# favonia/homotopy

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 ------------------------------------------------------------------------ -- My Code for Homotopy Type Theory -- -- Favonia ------------------------------------------------------------------------ -- Copyright (c) 2012 Favonia -- A large portion of code was copied from Nils Anders Danielsson' -- library released under BSD-like license -- http://www.cse.chalmers.se/~nad/repos/equality/ -- Copyright (c) 2011-2012 Nils Anders Danielsson -- See LICENSE for detailed copyright notice. {-# OPTIONS --without-K #-} module README where ------------------------------------------------------------------------ -- Common definitions import Prelude ------------------------------------------------------------------------ -- Maps, continuous functions between spaces -- Homotopy equivalence import Map.H-equivalence -- Injections import Map.Injection -- Surjections import Map.Surjection -- Homotopy Fiber import Map.H-fiber -- Weak equivalent import Map.WeakEquivalence ------------------------------------------------------------------------ -- Paths (propositional equalities in type theories) -- The definition of paths and trans, subst, cong import Path -- Some really basic lemmas for equivalence of paths import Path.Lemmas -- Fancy Unicode symbols for writing incomprehensible proofs -- (Subjecting to changes.) import Path.Symbol -- Higher-order paths and loops import Path.HigherOrder -- A short proof that Ω₂(A) is abelian for any space A import Path.Omega2-abelian -- Tools to compose/decompose paths in Σ-type import Path.Sum -- Tools to manipulate paths in Π-type (extensionality) --import Path.Prod -- Definition of H-level and some basic lemmas --import Path.H-level ------------------------------------------------------------------------ -- Space -- Kristina's theorem: hom is contractable iff we have a dependent -- eliminator. -- (Only the interesting direction.) import Space.Bool.Initial -- Basic facts about Fin import Space.Fin.Lemmas -- Definition of flowers import Space.Flower -- A proof that Ω₁(Flower) is FreeGroup (currently broken) -- import Space.Flower.Omega1 -- Definition of free groups (currently broken) -- import Space.FreeGroup -- Definition of integers import Space.Integer -- Definition of intervals import Space.Interval -- Basic facts about Fin import Space.List.Lemmas -- Some basic facts about Nat -- (Definition of Nat is in the Prelude) import Space.Nat.Lemmas -- Definition of spheres (base + loop) import Space.Sphere -- Alternative definition of spheres (two-point) import Space.Sphere.TwoPoints -- Definition of the Hopf junior (S₀ ↪ S₁ → S₁) -- and a proof that the total space is indeed S₁ import Space.Sphere.HopfJunior -- A proof that Ω₁(S₁) is ℤ import Space.Sphere.Omega1 -- Definition of torus import Space.Torus ------------------------------------------------------------------------ -- The Univalence axiom -- Definition of the Univalence axiom import Univalence -- A proof that the Univalence axiom implies extensionality for functions -- Might be moved to Path.Prod later --import Univalence.Extensionality -- Some basic lemmas implied by the Univalence axiom import Univalence.Lemmas