Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.Point.presheafFiberOfIsCofilteredCocone_pt

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.LocallySmall.{w, v, u} C]
  {N : Type u'} [inst_2 : CategoryTheory.Category.{v', u'} N] (p : CategoryTheory.Functor N C)
  [inst_3 : CategoryTheory.InitiallySmall N] {J : CategoryTheory.GrothendieckTopology C}
  [inst_4 : CategoryTheory.IsCofiltered N]
  (hp :
    ∀ ⦃X : C⦄,
      ∀ R ∈ J X,
        ∀ ⦃U : N⦄ (f : p.obj U ⟶ X),
          ∃ Y g,
            ∃ (_ : R.arrows g),
              ∃ V q a, CategoryTheory.CategoryStruct.comp a g = CategoryTheory.CategoryStruct.comp (p.map q) f)
  {A : Type u''} [inst_5 : CategoryTheory.Category.{v'', u''} A]
  [inst_6 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v'', u''} A] (P : CategoryTheory.Functor Cᵒᵖ A),
  (CategoryTheory.GrothendieckTopology.Point.presheafFiberOfIsCofilteredCocone p hp P).pt =
    (CategoryTheory.GrothendieckTopology.Point.ofIsCofiltered p hp).presheafFiber.obj P

Defined in

Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Cited by

0 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.CategoryCategoryTheory.InitiallySmallCategoryTheory.IsCofilteredCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfSize