Theorems · Theorem · category theory

TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {FC : C → C → Type u_2} {CC : C → Type u_3}
  [inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_2 : CategoryTheory.ConcreteCategory C FC]
  [CategoryTheory.Limits.HasLimitsOfSize.{x, x, v_1, u_1} C] [(CategoryTheory.forget C).ReflectsIsomorphisms]
  [CategoryTheory.Limits.PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)] {X : TopCat}
  (F : TopCat.Presheaf C X), F.IsSheaf ↔ F.IsSheafUniqueGluing

For presheaves valued in a concrete category, whose forgetful functor reflects isomorphisms and preserves limits, the sheaf condition in terms of unique gluings is equivalent to the usual one.

Defined in

Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Cited by

0 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.HasLimitsOfSizeCategoryTheory.Functor.ReflectsIsomorphismsCategoryTheory.Limits.PreservesLimitsOfSize