Theorems · Definition · category theory

CategoryTheory.Limits.isColimitOfHasPushoutOfPreservesColimit

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        (G : CategoryTheory.Functor C D) →
          {W X Y : C} →
            (f : W ⟶ X) →
              (g : W ⟶ Y) →
                [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span f g) G] →
                  [i : CategoryTheory.Limits.HasPushout f g] →
                    CategoryTheory.Limits.IsColimit
                      (CategoryTheory.Limits.PushoutCocone.mk (G.map (CategoryTheory.Limits.pushout.inl f g))
                        (G.map (CategoryTheory.Limits.pushout.inr f g)) ⋯)

If G preserves pushouts and C has them, then the pushout cocone constructed of the mapped morphisms of the pushout cocone is a colimit.

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Cited by

4 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.PreservesColimitCategoryTheory.Limits.HasPushout