Theorems · Definition · category theory

CategoryTheory.Localization.SmallHom.equiv

{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) →
            [L.IsLocalization W] →
              {X Y : C} →
                [inst_3 : CategoryTheory.Localization.HasSmallLocalizedHom W X Y] →
                  CategoryTheory.Localization.SmallHom W X Y ≃ (L.obj X ⟶ L.obj Y)

The canonical bijection SmallHom.{w} W X Y ≃ (L.obj X ⟶ L.obj Y) when L is a localization functor for W : MorphismProperty C and that HasSmallLocalizedHom.{w} W X Y holds.

Defined in

Mathlib.CategoryTheory.Localization.SmallHom

Cited by

25 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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