Theorems · Definition · category theory
CategoryTheory.Limits.PushoutCocone.isColimitEquivIsLimitUnop
{C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{X Y Z : Cᵒᵖ} →
{f : X ⟶ Y} →
{g : X ⟶ Z} →
(c : CategoryTheory.Limits.PushoutCocone f g) →
CategoryTheory.Limits.IsColimit c ≃ CategoryTheory.Limits.IsLimit c.unopA pushout cone is a colimit cocone in Cᵒᵖ if and only if the corresponding pullback cone
in C is a limit cone.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 37 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.
Cites32
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
- Equivstatement · cited by 8,337
- Oppositestatement and proof · cited by 8,081
- Opposite.unopstatement · cited by 2,231
- CategoryTheory.Limits.WalkingPairstatement · cited by 1,319
- CategoryTheory.Equivalence.functorproof · cited by 1,268
- CategoryTheory.Equivalence.inverseproof · cited by 1,130
- Quiver.Hom.unopstatement and proof · cited by 903
- CategoryTheory.Limits.IsColimitstatement and proof · cited by 773
- CategoryTheory.Limits.IsLimitstatement and proof · cited by 664
- CategoryTheory.Limits.WalkingCospanstatement · cited by 496
Cited by5
Results whose statement or proof uses this declaration.
- CategoryTheory.Limits.pullbackIsoUnopPushoutproof · cited by 9
- CategoryTheory.IsPushout.unopproof · cited by 4
- CategoryTheory.Limits.PullbackCone.isLimitEquivIsColimitOpproof · cited by 3
- CategoryTheory.Limits.pullbackIsoUnopPushout_inv_fstproof · cited by 1
- CategoryTheory.Limits.pullbackIsoUnopPushout_inv_sndproof · cited by 1