Theorems · Theorem · category theory

CategoryTheory.Localization.SmallHom.equiv_comp

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {W : CategoryTheory.MorphismProperty C} {D : Type u₂}
  [inst_1 : CategoryTheory.Category.{v₂, u₂} D] (L : CategoryTheory.Functor C D) [inst_2 : L.IsLocalization W]
  {X Y Z : C} [inst_3 : CategoryTheory.Localization.HasSmallLocalizedHom W X Y]
  [inst_4 : CategoryTheory.Localization.HasSmallLocalizedHom W Y Z]
  [inst_5 : CategoryTheory.Localization.HasSmallLocalizedHom W X Z] (α : CategoryTheory.Localization.SmallHom W X Y)
  (β : CategoryTheory.Localization.SmallHom W Y Z),
  (CategoryTheory.Localization.SmallHom.equiv W L) (α.comp β) =
    CategoryTheory.CategoryStruct.comp ((CategoryTheory.Localization.SmallHom.equiv W L) α)
      ((CategoryTheory.Localization.SmallHom.equiv W L) β)

Defined in

Mathlib.CategoryTheory.Localization.SmallHom

Cited by

11 results in Mathlib

Foundations

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

Assumes

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