Theorems · Inductive type · category theory

CategoryTheory.GrothendieckTopology.WEqualsLocallyBijective

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    CategoryTheory.GrothendieckTopology C →
      (A : Type u') →
        [inst : CategoryTheory.Category.{v', u'} A] →
          {FA : A → A → Type u_1} →
            {CA : A → Type w'} →
              [inst_1 : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] → [CategoryTheory.ConcreteCategory A FA] → Prop

Given a category C equipped with a Grothendieck topology J and a concrete category A, this property holds if a morphism in Cᵒᵖ ⥤ A satisfies J.W (i.e. becomes an iso after sheafification) iff it is both locally injective and locally surjective.

Defined in

Mathlib.CategoryTheory.Sites.LocallyBijective

Cited by

142 results in Mathlib

Foundations

Depth 4 from the axioms, rests on 7 definitions · uses no axioms

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategory