Lemma FiberExt_inverses {C1 C2 : Complex A} (f g : Morphism C1 C2) (H : ComplexHomot A C1 C2) (Comm : to_binop _ _ ((f : (ComplexPreCat_Additive A)⟦_, _⟧)) (to_inv ((g : (ComplexPreCat_Additive A)⟦_, _⟧))) = ComplexHomotMorphism A H) : is_inverse_in_precat (# (ComplexHomotFunctor A) (FiberExt f g H Comm)) (# (ComplexHomotFunctor A) (FiberExt g f (InvHomot H) (InvHomotEq f g H Comm))). Proof. use make_is_inverse_in_precat. - rewrite <- functor_id. rewrite <- functor_comp. apply maponpaths. exact (FiberExt_eq1 f g H Comm). - rewrite <- functor_id. rewrite <- functor_comp. apply maponpaths. exact (FiberExt_eq2 f g H Comm). Qed.
Require Import UniMath.Foundations.UnivalenceAxiom. Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.Algebra.Monoids. Require Import UniMath.NumberSystems.Integers. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Kernels. Require Import UniMath.CategoryTheory.Limits.Cokernels. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.BinDirectSums. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.PrecategoriesWithAbgrops. Require Import UniMath.CategoryTheory.PreAdditive. Require Import UniMath.CategoryTheory.Additive. Require Import UniMath.CategoryTheory.Abelian. Require Import UniMath.CategoryTheory.AbelianToAdditive. Require Import UniMath.CategoryTheory.AdditiveFunctors. Require Import UniMath.HomologicalAlgebra.Complexes. Require Import UniMath.HomologicalAlgebra.KA. Require Import UniMath.HomologicalAlgebra.TranslationFunctors.