Theorems · Theorem · category theory

TopCat.Sheaf.eq_app_of_locally_eq

∀ {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.Sheaf C X} {ι : Type u_4} {U : ι → TopologicalSpace.Opens ↑X} {V : TopologicalSpace.Opens ↑X}
  {G : TopCat.Sheaf C X} {f : F ⟶ G} {s : CategoryTheory.ToType (F.obj.obj (Opposite.op (iSup U)))}
  {t : CategoryTheory.ToType (G.obj.obj (Opposite.op V))}
  {sf : (i : ι) → CategoryTheory.ToType (F.obj.obj (Opposite.op (U i)))},
  TopCat.Presheaf.IsGluing F.obj U sf s →
    ∀ (hV : ∀ (i : ι), U i ≤ V),
      (∀ (i : ι),
          (CategoryTheory.ConcreteCategory.hom (f.hom.app (Opposite.op (U i)))) (sf i) =
            (CategoryTheory.ConcreteCategory.hom (G.obj.map (CategoryTheory.homOfLE ⋯).op)) t) →
        (CategoryTheory.ConcreteCategory.hom (f.hom.app (Opposite.op (iSup U)))) s =
          (CategoryTheory.ConcreteCategory.hom (G.obj.map (CategoryTheory.homOfLE ⋯).op)) t

Defined in

Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Cited by

2 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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