Theorems · Theorem · category theory

CategoryTheory.PresheafHom.IsSheafFor.exists_app

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {A : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} A]
  {F G : CategoryTheory.Functor Cᵒᵖ A} {X : C} {S : CategoryTheory.Sieve X}
  (hG :
    ⦃Y : C⦄ →
      (f : Y ⟶ X) → CategoryTheory.Limits.IsLimit (G.mapCone (CategoryTheory.Sieve.pullback f S).arrows.cocone.op))
  (x : CategoryTheory.Presieve.FamilyOfElements (CategoryTheory.presheafHom F G) S.arrows) {Y : C},
  x.Compatible →
    ∀ (g : Y ⟶ X),
      ∃ φ,
        ∀ {Z : C} (p : Z ⟶ Y) (hp : S.arrows (CategoryTheory.CategoryStruct.comp p g)),
          CategoryTheory.CategoryStruct.comp φ (G.map p.op) =
            CategoryTheory.CategoryStruct.comp (F.map p.op)
              ((x (CategoryTheory.CategoryStruct.comp p g) hp).app
                (Opposite.op (CategoryTheory.Over.mk (CategoryTheory.CategoryStruct.id Z))))

Defined in

Mathlib.CategoryTheory.Sites.SheafHom

Cited by

1 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Category