Theorems · Theorem · category theory

CategoryTheory.LocalizerMorphism.homMap_apply_assoc

∀ {C₁ : Type u_2} {C₂ : Type u_3} {D₁ : Type u_5} {D₂ : Type u_6} [inst : CategoryTheory.Category.{v_2, u_2} C₁]
  [inst_1 : CategoryTheory.Category.{v_3, u_3} C₂] [inst_2 : CategoryTheory.Category.{v_5, u_5} D₁]
  [inst_3 : CategoryTheory.Category.{v_6, u_6} D₂] {W₁ : CategoryTheory.MorphismProperty C₁}
  {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂)
  (L₁ : CategoryTheory.Functor C₁ D₁) [inst_4 : L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂)
  [inst_5 : L₂.IsLocalization W₂] {X Y : C₁} (G : CategoryTheory.Functor D₁ D₂) (e : Φ.functor.comp L₂ ≅ L₁.comp G)
  (f : L₁.obj X ⟶ L₁.obj Y) {Z : D₂} (h : L₂.obj (Φ.functor.obj Y) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (Φ.homMap L₁ L₂ f) h =
    CategoryTheory.CategoryStruct.comp (e.hom.app X)
      (CategoryTheory.CategoryStruct.comp (G.map f) (CategoryTheory.CategoryStruct.comp (e.inv.app Y) h))

Defined in

Mathlib.CategoryTheory.Localization.HomEquiv

Cited by

0 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.IsLocalization