Theorems · Theorem · category theory

CategoryTheory.Functor.IsLocalization.of_equivalence_source

∀ {C₁ : Type u_1} {C₂ : Type u_2} {D : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C₁]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_3, u_3} D]
  (L₁ : CategoryTheory.Functor C₁ D) (W₁ : CategoryTheory.MorphismProperty C₁) (L₂ : CategoryTheory.Functor C₂ D)
  (W₂ : CategoryTheory.MorphismProperty C₂) (E : C₁ ≌ C₂),
  W₁ ≤ W₂.isoClosure.inverseImage E.functor →
    W₂.IsInvertedBy L₂ → ∀ [L₁.IsLocalization W₁] (iso : E.functor.comp L₂ ≅ L₁), L₂.IsLocalization W₂

If L₁ : C₁ ⥤ D is a localization functor for W₁ : MorphismProperty C₁, then it is also the case of a functor L₂ : C₂ ⥤ D for a suitable W₂ : MorphismProperty C₂ when we have an equivalence of category E : C₁ ≌ C₂ and an isomorphism E.functor ⋙ L₂ ≅ L₁.

Defined in

Mathlib.CategoryTheory.Localization.Equivalence

Cited by

2 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalization