Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.diagramCompIso

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (J : CategoryTheory.GrothendieckTopology C) →
      {D : Type u_1} →
        [inst_1 : CategoryTheory.Category.{v_1, u_1} D] →
          {E : Type u_2} →
            [inst_2 : CategoryTheory.Category.{v_2, u_2} E] →
              (F : CategoryTheory.Functor D E) →
                [inst_3 :
                    ∀ (J : CategoryTheory.Limits.MulticospanShape),
                      CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.Limits.WalkingMulticospan J) D] →
                  [inst_4 :
                      ∀ (J : CategoryTheory.Limits.MulticospanShape),
                        CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.Limits.WalkingMulticospan J) E] →
                    [∀ (X : C) (W : J.Cover X) (P : CategoryTheory.Functor Cᵒᵖ D),
                          CategoryTheory.Limits.PreservesLimit (W.index P).multicospan F] →
                      (P : CategoryTheory.Functor Cᵒᵖ D) → (X : C) → (J.diagram P X).comp F ≅ J.diagram (P.comp F) X

The diagram used to define P⁺, composed with F, is isomorphic to the diagram used to define P ⋙ F.

Defined in

Mathlib.CategoryTheory.Sites.CompatiblePlus

Cited by

8 results in Mathlib

Foundations

Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.PreservesLimit