Theorems · Definition · category theory

CategoryTheory.Functor.colimitIsoOfIsLeftKanExtension

{C : Type u_1} →
  {H : Type u_3} →
    {D : Type u_4} →
      [inst : CategoryTheory.Category.{v_1, u_1} C] →
        [inst_1 : CategoryTheory.Category.{v_3, u_3} H] →
          [inst_2 : CategoryTheory.Category.{v_4, u_4} D] →
            (F' : CategoryTheory.Functor D H) →
              {L : CategoryTheory.Functor C D} →
                {F : CategoryTheory.Functor C H} →
                  (α : F ⟶ L.comp F') →
                    [F'.IsLeftKanExtension α] →
                      [inst_4 : CategoryTheory.Limits.HasColimit F] →
                        [inst_5 : CategoryTheory.Limits.HasColimit F'] →
                          CategoryTheory.Limits.colimit F' ≅ CategoryTheory.Limits.colimit F

If F' : D ⥤ H is a left Kan extension of F : C ⥤ H along L : C ⥤ D, the colimit over F' is isomorphic to the colimit over F.

Defined in

Mathlib.CategoryTheory.Functor.KanExtension.Basic

Cited by

8 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLeftKanExtensionCategoryTheory.Limits.HasColimitCategoryTheory.Limits.HasColimit