Theorems · Theorem · category theory

CategoryTheory.Adjunction.localization_counit_app

∀ {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [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} D₁]
  [inst_3 : CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁}
  (adj : G ⊣ F) (L₁ : CategoryTheory.Functor C₁ D₁) (W₁ : CategoryTheory.MorphismProperty C₁)
  [inst_4 : L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂) (W₂ : CategoryTheory.MorphismProperty C₂)
  [inst_5 : L₂.IsLocalization W₂] (G' : CategoryTheory.Functor D₁ D₂) (F' : CategoryTheory.Functor D₂ D₁)
  [inst_6 : CategoryTheory.CatCommSq G L₁ L₂ G'] [inst_7 : CategoryTheory.CatCommSq F L₂ L₁ F'] (X₂ : C₂),
  (adj.localization L₁ W₁ L₂ W₂ G' F').counit.app (L₂.obj X₂) =
    CategoryTheory.CategoryStruct.comp (G'.map ((CategoryTheory.CatCommSq.iso F L₂ L₁ F').inv.app X₂))
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.CatCommSq.iso G L₁ L₂ G').inv.app (F.obj X₂))
        (L₂.map (adj.counit.app X₂)))

Defined in

Mathlib.CategoryTheory.Localization.Adjunction

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.CatCommSq