Theorems · Theorem · category theory

TopCat.Presheaf.app_surjective_of_injective_of_locally_surjective

∀ {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)] [CategoryTheory.Limits.HasLimits C]
  [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)] [(CategoryTheory.forget C).ReflectsIsomorphisms]
  {F G : TopCat.Sheaf C X} (f : F ⟶ G) (U : TopologicalSpace.Opens ↑X),
  (∀ x ∈ U, Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom ((TopCat.Presheaf.stalkFunctor C x).map f.hom))) →
    (∀ (t : CC (G.obj.obj (Opposite.op U))),
        ∀ x ∈ U,
          ∃ V,
            ∃ (_ : x ∈ V),
              ∃ iVU s,
                (CategoryTheory.ConcreteCategory.hom (f.hom.app (Opposite.op V))) s =
                  (CategoryTheory.ConcreteCategory.hom (G.obj.map iVU.op)) t) →
      Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (f.hom.app (Opposite.op U)))

For surjectivity, we are given an arbitrary section t and need to find a preimage for it. We claim that it suffices to find preimages locally. That is, for each x : U we construct a neighborhood V ≤ U and a section s : F.obj (op V)) such that f.app (op V) s and t agree on V.

Defined in

Mathlib.Topology.Sheaves.Stalks

Cited by

1 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasColimitsFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesFilteredColimitsCategoryTheory.Limits.HasLimitsCategoryTheory.Limits.PreservesLimitsCategoryTheory.Functor.ReflectsIsomorphisms