Theorems · Definition · category theory

CategoryTheory.Presheaf.IsSheaf.amalgamate

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {J : CategoryTheory.GrothendieckTopology C} →
      {A : Type u₂} →
        [inst_1 : CategoryTheory.Category.{v₂, u₂} A] →
          {E : A} →
            {X : C} →
              {P : CategoryTheory.Functor Cᵒᵖ A} →
                CategoryTheory.Presheaf.IsSheaf J P →
                  (S : J.Cover X) →
                    (x : (I : S.Arrow) → E ⟶ P.obj (Opposite.op I.Y)) →
                      (∀ ⦃I₁ I₂ : S.Arrow⦄ (r : I₁.Relation I₂),
                          CategoryTheory.CategoryStruct.comp (x I₁) (P.map r.g₁.op) =
                            CategoryTheory.CategoryStruct.comp (x I₂) (P.map r.g₂.op)) →
                        (E ⟶ P.obj (Opposite.op X))

This is a wrapper around Presieve.IsSheafFor.amalgamate to be used below. If P is a sheaf, S is a cover of X, and x is a collection of morphisms from E to P evaluated at terms in the cover which are compatible, then we can amalgamate the xs to obtain a single morphism E ⟶ P.obj (op X).

Defined in

Mathlib.CategoryTheory.Sites.Sheaf

Cited by

5 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Category