Theorems · Theorem · category theory

TopCat.Presheaf.germ_eq_of_isBasis

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasColimits C] {X : TopCat}
  {FC : C → C → Type u_1} {CC : C → Type v} [inst_2 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)]
  [instCC : CategoryTheory.ConcreteCategory C FC]
  [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] {B : Set (TopologicalSpace.Opens ↑X)},
  TopologicalSpace.Opens.IsBasis B →
    ∀ (F : TopCat.Presheaf C X) {U V : TopologicalSpace.Opens ↑X} (x : ↑X) (mU : x ∈ U) (mV : x ∈ V)
      {s : CategoryTheory.ToType (F.obj (Opposite.op U))} {t : CategoryTheory.ToType (F.obj (Opposite.op V))},
      (CategoryTheory.ConcreteCategory.hom (F.germ U x mU)) s =
          (CategoryTheory.ConcreteCategory.hom (F.germ V x mV)) t →
        ∃ W,
          ∃ (_ : x ∈ W) (_ : W ∈ B) (hWU : W ≤ U) (hWV : W ≤ V),
            (CategoryTheory.ConcreteCategory.hom (F.map (CategoryTheory.homOfLE hWU).op)) s =
              (CategoryTheory.ConcreteCategory.hom (F.map (CategoryTheory.homOfLE hWV).op)) t

Defined in

Mathlib.Topology.Sheaves.Stalks

Cited by

1 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasColimitsFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesFilteredColimits