Theorems · Definition · category theory

CategoryTheory.Comma.coconeOfPreserves

{J : Type w} →
  [inst : CategoryTheory.Category.{w', w} J] →
    {A : Type u₁} →
      [inst_1 : CategoryTheory.Category.{v₁, u₁} A] →
        {B : Type u₂} →
          [inst_2 : CategoryTheory.Category.{v₂, u₂} B] →
            {T : Type u₃} →
              [inst_3 : CategoryTheory.Category.{v₃, u₃} T] →
                {L : CategoryTheory.Functor A T} →
                  {R : CategoryTheory.Functor B T} →
                    (F : CategoryTheory.Functor J (CategoryTheory.Comma L R)) →
                      [CategoryTheory.Limits.PreservesColimit (F.comp (CategoryTheory.Comma.fst L R)) L] →
                        {c₁ : CategoryTheory.Limits.Cocone (F.comp (CategoryTheory.Comma.fst L R))} →
                          CategoryTheory.Limits.IsColimit c₁ →
                            CategoryTheory.Limits.Cocone (F.comp (CategoryTheory.Comma.snd L R)) →
                              CategoryTheory.Limits.Cocone F

If L preserves the appropriate colimit, then given a colimit cocone for F ⋙ fst L R : J ⥤ L and a cocone for F ⋙ snd L R : J ⥤ R we can build a cocone for F which will turn out to be a colimit cocone.

Defined in

Mathlib.CategoryTheory.Limits.Comma

Cited by

5 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.PreservesColimit