Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.Point.presheafFiberOfIsCofilteredCocone

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [CategoryTheory.LocallySmall.{w, v, u} C] →
      {N : Type u'} →
        [inst_2 : CategoryTheory.Category.{v', u'} N] →
          (p : CategoryTheory.Functor N C) →
            [CategoryTheory.InitiallySmall N] →
              {J : CategoryTheory.GrothendieckTopology C} →
                [CategoryTheory.IsCofiltered N] →
                  (∀ ⦃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] →
                        [CategoryTheory.Limits.HasColimitsOfSize.{w, w, v'', u''} A] →
                          (P : CategoryTheory.Functor Cᵒᵖ A) → CategoryTheory.Limits.Cocone (p.op.comp P)

The (colimit) cocone which, for a point constructed using Point.ofIsCofiltered and a functor p : N ⥤ C expresses the fiber of a presheaf as a colimit indexed indexed by N.

Defined in

Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Cited by

2 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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