Theorems · Definition · category theory

CategoryTheory.CostructuredArrow.toOverCompCoyoneda

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (A : CategoryTheory.Functor Cᵒᵖ (Type v)) →
      (CategoryTheory.CostructuredArrow.toOver CategoryTheory.yoneda A).op.comp CategoryTheory.coyoneda ≅
        CategoryTheory.yoneda.op.comp
          (CategoryTheory.coyoneda.comp
            ((CategoryTheory.Functor.whiskeringLeft (CategoryTheory.Over A)
                  (CategoryTheory.Functor (CategoryTheory.CostructuredArrow CategoryTheory.yoneda A)ᵒᵖ (Type v))
                  (Type (max u v))).obj
              (CategoryTheory.overEquivPresheafCostructuredArrow A).functor))

This isomorphism says that hom-sets in the category Over A for a presheaf A where the domain is of the form (CostructuredArrow.toOver yoneda A).obj X can instead be interpreted as hom-sets in the category (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v where the domain is of the form yoneda.obj X after adjusting the codomain accordingly. This is desirable because in the latter case the Yoneda lemma can be applied.

Defined in

Mathlib.CategoryTheory.Comma.Presheaf.Basic

Cited by

2 results in Mathlib

Foundations

Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Category