Theorems · Definition · category theory

CategoryTheory.Limits.diagramIsoCospan

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (F : CategoryTheory.Functor CategoryTheory.Limits.WalkingCospan C) →
      F ≅
        CategoryTheory.Limits.cospan (F.map CategoryTheory.Limits.WalkingCospan.Hom.inl)
          (F.map CategoryTheory.Limits.WalkingCospan.Hom.inr)

Every diagram indexing a pullback is naturally isomorphic (actually, equal) to a cospan

Defined in: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan
Cited by: 17 results in Mathlib
Foundations: Depth 24 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category

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

Cites15

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
CategoryTheory.Functor.objstatement · cited by 19,642
CategoryTheory.Functorstatement and proof · cited by 16,252
CategoryTheory.Functor.mapstatement · cited by 8,698
CategoryTheory.Isostatement · cited by 3,963
CategoryTheory.Limits.WalkingPairstatement · cited by 1,319
CategoryTheory.Limits.WalkingCospanstatement and proof · cited by 496
CategoryTheory.Limits.cospanstatement · cited by 467
CategoryTheory.Limits.WalkingCospan.leftstatement · cited by 190
CategoryTheory.Limits.WalkingCospan.rightstatement · cited by 182
CategoryTheory.NatIso.ofComponentsproof · cited by 178
CategoryTheory.eqToIsoproof · cited by 97

Cited by22

Results whose statement or proof uses this declaration.

CategoryTheory.Limits.pullbackObjIsoproof · cited by 9
CategoryTheory.Limits.PullbackCone.isoMkstatement and proof · cited by 4
CategoryTheory.compatiblePreservingOfFlatproof · cited by 3
CategoryTheory.Limits.PullbackCone.ofConeproof · cited by 2
CategoryTheory.Limits.PullbackCone.isLimitMapConeEquivproof · cited by 2
CategoryTheory.Limits.Cone.ofPullbackConeproof · cited by 2
CategoryTheory.Limits.diagramIsoCospan_hom_appstatement and proof · cited by 2
CategoryTheory.IsPullback.of_isLimit_coneproof · cited by 1
CategoryTheory.Limits.pullbackObjIso_hom_comp_fstproof · cited by 1
CategoryTheory.Limits.pullbackObjIso_hom_comp_sndproof · cited by 1
CategoryTheory.Limits.pullbackObjIso_inv_comp_fstproof · cited by 1
CategoryTheory.Limits.pullbackObjIso_inv_comp_sndproof · cited by 1