Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {J : CategoryTheory.GrothendieckTopology C} →
      (Φ : J.Point) →
        {A : Type u'} →
          [inst_1 : CategoryTheory.Category.{v', u'} A] →
            [inst_2 : CategoryTheory.Limits.HasProducts A] →
              {P : CategoryTheory.Functor Cᵒᵖ A} →
                {M : A} →
                  [inst_3 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v', u'} A] →
                    (Φ.presheafFiber.obj P ⟶ M) ≃ (P ⟶ Φ.skyscraperPresheaf M)

If Φ is a point of a site (C, J), P : Cᵒᵖ ⥤ A and M : A, this is the bijection (Φ.presheafFiber.obj P ⟶ M) ≃ (P ⟶ Φ.skyscraperPresheaf M) that is part of the adjunction skyscraperPresheafAdjunction.

Defined in

Mathlib.CategoryTheory.Sites.Point.Skyscraper

Cited by

18 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasProductsCategoryTheory.Limits.HasColimitsOfSize