Theorems · Theorem · category theory

CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.W_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C}
  {P : CategoryTheory.ObjectProperty J.Point},
  P.IsConservativeFamilyOfPoints →
    ∀ {A : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} A] [CategoryTheory.LocallySmall.{w, v, u} C]
      [inst_3 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v', u'} A] {FC : A → A → Type u_1} {CC : A → Type w}
      [inst_4 : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)] [inst_5 : CategoryTheory.ConcreteCategory A FC]
      [(CategoryTheory.forget A).ReflectsIsomorphisms]
      [CategoryTheory.Limits.PreservesFilteredColimitsOfSize.{w, w, v', w, u', w + 1} (CategoryTheory.forget A)]
      [hJ : J.HasSheafCompose (CategoryTheory.forget A)] {F G : CategoryTheory.Functor Cᵒᵖ A} (f : F ⟶ G)
      [CategoryTheory.HasWeakSheafify J A] [CategoryTheory.Limits.HasProducts A],
      J.W f ↔ ∀ (Φ : P.FullSubcategory), CategoryTheory.IsIso (Φ.obj.presheafFiber.map f)

Defined in

Mathlib.CategoryTheory.Sites.Point.Conservative

Cited by

1 results in Mathlib

Foundations

Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.Limits.HasColimitsOfSizeFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Functor.ReflectsIsomorphismsCategoryTheory.Limits.PreservesFilteredColimitsOfSizeCategoryTheory.GrothendieckTopology.HasSheafComposeCategoryTheory.HasWeakSheafifyCategoryTheory.Limits.HasProducts