Theorems · Definition · category theory

CategoryTheory.Limits.pasteVertIsPushout

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {Y₃ Y₂ Y₁ X₃ : C} →
      {g₂ : Y₃ ⟶ Y₂} →
        {g₁ : Y₂ ⟶ Y₁} →
          {i₃ : Y₃ ⟶ X₃} →
            {t₁ : CategoryTheory.Limits.PushoutCocone g₂ i₃} →
              {i₂ : Y₂ ⟶ t₁.pt} →
                {t₂ : CategoryTheory.Limits.PushoutCocone g₁ i₂} →
                  (hi₂ : i₂ = t₁.inl) →
                    CategoryTheory.Limits.IsColimit t₁ →
                      CategoryTheory.Limits.IsColimit t₂ → CategoryTheory.Limits.IsColimit (t₁.pasteVert t₂ hi₂)

Given `` Y₃ - i₃ -> X₃ | | g₂ f₂ ∨ ∨ Y₂ - i₂ -> X₂ | | g₁ f₁ ∨ ∨ Y₁ - i₁ -> X₁ `` The big square is a pushout if both the small squares are.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Pullback.Pasting

Cited by

4 results in Mathlib

Foundations

Depth 35 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Category