Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.Point.toPresheafFiberOfIsCofiltered

{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] →
                          (U : N) →
                            (P : CategoryTheory.Functor Cᵒᵖ A) →
                              P.obj (Opposite.op (p.obj U)) ⟶
                                (CategoryTheory.GrothendieckTopology.Point.ofIsCofiltered p hp).presheafFiber.obj P

The canonical maps P.obj (op (p.obj U)) ⟶ (ofIsCofiltered p hp).presheafFiber.obj P that are part of the colimit cocone presheafFiberOfIsCofilteredCocone.

Defined in

Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Cited by

5 results in Mathlib

Foundations

Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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