Theorems · Theorem · category theory

CategoryTheory.Presheaf.isLocallyInjective_iff_injective_of_separated

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D]
  {FD : D → D → Type u_1} {CD : D → Type w} [inst_2 : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)]
  [inst_3 : CategoryTheory.ConcreteCategory D FD] (J : CategoryTheory.GrothendieckTopology C)
  {F₁ F₂ : CategoryTheory.Functor Cᵒᵖ D} (φ : F₁ ⟶ F₂),
  CategoryTheory.Presieve.IsSeparated J (F₁.comp (CategoryTheory.forget D)) →
    (CategoryTheory.Presheaf.IsLocallyInjective J φ ↔
      ∀ (X : Cᵒᵖ), Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom (φ.app X)))

Defined in

Mathlib.CategoryTheory.Sites.LocallyInjective

Cited by

1 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategory