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Mirror of https://github.com/cmu-phil/Spectral in case it ever disappears
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Floris van Doorn 5c9355c4c1 feat(chain_complex): give the construction of the LES of homotopy groups 
This commit defines "type_chain_complex" which is a typal variant of a chain complex, where the exactness condition is formulated without a propositional truncation in it. The fiber sequence of a pointed map is an instance of this structure.
It also defines "chain_complex" which is the usual notion of a chain complex: a sequence of pointed sets with pointed maps between them, such that the kernel and image of consecutive maps coincide.
The biggest part of this commit is the definition of the long exact sequence of homotopy groups of a pointed map. The definition uses the fiber sequence of a pointed map.
2016-02-22 20:53:48 -05:00
group_theory fix(*): fix broken files 2016-02-04 13:38:33 -05:00
homotopy feat(chain_complex): give the construction of the LES of homotopy groups 2016-02-22 20:53:48 -05:00
Notes add rough sketch of dependency graph 2016-01-21 14:18:00 -05:00
.gitignore add .gitignore and .project files 2015-11-20 17:55:33 -05:00
.project add .gitignore and .project files 2015-11-20 17:55:33 -05:00
README.md added HoTT Book Chapter 8 to do 2016-01-21 14:25:17 -05:00

README.md

Spectral Sequences

Formalization project of the CMU HoTT group towards formalizing the Serre spectral sequence.

Participants

Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Floris van Doorn, Clive Newstead, Egbert Rijke, Mike Shulman.

Resources

  • Mike's blog post at the HoTT blog.
  • Mike's blog post at the n-category café.
  • The Licata-Finster article about Eilenberg-Mac Lane spaces.
  • We learned about the Serre spectral sequence from Hatcher's chapter about spectral sequences.
  • Lang's algebra (revised 3rd edition) contains a chapter on general homology theory, with a section on spectral sequences. Thus, we can use this book at least as an outline for the algebraic part of the project.
  • Mac Lane's Homology contains a lot of homological algebra and a chapter on spectral sequences, including exact couples.

Things to do for Lean spectral sequences project

Algebra To Do:

Topology To Do:

  • HoTT Book chapter 8
  • fiber and cofiber sequences (is this in the library already?)
  • prespectra and spectra, suspension
  • spectrification
  • parametrized spectra, parametrized smash and hom between types and spectra
  • fiber and cofiber sequences of spectra, stability
  • long exact sequences from (co)fiber sequences of spectra
  • Eilenberg-MacLane spaces and spectra
  • Postnikov towers of spectra
  • exact couple of a tower of spectra

Already Done:

  • pointed types
  • definition of algebraic structures such as groups, rings, fields,
  • some algebra: quotient, product, free.