lean2/hott/hit/hit.md

hit
===

Declaration and theorems of higher inductive types in Lean. We take
two higher inductive types (hits) as primitive notions in Lean. We
define all other hits in terms of these two hits. The hits which are
primitive are n-truncation and quotients. These are defined in
[init.hit](../init/hit.hlean) and they have definitional computation
rules on the point constructors.

Here we find hits related to the basic structure theory of HoTT.  The
hits related to homotopy theory are defined in
[homotopy](../homotopy/homotopy.md).

Files in this folder:

* [quotient](quotient.hlean): quotients, primitive
* [trunc](trunc.hlean): truncation, primitive
* [colimit](colimit.hlean): Colimits of arbitrary diagrams and sequential colimits, defined using quotients
* [pushout](pushout.hlean): Pushouts, defined using quotients
* [coeq](coeq.hlean): Co-equalizers, defined using quotients
* [set_quotient](set_quotient.hlean): Set-quotients, defined using quotients and set-truncation

The following hits have also 2-constructors. They are defined only using quotients.
* [two_quotient](two_quotient.hlean): Quotients where you can also specify 2-paths. These are used for all hits which have 2-constructors, and they are almost fully general for such hits, as long as they are nonrecursive
* [refl_quotient](refl_quotient.hlean): Quotients where you can also specify 2-paths
feat(hott): add hit-markdown file 2015-04-25 00:13:08 +00:00			`hit`
			`===`

feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places 2015-05-14 02:01:48 +00:00			`Declaration and theorems of higher inductive types in Lean. We take`
			`two higher inductive types (hits) as primitive notions in Lean. We`
			`define all other hits in terms of these two hits. The hits which are`
renaming(hit): rename type_quotient to quotient, and quotient to set_quotient This renaming is because type_quotient is a nonstandard name. I've had a discussion with Egbert Rijke, Steve Awodey and Dan Licata, and the consensus for a better name was 'quotient'. I had to make changes in src/kernel/hits/hits.cpp, I renamed g_type_quotient* by g_hit_quotient* (to avoid name clash the standard library quotient, although I don't know whether that name clash would matter). 2015-06-04 19:57:00 +00:00			`primitive are n-truncation and quotients. These are defined in`
			`[init.hit](../init/hit.hlean) and they have definitional computation`
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places 2015-05-14 02:01:48 +00:00			`rules on the point constructors.`
feat(hott): add hit-markdown file 2015-04-25 00:13:08 +00:00
refactor(hott): move homotopy hits to new homotopy folder 2015-09-13 18:58:11 +00:00			`Here we find hits related to the basic structure theory of HoTT. The`
			`hits related to homotopy theory are defined in`
			`[homotopy](../homotopy/homotopy.md).`

feat(hott): add hit-markdown file 2015-04-25 00:13:08 +00:00			`Files in this folder:`

feat(hott): add recursor to refl_quotient 2015-11-22 08:08:31 +00:00			`* [quotient](quotient.hlean): quotients, primitive`
			`* [trunc](trunc.hlean): truncation, primitive`
			`* [colimit](colimit.hlean): Colimits of arbitrary diagrams and sequential colimits, defined using quotients`
			`* [pushout](pushout.hlean): Pushouts, defined using quotients`
			`* [coeq](coeq.hlean): Co-equalizers, defined using quotients`
			`* [set_quotient](set_quotient.hlean): Set-quotients, defined using quotients and set-truncation`

			`The following hits have also 2-constructors. They are defined only using quotients.`
			`* [two_quotient](two_quotient.hlean): Quotients where you can also specify 2-paths. These are used for all hits which have 2-constructors, and they are almost fully general for such hits, as long as they are nonrecursive`
			`* [refl_quotient](refl_quotient.hlean): Quotients where you can also specify 2-paths`