Mathlib Map

Theorems · Definition · category theory

CategoryTheory.Limits.pushoutIsPushout

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {X Y Z : C} →
      (f : X ⟶ Y) →
        (g : X ⟶ Z) →
          [inst_1 : CategoryTheory.Limits.HasPushout f g] →
            CategoryTheory.Limits.IsColimit
              (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Limits.pushout.inl f g)
                (CategoryTheory.Limits.pushout.inr f g) ⋯)

The pushout cocone built from the pushout coprojections is a pushout.

Defined in
Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
Cited by
7 results in Mathlib
Foundations
Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CategoryTheory.CategoryCategoryTheory.Limits.HasPushout
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

CategoryTheory.Limits.pushoutSymmetry · cited by 30Limits.pushoutSymmetryCategoryTheory.Limits.hasPushout_symmetry · cited by 18Limits.hasPushout_symmetrycolimitCoconeOfInitialAndPushouts · cited by 6colimitCoconeOfInitialAnd…CategoryTheory.Limits.inr_comp_pushoutSymmetry_hom · cited by 5Limits.inr_comp_pushoutSy…CategoryTheory.Limits.inl_comp_pushoutSymmetry_hom · cited by 4Limits.inl_comp_pushoutSy…CategoryTheory.Limits.isColimitOfHasPushoutOfPreservesColimit · cited by 4Limits.isColimitOfHasPush…CategoryTheory.Limits.pushoutPushoutLeftIsPushout · cited by 3Limits.pushoutPushoutLeft…CategoryTheory.Limits.pushoutPushoutRightIsPushout · cited by 3Limits.pushoutPushoutRigh…CategoryTheory.Limits.pushoutAssocSymmIsPushout · cited by 1Limits.pushoutAssocSymmIs…CategoryTheory.Limits.pushoutAssocIsPushout · cited by 1Limits.pushoutAssocIsPush…CategoryTheory.JointlyReflectIsomorphisms.epi · cited by 1JointlyReflectIsomorphism…CategoryTheory.adhesive_of_reflective · cited by 0CategoryTheory.adhesive_o…CategoryTheory.Limits.hasPushout_of_preservesPushout · cited by 0Limits.hasPushout_of_pres…CategoryTheory.Limits.PreservesPushout.of_iso_comparison · cited by 0PreservesPushout.of_iso_c…CategoryTheory.Category · cited by 32673CategoryTheory.CategoryQuiver.Hom · cited by 32603Quiver.HomCategoryTheory.Limits.WalkingPair · cited by 1319Limits.WalkingPairCategoryTheory.Limits.IsColimit · cited by 773Limits.IsColimitCategoryTheory.Limits.WalkingSpan · cited by 300Limits.WalkingSpanCategoryTheory.Limits.span · cited by 294Limits.spanCategoryTheory.Limits.pushout · cited by 284Limits.pushoutCategoryTheory.Limits.pushout.inr · cited by 193pushout.inrCategoryTheory.Limits.HasPushout · cited by 192Limits.HasPushoutCategoryTheory.Limits.pushout.inl · cited by 192pushout.inlCategoryTheory.Limits.PushoutCocone.mk · cited by 89PushoutCocone.mkCategoryTheory.Limits.PushoutCocone · cited by 66Limits.PushoutCoconeCategoryTheory.Limits.PushoutCocone.inl · cited by 58PushoutCocone.inlCategoryTheory.Limits.PushoutCocone.inr · cited by 58PushoutCocone.inrCategoryTheory.Limits.pushout.desc · cited by 46pushout.descLimits.pushoutIsPushoutCITED BYCITES

Cites18

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by14

Results whose statement or proof uses this declaration.