Theorems · Theorem · category theory

CategoryTheory.Localization.SmallHom.equiv_equiv_symm

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] (W : CategoryTheory.MorphismProperty C) {D : Type u₂}
  [inst_1 : CategoryTheory.Category.{v₂, u₂} D] {D' : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} D']
  (L : CategoryTheory.Functor C D) [inst_3 : L.IsLocalization W] (L' : CategoryTheory.Functor C D')
  [inst_4 : L'.IsLocalization W] (G : CategoryTheory.Functor D D') (e : L.comp G ≅ L') {X Y : C}
  [inst_5 : CategoryTheory.Localization.HasSmallLocalizedHom W X Y] (f : L.obj X ⟶ L.obj Y),
  (CategoryTheory.Localization.SmallHom.equiv W L') ((CategoryTheory.Localization.SmallHom.equiv W L).symm f) =
    CategoryTheory.CategoryStruct.comp (e.inv.app X) (CategoryTheory.CategoryStruct.comp (G.map f) (e.hom.app Y))

Defined in

Mathlib.CategoryTheory.Localization.SmallHom

Cited by

2 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.IsLocalizationCategoryTheory.Localization.HasSmallLocalizedHom