Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.IsLocalSite.pointPresheafFiberIso

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (J : CategoryTheory.GrothendieckTopology C) →
      [inst_1 : CategoryTheory.LocallySmall.{w, v, u} C] →
        [inst_2 : J.IsLocalSite] →
          {A : Type u'} →
            [inst_3 : CategoryTheory.Category.{v', u'} A] →
              [inst_4 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v', u'} A] →
                (P : CategoryTheory.Functor Cᵒᵖ A) →
                  (CategoryTheory.GrothendieckTopology.IsLocalSite.point J).presheafFiber.obj P ≅
                    P.obj (Opposite.op (⊤_ C))

The fibre of any presheaf P : Cᵒᵖ ⥤ A at point J is just P evaluated at the terminal object.

Defined in

Mathlib.CategoryTheory.Sites.LocalSite

Cited by

4 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.GrothendieckTopology.IsLocalSiteCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfSize