Theorems · Theorem · category theory

CategoryTheory.LocalizerMorphism.isEquivalence_imp

∀ {C₁ : Type u₁} {C₂ : Type u₂} {D₁ : Type u₄} {D₂ : Type u₅} [inst : CategoryTheory.Category.{v₁, u₁} C₁]
  [inst_1 : CategoryTheory.Category.{v₂, u₂} C₂] [inst_2 : CategoryTheory.Category.{v₄, u₄} D₁]
  [inst_3 : CategoryTheory.Category.{v₅, u₅} D₂] {W₁ : CategoryTheory.MorphismProperty C₁}
  {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂)
  (L₁ : CategoryTheory.Functor C₁ D₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂]
  (G : CategoryTheory.Functor D₁ D₂) [CategoryTheory.CatCommSq Φ.functor L₁ L₂ G] {D₁' : Type u₄'} {D₂' : Type u₅'}
  [inst_7 : CategoryTheory.Category.{v₄', u₄'} D₁'] [inst_8 : CategoryTheory.Category.{v₅', u₅'} D₂']
  (L₁' : CategoryTheory.Functor C₁ D₁') (L₂' : CategoryTheory.Functor C₂ D₂') [L₁'.IsLocalization W₁]
  [L₂'.IsLocalization W₂] (G' : CategoryTheory.Functor D₁' D₂') [CategoryTheory.CatCommSq Φ.functor L₁' L₂' G']
  [G.IsEquivalence], G'.IsEquivalence

If a localizer morphism induces an equivalence on some choice of localized categories, it will be so for any choice of localized categories.

Defined in

Mathlib.CategoryTheory.Localization.LocalizerMorphism

Cited by

1 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.IsLocalizationCategoryTheory.CatCommSqCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.IsLocalizationCategoryTheory.CatCommSqCategoryTheory.Functor.IsEquivalence