Theorems · Theorem · category theory

CategoryTheory.Sheaf.isLocallyBijective_iff_isIso

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {A : Type u'}
  [inst_1 : CategoryTheory.Category.{v', u'} A] {FA : A → A → Type u_1} {CA : A → Type w'}
  [inst_2 : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] [inst_3 : CategoryTheory.ConcreteCategory A FA]
  {F G : CategoryTheory.Sheaf J A} (f : F ⟶ G) [(CategoryTheory.forget A).ReflectsIsomorphisms]
  [J.HasSheafCompose (CategoryTheory.forget A)],
  CategoryTheory.Sheaf.IsLocallyInjective f ∧ CategoryTheory.Sheaf.IsLocallySurjective f ↔ CategoryTheory.IsIso f

Defined in

Mathlib.CategoryTheory.Sites.LocallyBijective

Cited by

1 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Functor.ReflectsIsomorphismsCategoryTheory.GrothendieckTopology.HasSheafCompose