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Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square

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 states 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.

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@[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.

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

The condition that a commutative square is a pullback square.

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

The condition that a commutative square is a pushout square.

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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) :

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

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 limit.

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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 colimit.

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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] :

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