Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.Point.ofIsCofiltered

{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) →
                    J.Point

Constructor for points of Grothendieck topologies J : GrothendieckTopology C that are given by a functor p : N ⥤ C from a cofiltered and initially small category N.

Defined in

Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered

Cited by

7 results in Mathlib

Foundations

Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.CategoryCategoryTheory.InitiallySmallCategoryTheory.IsCofiltered