lean2/hott/homotopy/homotopy.md

homotopy
========

Development of Homotopy Theory, including basic hits (higher inductive
types; see also [hit](../hit/hit.md)). The following files are in this
folder (sorted such that files only import previous files).

* [connectedness](connectedness.hlean) (n-Connectedness of types and functions)
* [cylinder](cylinder.hlean) (Mapping cylinders, defined using quotients)
* [susp](susp.hlean) (Suspensions, defined using pushouts)
* [sphere](sphere.hlean) (Higher spheres, defined recursively using suspensions)
* [circle](circle.hlean) (defined as sphere 1)
* [interval](interval.hlean) (defined as the suspension of unit)
* [cellcomplex](cellcomplex.hlean) (general cell complexes)
* [cofiber](cofiber.hlean)
* [wedge](wedge.hlean)
* [smash](smash.hlean)
* [join](join.hlean)
* [freudenthal](freudenthal.hlean) (The Freudenthal Suspension Theorem)
* [hopf](hopf.hlean) (the Hopf construction and delooping of coherent connected H-spaces)
* [complex_hopf](complex_hopf.hlean) (the complex Hopf fibration)
* [imaginaroid](imaginaroid.hlean) (imaginaroids as a variant of the Cayley-Dickson construction)
* [quaternionic_hopf](quaternionic_hopf.hlean) (the quaternionic Hopf fibration)
* [chain_complex](chain_complex.hlean)
* [LES_of_homotopy_groups](LES_of_homotopy_groups.hlean)
* [vankampen](vankampen.hlean) (the Seifert-van Kampen theorem)
* [homotopy_group](homotopy_group.hlean) (theorems about homotopy groups. The definition and basic facts about homotopy groups is in [algebra/homotopy_group](../algebra/homotopy_group.hlean))
* [sphere2](sphere2.hlean) (calculation of the homotopy group of spheres)


The following files depend on
[hit.two_quotient](../hit/two_quotient.hlean) which on turn depends on
[circle](circle.hlean).

* [red_susp](red_susp.hlean) (Reduced suspensions)
* [torus](torus.hlean) (defined as a two-quotient)
* [EM](EM.hlean): Eilenberg MacLane spaces
refactor(hott): move homotopy hits to new homotopy folder 2015-09-13 18:58:11 +00:00			`homotopy`
			`========`

			`Development of Homotopy Theory, including basic hits (higher inductive`
			`types; see also [hit](../hit/hit.md)). The following files are in this`
			`folder (sorted such that files only import previous files).`

feat(hott): update book.md and homotopy.md to reflect additions 2016-03-07 01:03:27 +00:00			`* [connectedness](connectedness.hlean) (n-Connectedness of types and functions)`
refactor(hott): move homotopy hits to new homotopy folder 2015-09-13 18:58:11 +00:00			`* [cylinder](cylinder.hlean) (Mapping cylinders, defined using quotients)`
			`* [susp](susp.hlean) (Suspensions, defined using pushouts)`
			`* [sphere](sphere.hlean) (Higher spheres, defined recursively using suspensions)`
			`* [circle](circle.hlean) (defined as sphere 1)`
			`* [interval](interval.hlean) (defined as the suspension of unit)`
			`* [cellcomplex](cellcomplex.hlean) (general cell complexes)`
fix(hott): add missing links to markdown files 2016-02-23 18:00:16 +00:00			`* [cofiber](cofiber.hlean)`
feat(hott): update book.md and homotopy.md to reflect additions 2016-03-07 01:03:27 +00:00			`* [wedge](wedge.hlean)`
fix(hott): add missing links to markdown files 2016-02-23 18:00:16 +00:00			`* [smash](smash.hlean)`
feat(hott): update book.md and homotopy.md to reflect additions 2016-03-07 01:03:27 +00:00			`* [join](join.hlean)`
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT. 2016-04-22 19:12:25 +00:00			`* [freudenthal](freudenthal.hlean) (The Freudenthal Suspension Theorem)`
feat(hott): update book.md and homotopy.md to reflect additions 2016-03-07 01:03:27 +00:00			`* [hopf](hopf.hlean) (the Hopf construction and delooping of coherent connected H-spaces)`
			`* [complex_hopf](complex_hopf.hlean) (the complex Hopf fibration)`
			`* [imaginaroid](imaginaroid.hlean) (imaginaroids as a variant of the Cayley-Dickson construction)`
			`* [quaternionic_hopf](quaternionic_hopf.hlean) (the quaternionic Hopf fibration)`
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT. 2016-04-22 19:12:25 +00:00			`* [chain_complex](chain_complex.hlean)`
			`* [LES_of_homotopy_groups](LES_of_homotopy_groups.hlean)`
feat(vankampen): prove the van Kampen theorem with basepoints 2016-06-23 20:09:18 +00:00			`* [vankampen](vankampen.hlean) (the Seifert-van Kampen theorem)`
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT. 2016-04-22 19:12:25 +00:00			`* [homotopy_group](homotopy_group.hlean) (theorems about homotopy groups. The definition and basic facts about homotopy groups is in [algebra/homotopy_group](../algebra/homotopy_group.hlean))`
			`* [sphere2](sphere2.hlean) (calculation of the homotopy group of spheres)`
feat(hott): update book.md and homotopy.md to reflect additions 2016-03-07 01:03:27 +00:00
feat(vankampen): prove the van Kampen theorem with basepoints 2016-06-23 20:09:18 +00:00
feat(hott): update book.md and homotopy.md to reflect additions 2016-03-07 01:03:27 +00:00			`The following files depend on`
			`[hit.two_quotient](../hit/two_quotient.hlean) which on turn depends on`
			`[circle](circle.hlean).`

			`* [red_susp](red_susp.hlean) (Reduced suspensions)`
			`* [torus](torus.hlean) (defined as a two-quotient)`
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT. 2016-04-22 19:12:25 +00:00			`* [EM](EM.hlean): Eilenberg MacLane spaces`