Commutative squares that are pushout or pullback squares

In this file, we translate the IsPushout and IsPullback API for the objects of the category Square C of commutative squares in a category C. We also obtain lemmas which state in this language that a pullback of a monomorphism is a monomorphism (and similarly for pushouts of epimorphisms).

@[reducible, inline]

abbrev CategoryTheory.Square.pullbackCone {C : Type u} [Category.{v, u} C] (sq : Square C) : Limits.PullbackCone sq.f₂₄ sq.f₃₄

The pullback cone attached to a commutative square.

Equations

• sq.pullbackCone = CategoryTheory.Limits.PullbackCone.mk sq.f₁₂ sq.f₁₃ ⋯

@[reducible, inline]

abbrev CategoryTheory.Square.pushoutCocone {C : Type u} [Category.{v, u} C] (sq : Square C) : Limits.PushoutCocone sq.f₁₂ sq.f₁₃

The pushout cocone attached to a commutative square.

Equations

• sq.pushoutCocone = CategoryTheory.Limits.PushoutCocone.mk sq.f₂₄ sq.f₃₄ ⋯

def CategoryTheory.Square.IsPullback {C : Type u} [Category.{v, u} C] (sq : Square C) : Prop

The condition that a commutative square is a pullback square.

Equations

• sq.IsPullback = CategoryTheory.IsPullback sq.f₁₂ sq.f₁₃ sq.f₂₄ sq.f₃₄

def CategoryTheory.Square.IsPushout {C : Type u} [Category.{v, u} C] (sq : Square C) : Prop

The condition that a commutative square is a pushout square.

Equations

• sq.IsPushout = CategoryTheory.IsPushout sq.f₁₂ sq.f₁₃ sq.f₂₄ sq.f₃₄

theorem CategoryTheory.Square.isPullback_iff {C : Type u} [Category.{v, u} C] (sq : Square C) : sq.IsPullback ↔ Nonempty (Limits.IsLimit sq.pullbackCone)

theorem CategoryTheory.Square.isPushout_iff {C : Type u} [Category.{v, u} C] (sq : Square C) : sq.IsPushout ↔ Nonempty (Limits.IsColimit sq.pushoutCocone)

theorem CategoryTheory.Square.IsPullback.mk {C : Type u} [Category.{v, u} C] (sq : Square C) (h : Limits.IsLimit sq.pullbackCone) : sq.IsPullback

theorem CategoryTheory.Square.IsPushout.mk {C : Type u} [Category.{v, u} C] (sq : Square C) (h : Limits.IsColimit sq.pushoutCocone) : sq.IsPushout

noncomputable def CategoryTheory.Square.IsPullback.isLimit {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPullback) : Limits.IsLimit sq.pullbackCone

If a commutative square sq is a pullback square, then sq.pullbackCone is a limit.

Equations

• h.isLimit = CategoryTheory.IsPullback.isLimit h

noncomputable def CategoryTheory.Square.IsPushout.isColimit {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPushout) : Limits.IsColimit sq.pushoutCocone

If a commutative square sq is a pushout square, then sq.pushoutCocone is a colimit.

Equations

• h.isColimit = CategoryTheory.IsPushout.isColimit h

theorem CategoryTheory.Square.IsPullback.of_iso {C : Type u} [Category.{v, u} C] {sq₁ sq₂ : Square C} (h : sq₁.IsPullback) (e : sq₁ ≅ sq₂) : sq₂.IsPullback

theorem CategoryTheory.Square.IsPullback.iff_of_iso {C : Type u} [Category.{v, u} C] {sq₁ sq₂ : Square C} (e : sq₁ ≅ sq₂) : sq₁.IsPullback ↔ sq₂.IsPullback

theorem CategoryTheory.Square.IsPushout.of_iso {C : Type u} [Category.{v, u} C] {sq₁ sq₂ : Square C} (h : sq₁.IsPushout) (e : sq₁ ≅ sq₂) : sq₂.IsPushout

theorem CategoryTheory.Square.IsPushout.iff_of_iso {C : Type u} [Category.{v, u} C] {sq₁ sq₂ : Square C} (e : sq₁ ≅ sq₂) : sq₁.IsPushout ↔ sq₂.IsPushout

theorem CategoryTheory.Square.IsPushout.op {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPushout) : sq.op.IsPullback

theorem CategoryTheory.Square.IsPushout.unop {C : Type u} [Category.{v, u} C] {sq : Square Cᵒᵖ} (h : sq.IsPushout) : sq.unop.IsPullback

theorem CategoryTheory.Square.IsPullback.op {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPullback) : sq.op.IsPushout

theorem CategoryTheory.Square.IsPullback.unop {C : Type u} [Category.{v, u} C] {sq : Square Cᵒᵖ} (h : sq.IsPullback) : sq.unop.IsPushout

theorem CategoryTheory.Square.IsPullback.flip {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPullback) : sq.flip.IsPullback

theorem CategoryTheory.Square.IsPullback.mono_f₁₃ {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPullback) [Mono sq.f₂₄] : Mono sq.f₁₃

theorem CategoryTheory.Square.IsPullback.mono_f₁₂ {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPullback) [Mono sq.f₃₄] : Mono sq.f₁₂

theorem CategoryTheory.Square.IsPushout.flip {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPushout) : sq.flip.IsPushout

theorem CategoryTheory.Square.IsPushout.epi_f₂₄ {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPushout) [Epi sq.f₁₃] : Epi sq.f₂₄

theorem CategoryTheory.Square.IsPushout.epi_f₃₄ {C : Type u} [Category.{v, u} C] {sq : Square C} (h : sq.IsPushout) [Epi sq.f₁₂] : Epi sq.f₃₄