Theorems · Definition · category theory

CategoryTheory.Presheaf.coconeCompShrinkYonedaHomEquiv

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {J : Type u'} →
      [inst_1 : CategoryTheory.Category.{v', u'} J] →
        [inst_2 : CategoryTheory.LocallySmall.{w, v, u} C] →
          {F : CategoryTheory.Functor J Cᵒᵖ} →
            {c' : CategoryTheory.Limits.Cocone (F.leftOp.comp CategoryTheory.shrinkYoneda.{w, v, u})} →
              CategoryTheory.Limits.IsColimit c' →
                {P : CategoryTheory.Functor Cᵒᵖ (Type w)} → (c'.pt ⟶ P) ≃ ↑(F.comp P).sections

Let F : J ⥤ Cᵒᵖ be a functor, c' a colimit cocone for F.leftOp ⋙ shrinkYoneda.{w}. For any P : Cᵒᵖ ⥤ Type w, this is the bijection between c'.pt ⟶ P and the type of sections of F ⋙ P.

Defined in

Mathlib.CategoryTheory.Limits.Types.PreservesLimit

Cited by

5 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.LocallySmall