\rcoh}[3]{\mathsf{rcoh}_{#1}(#2,#3)} +\newcommand{\hfib}[2]{{\mathsf{fib}}_{#1}(#2)} + +%%% Map on total spaces %%% +\newcommand{\total}[1]{\ensuremath{\mathsf{total}(#1)}} + +%%% Universe types %%% +%\newcommand{\type}{\ensuremath{\mathsf{Type}}\xspace} +\newcommand{\UU}{\ensuremath{\mathcal{U}}\xspace} +\let\bbU\UU +\let\type\UU +% Universes of truncated types +\newcommand{\typele}[1]{\ensuremath{{#1}\text-\mathsf{Type}}\xspace} +\newcommand{\typeleU}[1]{\ensuremath{{#1}\text-\mathsf{Type}_\UU}\xspace} +\newcommand{\typelep}[1]{\ensuremath{{(#1)}\text-\mathsf{Type}}\xspace} +\newcommand{\typelepU}[1]{\ensuremath{{(#1)}\text-\mathsf{Type}_\UU}\xspace} +\let\ntype\typele +\let\ntypeU\typeleU +\let\ntypep\typelep +\let\ntypepU\typelepU +\newcommand{\set}{\ensuremath{\mathsf{Set}}\xspace} +\newcommand{\setU}{\ensuremath{\mathsf{Set}_\UU}\xspace} +\newcommand{\prop}{\ensuremath{\mathsf{Prop}}\xspace} +\newcommand{\propU}{\ensuremath{\mathsf{Prop}_\UU}\xspace} +%Pointed types +\newcommand{\pointed}[1]{\ensuremath{#1_\bullet}} + +%%% Ordinals and cardinals +\newcommand{\card}{\ensuremath{\mathsf{Card}}\xspace} +\newcommand{\ord}{\ensuremath{\mathsf{Ord}}\xspace} +\newcommand{\ordsl}[2]{{#1}_{/#2}} + +%%% Univalence +\newcommand{\ua}{\ensuremath{\mathsf{ua}}\xspace} % the inverse of idtoeqv +\newcommand{\idtoeqv}{\ensuremath{\mathsf{idtoeqv}}\xspace} +\newcommand{\univalence}{\ensuremath{\mathsf{univalence}}\xspace} % the full axiom + +%%% Truncation levels +\newcommand{\iscontr}{\ensuremath{\mathsf{isContr}}} +\newcommand{\contr}{\ensuremath{\mathsf{contr}}} % The path to the center of contraction +\newcommand{\isset}{\ensuremath{\mathsf{isSet}}} +\newcommand{\isprop}{\ensuremath{\mathsf{isProp}}} +% h-propositions +% \newcommand{\anhprop}{a mere proposition\xspace} +% \newcommand{\hprops}{mere propositions\xspace} + +%%% Homotopy fibers %%% +%\newcommand{\hfiber}[2]{\ensuremath{\mathsf{hFiber}(#1,#2)}\xspace} +\let\hfiber\hfib + +%%% Bracket/squash/truncation types %%% +% \newcommand{\brck}[1]{\textsf{mere}(#1)} +% \newcommand{\Brck}[1]{\textsf{mere}\Big(#1\Big)} +% \newcommand{\trunc}[2]{\tau_{#1}(#2)} +% \newcommand{\Trunc}[2]{\tau_{#1}\Big(#2\Big)} +% \newcommand{\truncf}[1]{\tau_{#1}} +%\newcommand{\trunc}[2]{\Vert #2\Vert_{#1}} +\newcommand{\trunc}[2]{\mathopen{}\left\Vert #2\right\Vert_{#1}\mathclose{}} +\newcommand{\ttrunc}[2]{\bigl\Vert #2\bigr\Vert_{#1}} +\newcommand{\Trunc}[2]{\Bigl\Vert #2\Bigr\Vert_{#1}} +\newcommand{\truncf}[1]{\Vert \blank \Vert_{#1}} +\newcommand{\tproj}[3][]{\mathopen{}\left|#3\right|_{#2}^{#1}\mathclose{}} +\newcommand{\tprojf}[2][]{|\blank|_{#2}^{#1}} +\def\pizero{\trunc0} +%\newcommand{\brck}[1]{\trunc{-1}{#1}} +%\newcommand{\Brck}[1]{\Trunc{-1}{#1}} +%\newcommand{\bproj}[1]{\tproj{-1}{#1}} +%\newcommand{\bprojf}{\tprojf{-1}} + +\newcommand{\brck}[1]{\trunc{}{#1}} +\newcommand{\bbrck}[1]{\ttrunc{}{#1}} +\newcommand{\Brck}[1]{\Trunc{}{#1}} +\newcommand{\bproj}[1]{\tproj{}{#1}} +\newcommand{\bprojf}{\tprojf{}} + +% Big parentheses +\newcommand{\Parens}[1]{\Bigl(#1\Bigr)} + +% Projection and extension for truncations +\let\extendsmb\ext +\newcommand{\extend}[1]{\extendsmb(#1)} + +% +%%% The empty type +\newcommand{\emptyt}{\ensuremath{\mathbf{0}}\xspace} + +%%% The unit type +\newcommand{\unit}{\ensuremath{\mathbf{1}}\xspace} +\newcommand{\ttt}{\ensuremath{\star}\xspace} + +%%% The two-element type +\newcommand{\bool}{\ensuremath{\mathbf{2}}\xspace} +\newcommand{\btrue}{{1_{\bool}}} +\newcommand{\bfalse}{{0_{\bool}}} + +%%% Injections into binary sums and pushouts +\newcommand{\inlsym}{{\mathsf{inl}}} +\newcommand{\inrsym}{{\mathsf{inr}}} +\newcommand{\inl}{\ensuremath\inlsym\xspace} +\newcommand{\inr}{\ensuremath\inrsym\xspace} + +%%% The segment of the interval +\newcommand{\seg}{\ensuremath{\mathsf{seg}}\xspace} + +%%% Free groups +\newcommand{\freegroup}[1]{F(#1)} +\newcommand{\freegroupx}[1]{F'(#1)} % the "other" free group + +%%% Glue of a pushout +\newcommand{\glue}{\mathsf{glue}} + +%%% Circles and spheres +\newcommand{\Sn}{\mathbb{S}} +\newcommand{\base}{\ensuremath{\mathsf{base}}\xspace} +\newcommand{\lloop}{\ensuremath{\mathsf{loop}}\xspace} +\newcommand{\surf}{\ensuremath{\mathsf{surf}}\xspace} + +%%% Suspension +\newcommand{\susp}{\Sigma} +\newcommand{\north}{\mathsf{N}} +\newcommand{\south}{\mathsf{S}} +\newcommand{\merid}{\mathsf{merid}} + +%%% Blanks (shorthand for lambda abstractions) +\newcommand{\blank}{\mathord{\hspace{1pt}\text{--}\hspace{1pt}}} + +%%% Nameless objects +\newcommand{\nameless}{\mathord{\hspace{1pt}\underline{\hspace{1ex}}\hspace{1pt}}} + +%%% Some decorations +%\newcommand{\bbU}{\ensuremath{\mathbb{U}}\xspace} +% \newcommand{\bbB}{\ensuremath{\mathbb{B}}\xspace} +\newcommand{\bbP}{\ensuremath{\mathbb{P}}\xspace} + +%%% Some categories +\newcommand{\uset}{\ensuremath{\mathcal{S}et}\xspace} +\newcommand{\ucat}{\ensuremath{{\mathcal{C}at}}\xspace} +\newcommand{\urel}{\ensuremath{\mathcal{R}el}\xspace} +\newcommand{\uhilb}{\ensuremath{\mathcal{H}ilb}\xspace} +\newcommand{\utype}{\ensuremath{\mathcal{T}\!ype}\xspace} + +% Pullback corner +%\newbox\pbbox +%\setbox\pbbox=\hbox{\xy \POS(65,0)\ar@{-} (0,0) \ar@{-} (65,65)\endxy} +%\def\pb{\save[]+<3.5mm,-3.5mm>*{\copy\pbbox} \restore} + +% Macros for the categories chapter +% \newcommand{\inv}[1]{{#1}^{-1}} +\newcommand{\idtoiso}{\ensuremath{\mathsf{idtoiso}}\xspace} +\newcommand{\isotoid}{\ensuremath{\mathsf{isotoid}}\xspace} +\newcommand{\op}{^{\mathrm{op}}} +\newcommand{\y}{\ensuremath{\mathbf{y}}\xspace} +\newcommand{\dgr}[1]{{#1}^{\dagger}} +\newcommand{\unitaryiso}{\mathrel{\cong^\dagger}} +\newcommand{\cteqv}[2]{\ensuremath{#1 \simeq #2}\xspace} +\newcommand{\cteqvsym}{\simeq} % Symbol for equivalence of categories + +%%% Natural numbers +\newcommand{\N}{\ensuremath{\mathbb{N}}\xspace} +%\newcommand{\N}{\textbf{N}} +\let\nat\N +\newcommand{\natp}{\ensuremath{\nat'}\xspace} % alternative nat in induction chapter + +\newcommand{\zerop}{\ensuremath{0'}\xspace} % alternative zero in induction chapter +\newcommand{\suc}{\mathsf{succ}} +\newcommand{\sucp}{\ensuremath{\suc'}\xspace} % alternative suc in induction chapter +\newcommand{\add}{\mathsf{add}} +\newcommand{\ack}{\mathsf{ack}} +\newcommand{\ite}{\mathsf{iter}} +\newcommand{\assoc}{\mathsf{assoc}} +\newcommand{\dbl}{\ensuremath{\mathsf{double}}} +\newcommand{\dblp}{\ensuremath{\dbl'}\xspace} % alternative double in induction chapter + + +%%% Lists +\newcommand{\lst}[1]{\mathsf{List}(#1)} +\newcommand{\nil}{\mathsf{nil}} +\newcommand{\cons}{\mathsf{cons}} + +%%% Vectors of given length, used in induction chapter +\newcommand{\vect}[2]{\ensuremath{\mathsf{Vec}_{#1}(#2)}\xspace} + +%%% Integers +\newcommand{\Z}{\ensuremath{\mathbb{Z}}\xspace} +\newcommand{\Zsuc}{\mathsf{succ}} +\newcommand{\Zpred}{\mathsf{pred}} + +%%% Rationals +\newcommand{\Q}{\ensuremath{\mathbb{Q}}\xspace} + +%%% Function extensionality +\newcommand{\funext}{\mathsf{funext}} +\newcommand{\happly}{\mathsf{happly}} + +%%% A naturality lemma +\newcommand{\com}[3]{\mathsf{swap}_{#1,#2}(#3)} + +%%% Code/encode/decode +\newcommand{\code}{\ensuremath{\mathsf{code}}\xspace} +\newcommand{\encode}{\ensuremath{\mathsf{encode}}\xspace} +\newcommand{\decode}{\ensuremath{\mathsf{decode}}\xspace} + +% Function definition with domain and codomain +\newcommand{\function}[4]{\left\{\begin{array}{rcl}#1 & + \longrightarrow & #2 \\ #3 & \longmapsto & #4 \end{array}\right.} + +%%% Cones and cocones +\newcommand{\cone}[2]{\mathsf{cone}_{#1}(#2)} +\newcommand{\cocone}[2]{\mathsf{cocone}_{#1}(#2)} +% Apply a function to a cocone +\newcommand{\composecocone}[2]{#1\circ#2} +\newcommand{\composecone}[2]{#2\circ#1} +%%% Diagrams +\newcommand{\Ddiag}{\mathscr{D}} + +%%% (pointed) mapping spaces +\newcommand{\Map}{\mathsf{Map}} + +%%% The interval +\newcommand{\interval}{\ensuremath{I}\xspace} +\newcommand{\izero}{\ensuremath{0_{\interval}}\xspace} +\newcommand{\ione}{\ensuremath{1_{\interval}}\xspace} + +%%% Arrows +\newcommand{\epi}{\ensuremath{\twoheadrightarrow}} +\newcommand{\mono}{\ensuremath{\rightarrowtail}} + +%%% Sets +\newcommand{\bin}{\ensuremath{\mathrel{\widetilde{\in}}}} + +%%% Semigroup structure +\newcommand{\semigroupstrsym}{\ensuremath{\mathsf{SemigroupStr}}} +\newcommand{\semigroupstr}[1]{\ensuremath{\mathsf{SemigroupStr}}(#1)} +\newcommand{\semigroup}[0]{\ensuremath{\mathsf{Semigroup}}} + +%%% Macros for the formal type theory +\newcommand{\emptyctx}{\ensuremath{\cdot}} +\newcommand{\production}{\vcentcolon\vcentcolon=} +\newcommand{\conv}{\downarrow} +\newcommand{\ctx}{\ensuremath{\mathsf{ctx}}} +\newcommand{\wfctx}[1]{#1\ \ctx} +\newcommand{\oftp}[3]{#1 \vdash #2 : #3} +\newcommand{\jdeqtp}[4]{#1 \vdash #2 \jdeq #3 : #4} +\newcommand{\judg}[2]{#1 \vdash #2} +\newcommand{\tmtp}[2]{#1 \mathord{:} #2} + +% rule names +\newcommand{\form}{\textsc{form}} +\newcommand{\intro}{\textsc{intro}} +\newcommand{\elim}{\textsc{elim}} +\newcommand{\comp}{\textsc{comp}} +\newcommand{\uniq}{\textsc{uniq}} +\newcommand{\Weak}{\mathsf{Wkg}} +\newcommand{\Vble}{\mathsf{Vble}} +\newcommand{\Exch}{\mathsf{Exch}} +\newcommand{\Subst}{\mathsf{Subst}} + +%%% Macros for HITs +\newcommand{\cc}{\mathsf{c}} +\newcommand{\pp}{\mathsf{p}} +\newcommand{\cct}{\widetilde{\mathsf{c}}} +\newcommand{\ppt}{\widetilde{\mathsf{p}}} +\newcommand{\Wtil}{\ensuremath{\widetilde{W}}\xspace} + +%%% Macros for n-types +\newcommand{\istype}[1]{\mathsf{is}\mbox{-}{#1}\mbox{-}\mathsf{type}} +\newcommand{\nplusone}{\ensuremath{(n+1)}} +\newcommand{\nminusone}{\ensuremath{(n-1)}} +% \newcommand{\fact}{\mathsf{fact}} + +%%% Macros for homotopy +\newcommand{\kbar}{\overline{k}} % Used in van Kampen's theorem + +%%% Macros for induction +\newcommand{\natw}{\ensuremath{\mathbf{N^w}}\xspace} +\newcommand{\zerow}{\ensuremath{0^\mathbf{w}}\xspace} +\newcommand{\sucw}{\ensuremath{\mathbf{s^w}}\xspace} +\newcommand{\nalg}{\nat\mathsf{Alg}} +\newcommand{\nhom}{\nat\mathsf{Hom}} +\newcommand{\ishinitw}{\mathsf{isHinit}_{\mathsf{W}}} +\newcommand{\ishinitn}{\mathsf{isHinit}_\nat} +\newcommand{\w}{\mathsf{W}} +\newcommand{\walg}{\w\mathsf{Alg}} +\newcommand{\whom}{\w\mathsf{Hom}} + +%%% Macros for real numbers +\newcommand{\RC}{\ensuremath{\mathbb{R}_\mathsf{c}}\xspace} % Cauchy +\newcommand{\RD}{\ensuremath{\mathbb{R}_\mathsf{d}}\xspace} % Dedekind +\newcommand{\R}{\ensuremath{\mathbb{R}}\xspace} % Either +\newcommand{\barRD}{\ensuremath{\bar{\mathbb{R}}_\mathsf{d}}\xspace} % Dedekind completion of Dedekind + +\newcommand{\close}[1]{\sim_{#1}} % Relation of closeness +\newcommand{\closesym}{\mathord\sim} +\newcommand{\rclim}{\mathsf{lim}} % HIT constructor for Cauchy reals +\newcommand{\rcrat}{\mathsf{rat}} % Embedding of rationals into Cauchy reals +\newcommand{\rceq}{\mathsf{eq}_{\RC}} % HIT path constructor +\newcommand{\CAP}{\mathcal{C}} % The type of Cauchy approximations +\newcommand{\Qp}{\Q_{+}} +\newcommand{\apart}{\mathrel{\#}} % apartness +\newcommand{\dcut}{\mathsf{isCut}} % Dedekind cut +\newcommand{\cover}{\triangleleft} % inductive cover +\newcommand{\intfam}[3]{(#2, \lam{#1} #3)} % family of rational intervals + +% Macros for the Cauchy reals construction +\newcommand{\bsim}{\frown} +\newcommand{\bbsim}{\smile} + +\newcommand{\hapx}{\diamondsuit\approx} +\newcommand{\hapname}{\diamondsuit} +\newcommand{\hapxb}{\heartsuit\approx} +\newcommand{\hapbname}{\heartsuit} +\newcommand{\tap}[1]{\bullet\approx_{#1}\triangle} +\newcommand{\tapname}{\triangle} +\newcommand{\tapb}[1]{\bullet\approx_{#1}\square} +\newcommand{\tapbname}{\square} + +%%% Macros for surreals +\newcommand{\NO}{\ensuremath{\mathsf{No}}\xspace} +\newcommand{\surr}[2]{\{\,#1\,\big|\,#2\,\}} +\newcommand{\LL}{\mathcal{L}} +\newcommand{\RR}{\mathcal{R}} +\newcommand{\noeq}{\mathsf{eq}_{\NO}} % HIT path constructor + +\newcommand{\ble}{\trianglelefteqslant} +\newcommand{\blt}{\vartriangleleft} +\newcommand{\bble}{\sqsubseteq} +\newcommand{\bblt}{\sqsubset} + +\newcommand{\hle}{\diamondsuit\preceq} +\newcommand{\hlt}{\diamondsuit\prec} +\newcommand{\hlname}{\diamondsuit} +\newcommand{\hleb}{\heartsuit\preceq} +\newcommand{\hltb}{\heartsuit\prec} +\newcommand{\hlbname}{\heartsuit} +% \newcommand{\tle}{(\bullet\preceq\triangle)} +% \newcommand{\tlt}{(\bullet\prec\triangle)} +\newcommand{\tle}{\triangle\preceq} +\newcommand{\tlt}{\triangle\prec} +\newcommand{\tlname}{\triangle} +% \newcommand{\tleb}{(\bullet\preceq\square)} +% \newcommand{\tltb}{(\bullet\prec\square)} +\newcommand{\tleb}{\square\preceq} +\newcommand{\tltb}{\square\prec} +\newcommand{\tlbname}{\square} + +%%% Macros for set theory +\newcommand{\vset}{\mathsf{set}} % point constructor for cummulative hierarchy V +\def\cd{\tproj0} +\newcommand{\inj}{\ensuremath{\mathsf{inj}}} % type of injections +\newcommand{\acc}{\ensuremath{\mathsf{acc}}} % accessibility + +\newcommand{\atMostOne}{\mathsf{atMostOne}} + +\newcommand{\power}[1]{\mathcal{P}(#1)} % power set +\newcommand{\powerp}[1]{\mathcal{P}_+(#1)} % inhabited power set + +%%%% THEOREM ENVIRONMENTS %%%% + +% Hyperref includes the command \autoref{...} which is like \ref{...} +% except that it automatically inserts the type of the thing you're +% referring to, e.g. it produces "Theorem 3.8" instead of just "3.8" +% (and makes the whole thing a hyperlink). This saves a slight amount +% of typing, but more importantly it means that if you decide later on +% that 3.8 should be a Lemma or a Definition instead of a Theorem, you +% don't have to change the name in all the places you referred to it. + +% The following hack improves on this by using the same counter for +% all theorem-type environments, so that after Theorem 1.1 comes +% Corollary 1.2 rather than Corollary 1.1. This makes it much easier +% for the reader to find a particular theorem when flipping through +% the document. +\makeatletter +\def\defthm#1#2#3{% + %% Ensure all theorem types are numbered with the same counter + \newaliascnt{#1}{thm} + \newtheorem{#1}[#1]{#2} + \aliascntresetthe{#1} + %% This command tells cleveref's \cref what to call things + \crefname{#1}{#2}{#3}} + +% Now define a bunch of theorem-type environments. +\newtheorem{thm}{Theorem}[section] +\crefname{thm}{Theorem}{Theorems} +\defthm{propn}{Proposition}{Propositions} % Probably we shouldn't use "Proposition" in this way +\defthm{cor}{Corollary}{Corollaries} +\defthm{lem}{Lemma}{Lemmas} +\defthm{ax}{Axiom}{Axioms} +\defthm{conj}{Conjecture}{Conjectures} +% Since definitions and theorems in type theory are synonymous, should +% we actually use the same theoremstyle for them? +\theoremstyle{definition} +\defthm{defn}{Definition}{Definitions} +\theoremstyle{remark} +\defthm{rmk}{Remark}{Remarks} +\defthm{ex}{Example}{Examples} +\defthm{exs}{Examples}{Examples} +\defthm{notes}{Notes}{Notes} +% Number exercises within chapters, with their own counter. +%\newtheorem{ex}{Exercise}[chapter] +%\crefname{ex}{Exercise}{Exercises} + +% Display format for sections +\crefformat{section}{\S#2#1#3} +\Crefformat{section}{Section~#2#1#3} +\crefrangeformat{section}{\S\S#3#1#4--#5#2#6} +\Crefrangeformat{section}{Sections~#3#1#4--#5#2#6} +\crefmultiformat{section}{\S\S#2#1#3}{ and~#2#1#3}{, #2#1#3}{ and~#2#1#3} +\Crefmultiformat{section}{Sections~#2#1#3}{ and~#2#1#3}{, #2#1#3}{ and~#2#1#3} +\crefrangemultiformat{section}{\S\S#3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}{, #3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6} +\Crefrangemultiformat{section}{Sections~#3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}{, #3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6} + +% Display format for appendices +\crefformat{appendix}{Appendix~#2#1#3} +\Crefformat{appendix}{Appendix~#2#1#3} +\crefrangeformat{appendix}{Appendices~#3#1#4--#5#2#6} +\Crefrangeformat{appendix}{Appendices~#3#1#4--#5#2#6} +\crefmultiformat{appendix}{Appendices~#2#1#3}{ and~#2#1#3}{, #2#1#3}{ and~#2#1#3} +\Crefmultiformat{appendix}{Appendices~#2#1#3}{ and~#2#1#3}{, #2#1#3}{ and~#2#1#3} +\crefrangemultiformat{appendix}{Appendices~#3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}{, #3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6} +\Crefrangemultiformat{appendix}{Appendices~#3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6}{, #3#1#4--#5#2#6}{ and~#3#1#4--#5#2#6} + +\crefname{part}{Part}{Parts} + +% Number subsubsections +\setcounter{secnumdepth}{5} + +% Display format for figures +\crefname{figure}{Figure}{Figures} + +% Use cleveref instead of hyperref's \autoref +\let\autoref\cref + + +%%%% EQUATION NUMBERING %%%% + +% The following hack uses the single theorem counter to number +% equations as well, so that we don't have both Theorem 1.1 and +% equation (1.1). +\let\c@equation\c@thm +\numberwithin{equation}{section} + + +%%%% ENUMERATE NUMBERING %%%% + +% Number the first level of enumerates as (i), (ii), ... +\renewcommand{\theenumi}{(\roman{enumi})} +\renewcommand{\labelenumi}{\theenumi} + + +%%%% MARGINS %%%% + +% This is a matter of personal preference, but I think the left +% margins on enumerates and itemizes are too wide. +% \setitemize[1]{leftmargin=2em} +% \setenumerate[1]{leftmargin=*} + +% Likewise that they are too spaced out. +% \setitemize[1]{itemsep=-0.2em} +% \setenumerate[1]{itemsep=-0.2em} + +%%% Notes %%% +\def\noteson{% +\gdef\note##1{\mbox{}\marginpar{\color{blue}\textasteriskcentered\ ##1}}} +\gdef\notesoff{\gdef\note##1{\null}} +\noteson + +\newcommand{\Coq}{\textsc{Coq}\xspace} +\newcommand{\Agda}{\textsc{Agda}\xspace} +\newcommand{\NuPRL}{\textsc{NuPRL}\xspace} + +%%%% CITATIONS %%%% + +% \let \cite \citep + +%%%% INDEX %%%% + +\newcommand{\footstyle}[1]{{\hyperpage{#1}}n} % If you index something that is in a footnote +\newcommand{\defstyle}[1]{\textbf{\hyperpage{#1}}} % Style for pageref to a definition + +\newcommand{\indexdef}[1]{\index{#1|defstyle}} % Index a definition +\newcommand{\indexfoot}[1]{\index{#1|footstyle}} % Index a term in a footnote +\newcommand{\indexsee}[2]{\index{#1|see{#2}}} % Index "see also" + + +%%%% Standard phrasing or spelling of common phrases %%%% + +\newcommand{\ZF}{Zermelo--Fraenkel} +\newcommand{\CZF}{Constructive \ZF{} Set Theory} + +\newcommand{\LEM}[1]{\ensuremath{\mathsf{LEM}_{#1}}\xspace} +\newcommand{\choice}[1]{\ensuremath{\mathsf{AC}_{#1}}\xspace} + +%%%% MISC %%%% + +\newcommand{\mentalpause}{\medskip} % Use for "mental" pause, instead of \smallskip or \medskip + +%% Use \symlabel instead of \label to mark a pageref that you need in the index of symbols +\newcounter{symindex} +\newcommand{\symlabel}[1]{\refstepcounter{symindex}\label{#1}} + +% Local Variables: +% mode: latex +% TeX-master: "hott-online" +% End: