Theorems · Definition · category theory

CategoryTheory.Limits.colimitYonedaHomIsoLimit

{C : Type u₁} →
  [inst : CategoryTheory.Category.{u₂, u₁} C] →
    {I : Type v₁} →
      [inst_1 : CategoryTheory.Category.{v₂, v₁} I] →
        (D : CategoryTheory.Functor Iᵒᵖ Cᵒᵖ) →
          (F : CategoryTheory.Functor Cᵒᵖ (Type u₂)) →
            [inst_2 : CategoryTheory.Limits.HasColimit (D.unop.comp CategoryTheory.yoneda)] →
              [inst_3 : CategoryTheory.Limits.HasLimitsOfShape Iᵒᵖ (Type (max u₁ u₂))] →
                (CategoryTheory.Limits.colimit (D.unop.comp CategoryTheory.yoneda) ⟶ F) ≅
                  CategoryTheory.Limits.limit (D.comp (F.comp CategoryTheory.uliftFunctor.{u₁, u₂}))

Ind-Yoneda lemma: morphisms from colimit of yoneda of diagram D to F is limit of F evaluated at D. This version is for covariant diagrams, see colimitYonedaHomIsoLimit' for a contravariant version.

Defined in

Mathlib.CategoryTheory.Limits.IndYoneda

Cited by

1 results in Mathlib

Foundations

Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitCategoryTheory.Limits.HasLimitsOfShape