Theorems · Definition · category theory

CategoryTheory.yonedaYonedaColimit

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {J : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} J] →
        [inst_2 : CategoryTheory.Limits.HasColimitsOfShape J (Type v₁)] →
          [inst_3 : CategoryTheory.Limits.HasColimitsOfShape J (Type (max u₁ v₁))] →
            (F : CategoryTheory.Functor J (CategoryTheory.Functor Cᵒᵖ (Type v₁))) →
              CategoryTheory.yoneda.op.comp (CategoryTheory.yoneda.obj (CategoryTheory.Limits.colimit F)) ≅
                CategoryTheory.yoneda.op.comp (CategoryTheory.Limits.colimit (F.comp CategoryTheory.yoneda))

Naturally in X, we have Hom(YX, colim_i Fi) ≅ colim_i Hom(YX, Fi).

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Yoneda

Cited by

1 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfShapeCategoryTheory.Limits.HasColimitsOfShape