Theorems · Definition · category theory
isPushoutOfIsInitialIsCoproduct
{C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{W X Y Z : C} →
(f : X ⟶ Z) →
(g : Y ⟶ Z) →
(h : W ⟶ X) →
(k : W ⟶ Y) →
(H₁ : CategoryTheory.Limits.IsInitial W) →
CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk f g) →
CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk f g ⋯)The coproduct is the pushout under the initial object.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CategoryTheory.Category
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Categorystatement and proof · cited by 32,673
- Quiver.Homstatement and proof · cited by 32,603
- CategoryTheory.CategoryStruct.compstatement · cited by 17,999
- CategoryTheory.Discretestatement · cited by 2,447
- CategoryTheory.Limits.WalkingPairstatement · cited by 1,319
- CategoryTheory.Limits.IsColimitstatement and proof · cited by 773
- CategoryTheory.Limits.pairstatement · cited by 536
- CategoryTheory.Limits.WalkingSpanstatement · cited by 300
- CategoryTheory.Limits.spanstatement · cited by 294
- CategoryTheory.Limits.IsInitialstatement and proof · cited by 158
- CategoryTheory.Limits.PushoutCocone.mkstatement and proof · cited by 89
- CategoryTheory.Limits.BinaryCofan.mkstatement and proof · cited by 83
Cited by4
Results whose statement or proof uses this declaration.
- CategoryTheory.Limits.pushoutZeroZeroIsoproof · cited by 5
- CategoryTheory.IsPushout.of_is_coproductproof · cited by 3
- CategoryTheory.Limits.inr_pushoutZeroZeroIso_homproof · cited by 1
- CategoryTheory.Limits.inl_pushoutZeroZeroIso_homproof · cited by 1