Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.Point.isColimitPresheafFiberMapCocone

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {D : Type u'} →
      [inst_1 : CategoryTheory.Category.{v', u'} D] →
        {J : CategoryTheory.GrothendieckTopology C} →
          (Φ : J.Point) →
            (F : CategoryTheory.Functor C D) →
              (K : CategoryTheory.GrothendieckTopology D) →
                [inst_2 : F.IsCocontinuous J K] →
                  [inst_3 : CategoryTheory.LocallySmall.{w, v', u'} D] →
                    {A : Type u''} →
                      [inst_4 : CategoryTheory.Category.{v'', u''} A] →
                        [inst_5 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v'', u''} A] →
                          (P : CategoryTheory.Functor Dᵒᵖ A) →
                            CategoryTheory.Limits.IsColimit (Φ.presheafFiberMapCocone F K P)

Given a cocontinuous functor F : C ⥤ D between sites (C, J) and (D, K), P a presheaf on D, then (Φ.map F K).presheafFiber.obj P is a colimit of P.obj (op (F.obj X)) for X : C, x : Φ.fiber.obj X.

Defined in

Mathlib.CategoryTheory.Sites.Point.Map

Cited by

2 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsCocontinuousCategoryTheory.LocallySmallCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfSize