Theorems · Theorem · category theory

CategoryTheory.Presieve.isSheafFor_of_preservesProduct

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor Cᵒᵖ (Type w)) {α : Type u_1}
  [Small.{w, u_1} α] {X : α → C} (c : CategoryTheory.Limits.Cofan X) (hc : CategoryTheory.Limits.IsColimit c)
  [(CategoryTheory.Presieve.ofArrows X c.inj).HasPairwisePullbacks]
  [CategoryTheory.Limits.PreservesLimit (CategoryTheory.Discrete.functor fun x => Opposite.op (X x)) F],
  CategoryTheory.Presieve.IsSheafFor F (CategoryTheory.Presieve.ofArrows X c.inj)

If F preserves a particular product, then it IsSheafFor the corresponding presieve of arrows.

Defined in

Mathlib.CategoryTheory.Sites.Preserves

Cited by

2 results in Mathlib

Foundations

Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategorySmallCategoryTheory.Presieve.HasPairwisePullbacksCategoryTheory.Limits.PreservesLimit