Theorems · Theorem · category theory

CategoryTheory.Presheaf.isLocallyInjective_iff_equalizerSieve_mem_imp

∀ {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.Presheaf.IsLocallyInjective J φ ↔
    ∀ ⦃X : Cᵒᵖ⦄ (x y : CategoryTheory.ToType (F₁.obj X)),
      CategoryTheory.Presheaf.equalizerSieve ((CategoryTheory.ConcreteCategory.hom (φ.app X)) x)
            ((CategoryTheory.ConcreteCategory.hom (φ.app X)) y) ∈
          J (Opposite.unop X) →
        CategoryTheory.Presheaf.equalizerSieve x y ∈ J (Opposite.unop X)

Defined in

Mathlib.CategoryTheory.Sites.LocallyInjective

Cited by

1 results in Mathlib

Foundations

Depth 35 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategory