# Pullback and pushout squares, and bicartesian squares #

We provide another API for pullbacks and pushouts.

IsPullback fst snd f g is the proposition that

  P --fst--> X
|          |
snd         f
|          |
v          v
Y ---g---> Zfst



is a pullback square.

(And similarly for IsPushout.)

We provide the glue to go back and forth to the usual IsLimit API for pullbacks, and prove IsPullback (pullback.fst f g) (pullback.snd f g) f g for the usual pullback f g provided by the HasLimit API.

We don't attempt to restate everything we know about pullbacks in this language, but do restate the pasting lemmas.

We define bicartesian squares, and show that the pullback and pushout squares for a biproduct are bicartesian.

def CategoryTheory.CommSq.cone {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :

The (not necessarily limiting) PullbackCone h i implicit in the statement that we have CommSq f g h i.

Equations
• s.cone =
Instances For
def CategoryTheory.CommSq.cocone {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :

The (not necessarily limiting) PushoutCocone f g implicit in the statement that we have CommSq f g h i.

Equations
• s.cocone =
Instances For
@[simp]
theorem CategoryTheory.CommSq.cone_fst {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :
s.cone.fst = f
@[simp]
theorem CategoryTheory.CommSq.cone_snd {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :
s.cone.snd = g
@[simp]
theorem CategoryTheory.CommSq.cocone_inl {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :
s.cocone.inl = h
@[simp]
theorem CategoryTheory.CommSq.cocone_inr {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :
s.cocone.inr = i
def CategoryTheory.CommSq.coneOp {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (p : ) :
p.cone.op .cocone

The pushout cocone in the opposite category associated to the cone of a commutative square identifies to the cocone of the flipped commutative square in the opposite category

Equations
Instances For
def CategoryTheory.CommSq.coconeOp {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (p : ) :
p.cocone.op .cone

The pullback cone in the opposite category associated to the cocone of a commutative square identifies to the cone of the flipped commutative square in the opposite category

Equations
Instances For
def CategoryTheory.CommSq.coneUnop {C : Type u₁} [] {W : Cᵒᵖ} {X : Cᵒᵖ} {Y : Cᵒᵖ} {Z : Cᵒᵖ} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (p : ) :
p.cone.unop .cocone

The pushout cocone obtained from the pullback cone associated to a commutative square in the opposite category identifies to the cocone associated to the flipped square.

Equations
Instances For
def CategoryTheory.CommSq.coconeUnop {C : Type u₁} [] {W : Cᵒᵖ} {X : Cᵒᵖ} {Y : Cᵒᵖ} {Z : Cᵒᵖ} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (p : ) :
p.cocone.unop .cone

The pullback cone obtained from the pushout cone associated to a commutative square in the opposite category identifies to the cone associated to the flipped square.

Equations
Instances For
structure CategoryTheory.IsPullback {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} (fst : P X) (snd : P Y) (f : X Z) (g : Y Z) extends :

The proposition that a square

  P --fst--> X
|          |
snd         f
|          |
v          v
Y ---g---> Z



is a pullback square. (Also known as a fibered product or cartesian square.)

• isLimit' :

the pullback cone is a limit

Instances For
theorem CategoryTheory.IsPullback.isLimit' {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (self : CategoryTheory.IsPullback fst snd f g) :

the pullback cone is a limit

structure CategoryTheory.IsPushout {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} (f : Z X) (g : Z Y) (inl : X P) (inr : Y P) extends :

The proposition that a square

  Z ---f---> X
|          |
g         inl
|          |
v          v
Y --inr--> P



is a pushout square. (Also known as a fiber coproduct or cocartesian square.)

• isColimit' :

the pushout cocone is a colimit

Instances For
theorem CategoryTheory.IsPushout.isColimit' {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (self : CategoryTheory.IsPushout f g inl inr) :

the pushout cocone is a colimit

structure CategoryTheory.BicartesianSq {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} (f : W X) (g : W Y) (h : X Z) (i : Y Z) extends :

A bicartesian square is a commutative square

  W ---f---> X
|          |
g          h
|          |
v          v
Y ---i---> Z



that is both a pullback square and a pushout square.

• isLimit' :
• isColimit' :

the pushout cocone is a colimit

Instances For

We begin by providing some glue between IsPullback and the IsLimit and HasLimit APIs. (And similarly for IsPushout.)

def CategoryTheory.IsPullback.cone {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :

The (limiting) PullbackCone f g implicit in the statement that we have an IsPullback fst snd f g.

Equations
• h.cone = .cone
Instances For
@[simp]
theorem CategoryTheory.IsPullback.cone_fst {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
h.cone.fst = fst
@[simp]
theorem CategoryTheory.IsPullback.cone_snd {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
h.cone.snd = snd
noncomputable def CategoryTheory.IsPullback.isLimit {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :

The cone obtained from IsPullback fst snd f g is a limit cone.

Equations
• h.isLimit = .some
Instances For
noncomputable def CategoryTheory.IsPullback.lift {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (hP : CategoryTheory.IsPullback fst snd f g) {W : C} (h : W X) (k : W Y) :
W P

API for PullbackCone.IsLimit.lift for IsPullback

Equations
Instances For
@[simp]
theorem CategoryTheory.IsPullback.lift_fst_assoc {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} (hP : CategoryTheory.IsPullback fst snd f g) {W : C} (h : W X) (k : W Y) {Z : C} (h : X Z) :
@[simp]
theorem CategoryTheory.IsPullback.lift_fst {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (hP : CategoryTheory.IsPullback fst snd f g) {W : C} (h : W X) (k : W Y) :
@[simp]
theorem CategoryTheory.IsPullback.lift_snd_assoc {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} (hP : CategoryTheory.IsPullback fst snd f g) {W : C} (h : W X) (k : W Y) {Z : C} (h : Y Z) :
@[simp]
theorem CategoryTheory.IsPullback.lift_snd {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (hP : CategoryTheory.IsPullback fst snd f g) {W : C} (h : W X) (k : W Y) :
theorem CategoryTheory.IsPullback.hom_ext {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (hP : CategoryTheory.IsPullback fst snd f g) {W : C} {k : W P} {l : W P} (h₀ : ) (h₁ : ) :
k = l
theorem CategoryTheory.IsPullback.of_isLimit {C : Type u₁} [] {X : C} {Y : C} {Z : C} {f : X Z} {g : Y Z} {c : } :

If c is a limiting pullback cone, then we have an IsPullback c.fst c.snd f g.

theorem CategoryTheory.IsPullback.of_isLimit' {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (w : CategoryTheory.CommSq fst snd f g) (h : ) :

A variant of of_isLimit that is more useful with apply.

theorem CategoryTheory.IsPullback.of_hasPullback {C : Type u₁} [] {X : C} {Y : C} {Z : C} (f : X Z) (g : Y Z) :

The pullback provided by HasPullback f g fits into an IsPullback.

theorem CategoryTheory.IsPullback.of_is_product {C : Type u₁} [] {X : C} {Y : C} {Z : C} {c : } :
CategoryTheory.IsPullback c.fst c.snd (t.from (.obj )) (t.from (.obj ))

If c is a limiting binary product cone, and we have a terminal object, then we have IsPullback c.fst c.snd 0 0 (where each 0 is the unique morphism to the terminal object).

theorem CategoryTheory.IsPullback.of_is_product' {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} (h : ) :
CategoryTheory.IsPullback fst snd (t.from X) (t.from Y)

A variant of of_is_product that is more useful with apply.

theorem CategoryTheory.IsPullback.of_hasBinaryProduct' {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPullback CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.snd
theorem CategoryTheory.IsPullback.of_hasBinaryProduct {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPullback CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.snd 0 0
noncomputable def CategoryTheory.IsPullback.isoIsPullback {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) :
P P'

Any object at the top left of a pullback square is isomorphic to the object at the top left of any other pullback square with the same cospan.

Equations
• = h.isLimit.conePointUniqueUpToIso h'.isLimit
Instances For
@[simp]
theorem CategoryTheory.IsPullback.isoIsPullback_hom_fst_assoc {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) {Z : C} (h : X Z) :
=
@[simp]
theorem CategoryTheory.IsPullback.isoIsPullback_hom_fst {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) :
@[simp]
theorem CategoryTheory.IsPullback.isoIsPullback_hom_snd_assoc {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) {Z : C} (h : Y Z) :
=
@[simp]
theorem CategoryTheory.IsPullback.isoIsPullback_hom_snd {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) :
@[simp]
theorem CategoryTheory.IsPullback.isoIsPullback_inv_fst_assoc {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) {Z : C} (h : X Z) :
=
@[simp]
theorem CategoryTheory.IsPullback.isoIsPullback_inv_fst {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) :
= fst'
@[simp]
theorem CategoryTheory.IsPullback.isoIsPullback_inv_snd_assoc {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) {Z : C} (h : Y Z) :
=
@[simp]
theorem CategoryTheory.IsPullback.isoIsPullback_inv_snd {C : Type u₁} [] {P : C} (X : C) (Y : C) {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} {P' : C} {fst' : P' X} {snd' : P' Y} (h : CategoryTheory.IsPullback fst snd f g) (h' : CategoryTheory.IsPullback fst' snd' f g) :
= snd'
noncomputable def CategoryTheory.IsPullback.isoPullback {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :

Any object at the top left of a pullback square is isomorphic to the pullback provided by the HasLimit API.

Equations
Instances For
@[simp]
theorem CategoryTheory.IsPullback.isoPullback_hom_fst_assoc {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} (h : CategoryTheory.IsPullback fst snd f g) {Z : C} (h : X Z) :
@[simp]
theorem CategoryTheory.IsPullback.isoPullback_hom_fst {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
@[simp]
theorem CategoryTheory.IsPullback.isoPullback_hom_snd_assoc {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} (h : CategoryTheory.IsPullback fst snd f g) {Z : C} (h : Y Z) :
@[simp]
theorem CategoryTheory.IsPullback.isoPullback_hom_snd {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
@[simp]
theorem CategoryTheory.IsPullback.isoPullback_inv_fst_assoc {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} (h : CategoryTheory.IsPullback fst snd f g) {Z : C} (h : X Z) :
@[simp]
theorem CategoryTheory.IsPullback.isoPullback_inv_fst {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
CategoryTheory.CategoryStruct.comp h.isoPullback.inv fst =
@[simp]
theorem CategoryTheory.IsPullback.isoPullback_inv_snd_assoc {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z✝} {g : Y Z✝} (h : CategoryTheory.IsPullback fst snd f g) {Z : C} (h : Y Z) :
@[simp]
theorem CategoryTheory.IsPullback.isoPullback_inv_snd {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
CategoryTheory.CategoryStruct.comp h.isoPullback.inv snd =
theorem CategoryTheory.IsPullback.of_iso_pullback {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.CommSq fst snd f g) (i : ) (w₁ : ) (w₂ : ) :
theorem CategoryTheory.IsPullback.of_horiz_isIso {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} [] (sq : CategoryTheory.CommSq fst snd f g) :
theorem CategoryTheory.IsPullback.of_iso {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) {P' : C} {X' : C} {Y' : C} {Z' : C} {fst' : P' X'} {snd' : P' Y'} {f' : X' Z'} {g' : Y' Z'} (e₁ : P P') (e₂ : X X') (e₃ : Y Y') (e₄ : Z Z') (commfst : CategoryTheory.CategoryStruct.comp fst e₂.hom = CategoryTheory.CategoryStruct.comp e₁.hom fst') (commsnd : CategoryTheory.CategoryStruct.comp snd e₃.hom = CategoryTheory.CategoryStruct.comp e₁.hom snd') (commf : = ) (commg : = ) :
def CategoryTheory.IsPushout.cocone {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :

The (colimiting) PushoutCocone f g implicit in the statement that we have an IsPushout f g inl inr.

Equations
• h.cocone = .cocone
Instances For
@[simp]
theorem CategoryTheory.IsPushout.cocone_inl {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
h.cocone.inl = inl
@[simp]
theorem CategoryTheory.IsPushout.cocone_inr {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
h.cocone.inr = inr
noncomputable def CategoryTheory.IsPushout.isColimit {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :

The cocone obtained from IsPushout f g inl inr is a colimit cocone.

Equations
• h.isColimit = .some
Instances For
noncomputable def CategoryTheory.IsPushout.desc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (hP : CategoryTheory.IsPushout f g inl inr) {W : C} (h : X W) (k : Y W) :
P W

API for PushoutCocone.IsColimit.lift for IsPushout

Equations
Instances For
@[simp]
theorem CategoryTheory.IsPushout.inl_desc_assoc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} (hP : CategoryTheory.IsPushout f g inl inr) {W : C} (h : X W) (k : Y W) {Z : C} (h : W Z) :
@[simp]
theorem CategoryTheory.IsPushout.inl_desc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (hP : CategoryTheory.IsPushout f g inl inr) {W : C} (h : X W) (k : Y W) :
@[simp]
theorem CategoryTheory.IsPushout.inr_desc_assoc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} (hP : CategoryTheory.IsPushout f g inl inr) {W : C} (h : X W) (k : Y W) {Z : C} (h : W Z) :
@[simp]
theorem CategoryTheory.IsPushout.inr_desc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (hP : CategoryTheory.IsPushout f g inl inr) {W : C} (h : X W) (k : Y W) :
theorem CategoryTheory.IsPushout.hom_ext {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (hP : CategoryTheory.IsPushout f g inl inr) {W : C} {k : P W} {l : P W} (h₀ : ) (h₁ : ) :
k = l
theorem CategoryTheory.IsPushout.of_isColimit {C : Type u₁} [] {Z : C} {X : C} {Y : C} {f : Z X} {g : Z Y} {c : } :
CategoryTheory.IsPushout f g c.inl c.inr

If c is a colimiting pushout cocone, then we have an IsPushout f g c.inl c.inr.

theorem CategoryTheory.IsPushout.of_isColimit' {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (w : CategoryTheory.CommSq f g inl inr) (h : CategoryTheory.Limits.IsColimit w.cocone) :

A variant of of_isColimit that is more useful with apply.

theorem CategoryTheory.IsPushout.of_hasPushout {C : Type u₁} [] {Z : C} {X : C} {Y : C} (f : Z X) (g : Z Y) :

The pushout provided by HasPushout f g fits into an IsPushout.

theorem CategoryTheory.IsPushout.of_is_coproduct {C : Type u₁} [] {Z : C} {X : C} {Y : C} {c : } :
CategoryTheory.IsPushout (t.to (.obj )) (t.to (.obj )) c.inl c.inr

If c is a colimiting binary coproduct cocone, and we have an initial object, then we have IsPushout 0 0 c.inl c.inr (where each 0 is the unique morphism from the initial object).

theorem CategoryTheory.IsPushout.of_is_coproduct' {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {inl : X P} {inr : Y P} :
CategoryTheory.IsPushout (t.to X) (t.to Y) inl inr

A variant of of_is_coproduct that is more useful with apply.

theorem CategoryTheory.IsPushout.of_hasBinaryCoproduct' {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPushout CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr
theorem CategoryTheory.IsPushout.of_hasBinaryCoproduct {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPushout 0 0 CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr
noncomputable def CategoryTheory.IsPushout.isoIsPushout {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') :
P P'

Any object at the bottom right of a pushout square is isomorphic to the object at the bottom right of any other pushout square with the same span.

Equations
• = h.isColimit.coconePointUniqueUpToIso h'.isColimit
Instances For
@[simp]
theorem CategoryTheory.IsPushout.inl_isoIsPushout_hom_assoc {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') {Z : C} (h : P' Z) :
=
@[simp]
theorem CategoryTheory.IsPushout.inl_isoIsPushout_hom {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') :
@[simp]
theorem CategoryTheory.IsPushout.inr_isoIsPushout_hom_assoc {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') {Z : C} (h : P' Z) :
=
@[simp]
theorem CategoryTheory.IsPushout.inr_isoIsPushout_hom {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') :
@[simp]
theorem CategoryTheory.IsPushout.inl_isoIsPushout_inv_assoc {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') {Z : C} (h : P Z) :
=
@[simp]
theorem CategoryTheory.IsPushout.inl_isoIsPushout_inv {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') :
@[simp]
theorem CategoryTheory.IsPushout.inr_isoIsPushout_inv_assoc {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') {Z : C} (h : P Z) :
=
@[simp]
theorem CategoryTheory.IsPushout.inr_isoIsPushout_inv {C : Type u₁} [] {Z : C} (X : C) (Y : C) {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} {P' : C} {inl' : X P'} {inr' : Y P'} (h : CategoryTheory.IsPushout f g inl inr) (h' : CategoryTheory.IsPushout f g inl' inr') :
noncomputable def CategoryTheory.IsPushout.isoPushout {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :

Any object at the top left of a pullback square is isomorphic to the pullback provided by the HasLimit API.

Equations
Instances For
@[simp]
theorem CategoryTheory.IsPushout.inl_isoPushout_inv_assoc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) {Z : C} (h : P Z) :
@[simp]
theorem CategoryTheory.IsPushout.inl_isoPushout_inv {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
@[simp]
theorem CategoryTheory.IsPushout.inr_isoPushout_inv_assoc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) {Z : C} (h : P Z) :
@[simp]
theorem CategoryTheory.IsPushout.inr_isoPushout_inv {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
@[simp]
theorem CategoryTheory.IsPushout.inl_isoPushout_hom_assoc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) {Z : C} (h : ) :
@[simp]
theorem CategoryTheory.IsPushout.inl_isoPushout_hom {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
@[simp]
theorem CategoryTheory.IsPushout.inr_isoPushout_hom_assoc {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z✝ X} {g : Z✝ Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) {Z : C} (h : ) :
@[simp]
theorem CategoryTheory.IsPushout.inr_isoPushout_hom {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
theorem CategoryTheory.IsPushout.of_iso_pushout {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.CommSq f g inl inr) (i : ) (w₁ : ) (w₂ : ) :
theorem CategoryTheory.IsPushout.of_iso {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) {Z' : C} {X' : C} {Y' : C} {P' : C} {f' : Z' X'} {g' : Z' Y'} {inl' : X' P'} {inr' : Y' P'} (e₁ : Z Z') (e₂ : X X') (e₃ : Y Y') (e₄ : P P') (commf : = ) (commg : = ) (comminl : CategoryTheory.CategoryStruct.comp inl e₄.hom = CategoryTheory.CategoryStruct.comp e₂.hom inl') (comminr : CategoryTheory.CategoryStruct.comp inr e₄.hom = CategoryTheory.CategoryStruct.comp e₃.hom inr') :
CategoryTheory.IsPushout f' g' inl' inr'
theorem CategoryTheory.IsPullback.flip {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
theorem CategoryTheory.IsPullback.flip_iff {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} :
@[simp]
theorem CategoryTheory.IsPullback.zero_left {C : Type u₁} [] (X : C) :

The square with 0 : 0 ⟶ 0 on the left and 𝟙 X on the right is a pullback square.

@[simp]
theorem CategoryTheory.IsPullback.zero_top {C : Type u₁} [] (X : C) :

The square with 0 : 0 ⟶ 0 on the top and 𝟙 X on the bottom is a pullback square.

@[simp]
theorem CategoryTheory.IsPullback.zero_right {C : Type u₁} [] (X : C) :

The square with 0 : 0 ⟶ 0 on the right and 𝟙 X on the left is a pullback square.

@[simp]
theorem CategoryTheory.IsPullback.zero_bot {C : Type u₁} [] (X : C) :

The square with 0 : 0 ⟶ 0 on the bottom and 𝟙 X on the top is a pullback square.

theorem CategoryTheory.IsPullback.paste_vert {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₂₁ : C} {X₂₂ : C} {X₃₁ : C} {X₃₂ : C} {h₁₁ : X₁₁ X₁₂} {h₂₁ : X₂₁ X₂₂} {h₃₁ : X₃₁ X₃₂} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₂₁ : X₂₁ X₃₁} {v₂₂ : X₂₂ X₃₂} (s : CategoryTheory.IsPullback h₁₁ v₁₁ v₁₂ h₂₁) (t : CategoryTheory.IsPullback h₂₁ v₂₁ v₂₂ h₃₁) :

Paste two pullback squares "vertically" to obtain another pullback square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂
|            |
v₁₁          v₁₂
↓            ↓
X₂₁ - h₂₁ -> X₂₂
|            |
v₂₁          v₂₂
↓            ↓
X₃₁ - h₃₁ -> X₃₂

theorem CategoryTheory.IsPullback.paste_horiz {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₁₃ : C} {X₂₁ : C} {X₂₂ : C} {X₂₃ : C} {h₁₁ : X₁₁ X₁₂} {h₁₂ : X₁₂ X₁₃} {h₂₁ : X₂₁ X₂₂} {h₂₂ : X₂₂ X₂₃} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₃ X₂₃} (s : CategoryTheory.IsPullback h₁₁ v₁₁ v₁₂ h₂₁) (t : CategoryTheory.IsPullback h₁₂ v₁₂ v₁₃ h₂₂) :
CategoryTheory.IsPullback v₁₁ v₁₃

Paste two pullback squares "horizontally" to obtain another pullback square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂ - h₁₂ -> X₁₃
|            |            |
v₁₁          v₁₂          v₁₃
↓            ↓            ↓
X₂₁ - h₂₁ -> X₂₂ - h₂₂ -> X₂₃

theorem CategoryTheory.IsPullback.of_bot {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₂₁ : C} {X₂₂ : C} {X₃₁ : C} {X₃₂ : C} {h₁₁ : X₁₁ X₁₂} {h₂₁ : X₂₁ X₂₂} {h₃₁ : X₃₁ X₃₂} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₂₁ : X₂₁ X₃₁} {v₂₂ : X₂₂ X₃₂} (s : CategoryTheory.IsPullback h₁₁ h₃₁) (p : ) (t : CategoryTheory.IsPullback h₂₁ v₂₁ v₂₂ h₃₁) :
CategoryTheory.IsPullback h₁₁ v₁₁ v₁₂ h₂₁

Given a pullback square assembled from a commuting square on the top and a pullback square on the bottom, the top square is a pullback square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂
|            |
v₁₁          v₁₂
↓            ↓
X₂₁ - h₂₁ -> X₂₂
|            |
v₂₁          v₂₂
↓            ↓
X₃₁ - h₃₁ -> X₃₂

theorem CategoryTheory.IsPullback.of_right {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₁₃ : C} {X₂₁ : C} {X₂₂ : C} {X₂₃ : C} {h₁₁ : X₁₁ X₁₂} {h₁₂ : X₁₂ X₁₃} {h₂₁ : X₂₁ X₂₂} {h₂₂ : X₂₂ X₂₃} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₃ X₂₃} (s : CategoryTheory.IsPullback v₁₁ v₁₃ ) (p : ) (t : CategoryTheory.IsPullback h₁₂ v₁₂ v₁₃ h₂₂) :
CategoryTheory.IsPullback h₁₁ v₁₁ v₁₂ h₂₁

Given a pullback square assembled from a commuting square on the left and a pullback square on the right, the left square is a pullback square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂ - h₁₂ -> X₁₃
|            |            |
v₁₁          v₁₂          v₁₃
↓            ↓            ↓
X₂₁ - h₂₁ -> X₂₂ - h₂₂ -> X₂₃

theorem CategoryTheory.IsPullback.paste_vert_iff {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₂₁ : C} {X₂₂ : C} {X₃₁ : C} {X₃₂ : C} {h₁₁ : X₁₁ X₁₂} {h₂₁ : X₂₁ X₂₂} {h₃₁ : X₃₁ X₃₂} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₂₁ : X₂₁ X₃₁} {v₂₂ : X₂₂ X₃₂} (s : CategoryTheory.IsPullback h₂₁ v₂₁ v₂₂ h₃₁) (e : ) :
CategoryTheory.IsPullback h₁₁ h₃₁ CategoryTheory.IsPullback h₁₁ v₁₁ v₁₂ h₂₁
theorem CategoryTheory.IsPullback.paste_horiz_iff {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₁₃ : C} {X₂₁ : C} {X₂₂ : C} {X₂₃ : C} {h₁₁ : X₁₁ X₁₂} {h₁₂ : X₁₂ X₁₃} {h₂₁ : X₂₁ X₂₂} {h₂₂ : X₂₂ X₂₃} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₃ X₂₃} (s : CategoryTheory.IsPullback h₁₂ v₁₂ v₁₃ h₂₂) (e : ) :
CategoryTheory.IsPullback v₁₁ v₁₃ CategoryTheory.IsPullback h₁₁ v₁₁ v₁₂ h₂₁
theorem CategoryTheory.IsPullback.of_right' {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₁₃ : C} {X₂₁ : C} {X₂₂ : C} {X₂₃ : C} {h₁₂ : X₁₂ X₁₃} {h₂₁ : X₂₁ X₂₂} {h₂₂ : X₂₂ X₂₃} {h₁₃ : X₁₁ X₁₃} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₃ X₂₃} (s : CategoryTheory.IsPullback h₁₃ v₁₁ v₁₃ ) (t : CategoryTheory.IsPullback h₁₂ v₁₂ v₁₃ h₂₂) :
CategoryTheory.IsPullback (t.lift h₁₃ ) v₁₁ v₁₂ h₂₁

Variant of IsPullback.of_right where h₁₁ is induced from a morphism h₁₃ : X₁₁ ⟶ X₁₃, and the universal property of the right square.

The objects fit in the following diagram:

X₁₁ - h₁₁ -> X₁₂ - h₁₂ -> X₁₃
|            |            |
v₁₁          v₁₂          v₁₃
↓            ↓            ↓
X₂₁ - h₂₁ -> X₂₂ - h₂₂ -> X₂₃

theorem CategoryTheory.IsPullback.of_bot' {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₂₁ : C} {X₂₂ : C} {X₃₁ : C} {X₃₂ : C} {h₁₁ : X₁₁ X₁₂} {h₂₁ : X₂₁ X₂₂} {h₃₁ : X₃₁ X₃₂} {v₃₁ : X₁₁ X₃₁} {v₁₂ : X₁₂ X₂₂} {v₂₁ : X₂₁ X₃₁} {v₂₂ : X₂₂ X₃₂} (s : CategoryTheory.IsPullback h₁₁ v₃₁ h₃₁) (t : CategoryTheory.IsPullback h₂₁ v₂₁ v₂₂ h₃₁) :
CategoryTheory.IsPullback h₁₁ (t.lift v₃₁ ) v₁₂ h₂₁

Variant of IsPullback.of_bot, where v₁₁ is induced from a morphism v₃₁ : X₁₁ ⟶ X₃₁, and the universal property of the bottom square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂
|            |
v₁₁          v₁₂
↓            ↓
X₂₁ - h₂₁ -> X₂₂
|            |
v₂₁          v₂₂
↓            ↓
X₃₁ - h₃₁ -> X₃₂

theorem CategoryTheory.IsPullback.of_isBilimit {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
@[simp]
theorem CategoryTheory.IsPullback.of_has_biproduct {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPullback CategoryTheory.Limits.biprod.fst CategoryTheory.Limits.biprod.snd 0 0
theorem CategoryTheory.IsPullback.inl_snd' {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
@[simp]
theorem CategoryTheory.IsPullback.inl_snd {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPullback CategoryTheory.Limits.biprod.inl 0 CategoryTheory.Limits.biprod.snd 0

The square

  X --inl--> X ⊞ Y
|            |
0           snd
|            |
v            v
0 ---0-----> Y


is a pullback square.

theorem CategoryTheory.IsPullback.inr_fst' {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
@[simp]
theorem CategoryTheory.IsPullback.inr_fst {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPullback CategoryTheory.Limits.biprod.inr 0 CategoryTheory.Limits.biprod.fst 0

The square

  Y --inr--> X ⊞ Y
|            |
0           fst
|            |
v            v
0 ---0-----> X


is a pullback square.

theorem CategoryTheory.IsPullback.of_is_bilimit' {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
theorem CategoryTheory.IsPullback.of_hasBinaryBiproduct {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPullback 0 0 CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr
instance CategoryTheory.IsPullback.hasPullback_biprod_fst_biprod_snd {C : Type u₁} [] {X : C} {Y : C} :
CategoryTheory.Limits.HasPullback CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr
Equations
• =
def CategoryTheory.IsPullback.pullbackBiprodInlBiprodInr {C : Type u₁} [] {X : C} {Y : C} :
CategoryTheory.Limits.pullback CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr 0

The pullback of biprod.inl and biprod.inr is the zero object.

Equations
Instances For
theorem CategoryTheory.IsPullback.op {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
CategoryTheory.IsPushout g.op f.op snd.op fst.op
theorem CategoryTheory.IsPullback.unop {C : Type u₁} [] {P : Cᵒᵖ} {X : Cᵒᵖ} {Y : Cᵒᵖ} {Z : Cᵒᵖ} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} (h : CategoryTheory.IsPullback fst snd f g) :
CategoryTheory.IsPushout g.unop f.unop snd.unop fst.unop
theorem CategoryTheory.IsPullback.of_vert_isIso {C : Type u₁} [] {P : C} {X : C} {Y : C} {Z : C} {fst : P X} {snd : P Y} {f : X Z} {g : Y Z} [] (sq : CategoryTheory.CommSq fst snd f g) :
theorem CategoryTheory.IsPullback.of_id_fst {C : Type u₁} [] {X : C} {Z : C} {f : X Z} :
theorem CategoryTheory.IsPullback.of_id_snd {C : Type u₁} [] {X : C} {Z : C} {f : X Z} :
theorem CategoryTheory.IsPullback.id_vert {C : Type u₁} [] {X : C} {Z : C} (f : X Z) :

The following diagram is a pullback

X --f--> Z
|        |
id       id
v        v
X --f--> Z

theorem CategoryTheory.IsPullback.id_horiz {C : Type u₁} [] {X : C} {Z : C} (f : X Z) :

The following diagram is a pullback

X --id--> X
|         |
f         f
v         v
Z --id--> Z

theorem CategoryTheory.IsPushout.flip {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
theorem CategoryTheory.IsPushout.flip_iff {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} :
@[simp]
theorem CategoryTheory.IsPushout.zero_right {C : Type u₁} [] (X : C) :

The square with 0 : 0 ⟶ 0 on the right and 𝟙 X on the left is a pushout square.

@[simp]
theorem CategoryTheory.IsPushout.zero_bot {C : Type u₁} [] (X : C) :

The square with 0 : 0 ⟶ 0 on the bottom and 𝟙 X on the top is a pushout square.

@[simp]
theorem CategoryTheory.IsPushout.zero_left {C : Type u₁} [] (X : C) :

The square with 0 : 0 ⟶ 0 on the right left 𝟙 X on the right is a pushout square.

@[simp]
theorem CategoryTheory.IsPushout.zero_top {C : Type u₁} [] (X : C) :

The square with 0 : 0 ⟶ 0 on the top and 𝟙 X on the bottom is a pushout square.

theorem CategoryTheory.IsPushout.paste_vert {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₂₁ : C} {X₂₂ : C} {X₃₁ : C} {X₃₂ : C} {h₁₁ : X₁₁ X₁₂} {h₂₁ : X₂₁ X₂₂} {h₃₁ : X₃₁ X₃₂} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₂₁ : X₂₁ X₃₁} {v₂₂ : X₂₂ X₃₂} (s : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁) (t : CategoryTheory.IsPushout h₂₁ v₂₁ v₂₂ h₃₁) :
CategoryTheory.IsPushout h₁₁ h₃₁

Paste two pushout squares "vertically" to obtain another pushout square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂
|            |
v₁₁          v₁₂
↓            ↓
X₂₁ - h₂₁ -> X₂₂
|            |
v₂₁          v₂₂
↓            ↓
X₃₁ - h₃₁ -> X₃₂

theorem CategoryTheory.IsPushout.paste_horiz {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₁₃ : C} {X₂₁ : C} {X₂₂ : C} {X₂₃ : C} {h₁₁ : X₁₁ X₁₂} {h₁₂ : X₁₂ X₁₃} {h₂₁ : X₂₁ X₂₂} {h₂₂ : X₂₂ X₂₃} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₃ X₂₃} (s : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁) (t : CategoryTheory.IsPushout h₁₂ v₁₂ v₁₃ h₂₂) :
CategoryTheory.IsPushout v₁₁ v₁₃

Paste two pushout squares "horizontally" to obtain another pushout square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂ - h₁₂ -> X₁₃
|            |            |
v₁₁          v₁₂          v₁₃
↓            ↓            ↓
X₂₁ - h₂₁ -> X₂₂ - h₂₂ -> X₂₃

theorem CategoryTheory.IsPushout.of_top {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₂₁ : C} {X₂₂ : C} {X₃₁ : C} {X₃₂ : C} {h₁₁ : X₁₁ X₁₂} {h₂₁ : X₂₁ X₂₂} {h₃₁ : X₃₁ X₃₂} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₂₁ : X₂₁ X₃₁} {v₂₂ : X₂₂ X₃₂} (s : CategoryTheory.IsPushout h₁₁ h₃₁) (p : ) (t : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁) :
CategoryTheory.IsPushout h₂₁ v₂₁ v₂₂ h₃₁

Given a pushout square assembled from a pushout square on the top and a commuting square on the bottom, the bottom square is a pushout square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂
|            |
v₁₁          v₁₂
↓            ↓
X₂₁ - h₂₁ -> X₂₂
|            |
v₂₁          v₂₂
↓            ↓
X₃₁ - h₃₁ -> X₃₂

theorem CategoryTheory.IsPushout.of_left {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₁₃ : C} {X₂₁ : C} {X₂₂ : C} {X₂₃ : C} {h₁₁ : X₁₁ X₁₂} {h₁₂ : X₁₂ X₁₃} {h₂₁ : X₂₁ X₂₂} {h₂₂ : X₂₂ X₂₃} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₃ X₂₃} (s : CategoryTheory.IsPushout v₁₁ v₁₃ ) (p : ) (t : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁) :
CategoryTheory.IsPushout h₁₂ v₁₂ v₁₃ h₂₂

Given a pushout square assembled from a pushout square on the left and a commuting square on the right, the right square is a pushout square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂ - h₁₂ -> X₁₃
|            |            |
v₁₁          v₁₂          v₁₃
↓            ↓            ↓
X₂₁ - h₂₁ -> X₂₂ - h₂₂ -> X₂₃

theorem CategoryTheory.IsPushout.paste_vert_iff {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₂₁ : C} {X₂₂ : C} {X₃₁ : C} {X₃₂ : C} {h₁₁ : X₁₁ X₁₂} {h₂₁ : X₂₁ X₂₂} {h₃₁ : X₃₁ X₃₂} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₂₁ : X₂₁ X₃₁} {v₂₂ : X₂₂ X₃₂} (s : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁) (e : ) :
CategoryTheory.IsPushout h₁₁ h₃₁ CategoryTheory.IsPushout h₂₁ v₂₁ v₂₂ h₃₁
theorem CategoryTheory.IsPushout.paste_horiz_iff {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₁₃ : C} {X₂₁ : C} {X₂₂ : C} {X₂₃ : C} {h₁₁ : X₁₁ X₁₂} {h₁₂ : X₁₂ X₁₃} {h₂₁ : X₂₁ X₂₂} {h₂₂ : X₂₂ X₂₃} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₃ X₂₃} (s : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁) (e : ) :
CategoryTheory.IsPushout v₁₁ v₁₃ CategoryTheory.IsPushout h₁₂ v₁₂ v₁₃ h₂₂
theorem CategoryTheory.IsPushout.of_top' {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₂₁ : C} {X₂₂ : C} {X₃₁ : C} {X₃₂ : C} {h₁₁ : X₁₁ X₁₂} {h₂₁ : X₂₁ X₂₂} {h₃₁ : X₃₁ X₃₂} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₂ X₃₂} {v₂₁ : X₂₁ X₃₁} (s : CategoryTheory.IsPushout h₁₁ v₁₃ h₃₁) (t : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁) :
CategoryTheory.IsPushout h₂₁ v₂₁ (t.desc v₁₃ ) h₃₁

Variant of IsPushout.of_top where v₂₂ is induced from a morphism v₁₃ : X₁₂ ⟶ X₃₂, and the universal property of the top square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂
|            |
v₁₁          v₁₂
↓            ↓
X₂₁ - h₂₁ -> X₂₂
|            |
v₂₁          v₂₂
↓            ↓
X₃₁ - h₃₁ -> X₃₂

theorem CategoryTheory.IsPushout.of_left' {C : Type u₁} [] {X₁₁ : C} {X₁₂ : C} {X₁₃ : C} {X₂₁ : C} {X₂₂ : C} {X₂₃ : C} {h₁₁ : X₁₁ X₁₂} {h₁₂ : X₁₂ X₁₃} {h₂₁ : X₂₁ X₂₂} {h₂₃ : X₂₁ X₂₃} {v₁₁ : X₁₁ X₂₁} {v₁₂ : X₁₂ X₂₂} {v₁₃ : X₁₃ X₂₃} (s : CategoryTheory.IsPushout v₁₁ v₁₃ h₂₃) (t : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁) :
CategoryTheory.IsPushout h₁₂ v₁₂ v₁₃ (t.desc h₂₃ )

Variant of IsPushout.of_right where h₂₂ is induced from a morphism h₂₃ : X₂₁ ⟶ X₂₃, and the universal property of the left square.

The objects in the statement fit into the following diagram:

X₁₁ - h₁₁ -> X₁₂ - h₁₂ -> X₁₃
|            |            |
v₁₁          v₁₂          v₁₃
↓            ↓            ↓
X₂₁ - h₂₁ -> X₂₂ - h₂₂ -> X₂₃

theorem CategoryTheory.IsPushout.of_isBilimit {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
CategoryTheory.IsPushout 0 0 b.inl b.inr
@[simp]
theorem CategoryTheory.IsPushout.of_has_biproduct {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPushout 0 0 CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr
theorem CategoryTheory.IsPushout.inl_snd' {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
CategoryTheory.IsPushout b.inl 0 b.snd 0
theorem CategoryTheory.IsPushout.inl_snd {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPushout CategoryTheory.Limits.biprod.inl 0 CategoryTheory.Limits.biprod.snd 0

The square

  X --inl--> X ⊞ Y
|            |
0           snd
|            |
v            v
0 ---0-----> Y


is a pushout square.

theorem CategoryTheory.IsPushout.inr_fst' {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
CategoryTheory.IsPushout b.inr 0 b.fst 0
theorem CategoryTheory.IsPushout.inr_fst {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPushout CategoryTheory.Limits.biprod.inr 0 CategoryTheory.Limits.biprod.fst 0

The square

  Y --inr--> X ⊞ Y
|            |
0           fst
|            |
v            v
0 ---0-----> X


is a pushout square.

theorem CategoryTheory.IsPushout.of_is_bilimit' {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
CategoryTheory.IsPushout b.fst b.snd 0 0
theorem CategoryTheory.IsPushout.of_hasBinaryBiproduct {C : Type u₁} [] (X : C) (Y : C) :
CategoryTheory.IsPushout CategoryTheory.Limits.biprod.fst CategoryTheory.Limits.biprod.snd 0 0
instance CategoryTheory.IsPushout.hasPushout_biprod_fst_biprod_snd {C : Type u₁} [] {X : C} {Y : C} :
CategoryTheory.Limits.HasPushout CategoryTheory.Limits.biprod.fst CategoryTheory.Limits.biprod.snd
Equations
• =
def CategoryTheory.IsPushout.pushoutBiprodFstBiprodSnd {C : Type u₁} [] {X : C} {Y : C} :
CategoryTheory.Limits.pushout CategoryTheory.Limits.biprod.fst CategoryTheory.Limits.biprod.snd 0

The pushout of biprod.fst and biprod.snd is the zero object.

Equations
Instances For
theorem CategoryTheory.IsPushout.op {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
CategoryTheory.IsPullback inr.op inl.op g.op f.op
theorem CategoryTheory.IsPushout.unop {C : Type u₁} [] {Z : Cᵒᵖ} {X : Cᵒᵖ} {Y : Cᵒᵖ} {P : Cᵒᵖ} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} (h : CategoryTheory.IsPushout f g inl inr) :
CategoryTheory.IsPullback inr.unop inl.unop g.unop f.unop
theorem CategoryTheory.IsPushout.of_horiz_isIso {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} [] (sq : CategoryTheory.CommSq f g inl inr) :
theorem CategoryTheory.IsPushout.of_vert_isIso {C : Type u₁} [] {Z : C} {X : C} {Y : C} {P : C} {f : Z X} {g : Z Y} {inl : X P} {inr : Y P} [] (sq : CategoryTheory.CommSq f g inl inr) :
theorem CategoryTheory.IsPushout.of_id_fst {C : Type u₁} [] {Z : C} {X : C} {f : Z X} :
theorem CategoryTheory.IsPushout.of_id_snd {C : Type u₁} [] {Z : C} {X : C} {f : Z X} :
theorem CategoryTheory.IsPushout.id_vert {C : Type u₁} [] {Z : C} {X : C} (f : X Z) :

The following diagram is a pullback

X --f--> Z
|        |
id       id
v        v
X --f--> Z

theorem CategoryTheory.IsPushout.id_horiz {C : Type u₁} [] {Z : C} {X : C} (f : X Z) :

The following diagram is a pullback

X --id--> X
|         |
f         f
v         v
Z --id--> Z

noncomputable def CategoryTheory.IsPullback.isLimitFork {C : Type u₁} [] {X : C} {Y : C} {Z : C} {f : X Y} {g : Y Z} {g' : Y Z} (H : ) :

If f : X ⟶ Y, g g' : Y ⟶ Z forms a pullback square, then f is the equalizer of g and g'.

Equations
• One or more equations did not get rendered due to their size.
Instances For
noncomputable def CategoryTheory.IsPushout.isLimitFork {C : Type u₁} [] {X : C} {Y : C} {Z : C} {f : X Y} {f' : X Y} {g : Y Z} (H : ) :

If f f' : X ⟶ Y, g : Y ⟶ Z forms a pushout square, then g is the coequalizer of f and f'.

Equations
• One or more equations did not get rendered due to their size.
Instances For
theorem CategoryTheory.BicartesianSq.of_isPullback_isPushout {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (p₁ : ) (p₂ : ) :
theorem CategoryTheory.BicartesianSq.flip {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (p : ) :
theorem CategoryTheory.BicartesianSq.of_is_biproduct₁ {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
 X ⊞ Y --fst--> X
|            |
snd           0
|            |
v            v
Y -----0---> 0


is a bicartesian square.

theorem CategoryTheory.BicartesianSq.of_is_biproduct₂ {C : Type u₁} [] {X : C} {Y : C} {b : } (h : b.IsBilimit) :
   0 -----0---> X
|            |
0           inl
|            |
v            v
Y --inr--> X ⊞ Y


is a bicartesian square.

@[simp]
theorem CategoryTheory.BicartesianSq.of_has_biproduct₁ {C : Type u₁} [] {X : C} {Y : C} :
CategoryTheory.BicartesianSq CategoryTheory.Limits.biprod.fst CategoryTheory.Limits.biprod.snd 0 0
 X ⊞ Y --fst--> X
|            |
snd           0
|            |
v            v
Y -----0---> 0


is a bicartesian square.

@[simp]
theorem CategoryTheory.BicartesianSq.of_has_biproduct₂ {C : Type u₁} [] {X : C} {Y : C} :
CategoryTheory.BicartesianSq 0 0 CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr
   0 -----0---> X
|            |
0           inl
|            |
v            v
Y --inr--> X ⊞ Y


is a bicartesian square.

theorem CategoryTheory.Functor.map_isPullback {C : Type u₁} [] {D : Type u₂} [] (F : ) {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :
CategoryTheory.IsPullback (F.map f) (F.map g) (F.map h) (F.map i)
theorem CategoryTheory.Functor.map_isPushout {C : Type u₁} [] {D : Type u₂} [] (F : ) {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :
CategoryTheory.IsPushout (F.map f) (F.map g) (F.map h) (F.map i)
theorem CategoryTheory.IsPullback.map {C : Type u₁} [] {D : Type u₂} [] (F : ) {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :
CategoryTheory.IsPullback (F.map f) (F.map g) (F.map h) (F.map i)

Alias of CategoryTheory.Functor.map_isPullback.

theorem CategoryTheory.IsPushout.map {C : Type u₁} [] {D : Type u₂} [] (F : ) {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (s : ) :
CategoryTheory.IsPushout (F.map f) (F.map g) (F.map h) (F.map i)

Alias of CategoryTheory.Functor.map_isPushout.

theorem CategoryTheory.IsPullback.of_map {C : Type u₁} [] {D : Type u₂} [] (F : ) {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (H : CategoryTheory.IsPullback (F.map f) (F.map g) (F.map h) (F.map i)) :
theorem CategoryTheory.IsPullback.of_map_of_faithful {C : Type u₁} [] {D : Type u₂} [] (F : ) {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} [F.Faithful] (H : CategoryTheory.IsPullback (F.map f) (F.map g) (F.map h) (F.map i)) :
theorem CategoryTheory.IsPullback.map_iff {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} {D : Type u_1} [] (F : ) :
CategoryTheory.IsPullback (F.map f) (F.map g) (F.map h) (F.map i)
theorem CategoryTheory.IsPushout.of_map {C : Type u₁} [] {D : Type u₂} [] (F : ) {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} (H : CategoryTheory.IsPushout (F.map f) (F.map g) (F.map h) (F.map i)) :
theorem CategoryTheory.IsPushout.of_map_of_faithful {C : Type u₁} [] {D : Type u₂} [] (F : ) {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} [F.Faithful] (H : CategoryTheory.IsPushout (F.map f) (F.map g) (F.map h) (F.map i)) :
theorem CategoryTheory.IsPushout.map_iff {C : Type u₁} [] {W : C} {X : C} {Y : C} {Z : C} {f : W X} {g : W Y} {h : X Z} {i : Y Z} {D : Type u_1} [] (F : ) :
CategoryTheory.IsPushout (F.map f) (F.map g) (F.map h) (F.map i)