Theorems · Theorem · category theory

CategoryTheory.is_coprod_iff_isPushout

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X E Y YE : C} (c : CategoryTheory.Limits.BinaryCofan X E)
  (hc : CategoryTheory.Limits.IsColimit c) {f : X ⟶ Y} {iY : Y ⟶ YE} {fE : c.pt ⟶ YE},
  CategoryTheory.CommSq f c.inl iY fE →
    (Nonempty
        (CategoryTheory.Limits.IsColimit
          (CategoryTheory.Limits.BinaryCofan.mk (CategoryTheory.CategoryStruct.comp c.inr fE) iY)) ↔
      CategoryTheory.IsPushout f c.inl iY fE)

Defined in: Mathlib.CategoryTheory.Adhesive.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category

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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.Functor.objstatement and proof · cited by 19,642
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.Functorstatement · cited by 16,252
CategoryTheory.NatTrans.appproof · cited by 7,406
CategoryTheory.Category.assocproof · cited by 6,433
CategoryTheory.Discretestatement · cited by 2,447
CategoryTheory.Limits.Cocone.ptstatement and proof · cited by 1,354
CategoryTheory.Limits.WalkingPairstatement · cited by 1,319
CategoryTheory.Functor.conststatement · cited by 1,264
CategoryTheory.Limits.IsColimitstatement and proof · cited by 773

Cited by1

Results whose statement or proof uses this declaration.

CategoryTheory.IsPushout.isVanKampen_inlproof · cited by 0