Theorems · Definition · category theory

CategoryTheory.CostructuredArrow.toOverCompYoneda

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

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

Assumes

CategoryTheory.Category