Theorems · Theorem · category theory

CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.jointly_reflect_ofArrows_mem_of_small

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C}
  {P : CategoryTheory.ObjectProperty J.Point} [CategoryTheory.LocallySmall.{w, v, u} C]
  [CategoryTheory.HasSheafify J (Type w)] [J.WEqualsLocallyBijective (Type w)],
  P.IsConservativeFamilyOfPoints →
    ∀ [CategoryTheory.ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P] {X : C} {ι : Type u_1} {U : ι → C}
      (f : (i : ι) → U i ⟶ X),
      CategoryTheory.Sieve.ofArrows U f ∈ J X ↔
        ∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
          ∃ i y, (CategoryTheory.ConcreteCategory.hom (Φ.obj.fiber.map (f i))) y = x

Defined in

Mathlib.CategoryTheory.Sites.Point.Conservative

Cited by

1 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.HasSheafifyCategoryTheory.GrothendieckTopology.WEqualsLocallyBijectiveCategoryTheory.ObjectProperty.Small