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---> Z
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}
[Category.{v₁, u₁} C]
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
(s : CommSq f g h i)
:
The (not necessarily limiting) PullbackCone h i implicit in the statement
that we have CommSq f g h i.
Equations
Instances For
def
CategoryTheory.CommSq.cocone
{C : Type u₁}
[Category.{v₁, u₁} C]
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
(s : CommSq f g h i)
:
The (not necessarily limiting) PushoutCocone f g implicit in the statement
that we have CommSq f g h i.
Equations
Instances For
def
CategoryTheory.CommSq.coneOp
{C : Type u₁}
[Category.{v₁, u₁} C]
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
(p : CommSq f g h i)
:
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}
[Category.{v₁, u₁} C]
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
(p : CommSq f g h i)
:
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}
[Category.{v₁, u₁} C]
{W X Y Z : Cᵒᵖ}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
(p : CommSq f g h i)
:
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}
[Category.{v₁, u₁} C]
{W X Y Z : Cᵒᵖ}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
(p : CommSq f g h i)
:
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}
[Category.{v₁, u₁} C]
{P X Y Z : C}
(fst : P ⟶ X)
(snd : P ⟶ Y)
(f : X ⟶ Z)
(g : Y ⟶ Z)
extends CategoryTheory.CommSq fst snd f g :
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' : Nonempty (Limits.IsLimit (Limits.PullbackCone.mk fst snd ⋯))
the pullback cone is a limit
Instances For
structure
CategoryTheory.IsPushout
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
(f : Z ⟶ X)
(g : Z ⟶ Y)
(inl : X ⟶ P)
(inr : Y ⟶ P)
extends CategoryTheory.CommSq f g inl inr :
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' : Nonempty (Limits.IsColimit (Limits.PushoutCocone.mk inl inr ⋯))
the pushout cocone is a colimit
Instances For
structure
CategoryTheory.BicartesianSq
{C : Type u₁}
[Category.{v₁, u₁} C]
{W X Y Z : C}
(f : W ⟶ X)
(g : W ⟶ Y)
(h : X ⟶ Z)
(i : Y ⟶ Z)
extends CategoryTheory.IsPullback f g h i, CategoryTheory.IsPushout f g h i :
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' : Nonempty (Limits.IsLimit (Limits.PullbackCone.mk f g ⋯))
- isColimit' : Nonempty (Limits.IsColimit (Limits.PushoutCocone.mk h i ⋯))
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}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
:
The (limiting) PullbackCone f g implicit in the statement
that we have an IsPullback fst snd f g.
Instances For
@[simp]
theorem
CategoryTheory.IsPullback.cone_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
:
@[simp]
theorem
CategoryTheory.IsPullback.cone_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
:
noncomputable def
CategoryTheory.IsPullback.isLimit
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
:
The cone obtained from IsPullback fst snd f g is a limit cone.
Instances For
noncomputable def
CategoryTheory.IsPullback.lift
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(hP : IsPullback fst snd f g)
{W : C}
(h : W ⟶ X)
(k : W ⟶ Y)
(w : CategoryStruct.comp h f = CategoryStruct.comp k g)
:
API for PullbackCone.IsLimit.lift for IsPullback
Equations
- hP.lift h k w = CategoryTheory.Limits.PullbackCone.IsLimit.lift hP.isLimit h k w
Instances For
@[simp]
theorem
CategoryTheory.IsPullback.lift_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(hP : IsPullback fst snd f g)
{W : C}
(h : W ⟶ X)
(k : W ⟶ Y)
(w : CategoryStruct.comp h f = CategoryStruct.comp k g)
:
@[simp]
theorem
CategoryTheory.IsPullback.lift_fst_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(hP : IsPullback fst snd f g)
{W : C}
(h : W ⟶ X)
(k : W ⟶ Y)
(w : CategoryStruct.comp h f = CategoryStruct.comp k g)
{Z✝ : C}
(h✝ : X ⟶ Z✝)
:
@[simp]
theorem
CategoryTheory.IsPullback.lift_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(hP : IsPullback fst snd f g)
{W : C}
(h : W ⟶ X)
(k : W ⟶ Y)
(w : CategoryStruct.comp h f = CategoryStruct.comp k g)
:
@[simp]
theorem
CategoryTheory.IsPullback.lift_snd_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(hP : IsPullback fst snd f g)
{W : C}
(h : W ⟶ X)
(k : W ⟶ Y)
(w : CategoryStruct.comp h f = CategoryStruct.comp k g)
{Z✝ : C}
(h✝ : Y ⟶ Z✝)
:
theorem
CategoryTheory.IsPullback.hom_ext
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(hP : IsPullback fst snd f g)
{W : C}
{k l : W ⟶ P}
(h₀ : CategoryStruct.comp k fst = CategoryStruct.comp l fst)
(h₁ : CategoryStruct.comp k snd = CategoryStruct.comp l snd)
:
theorem
CategoryTheory.IsPullback.of_isLimit
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y Z : C}
{f : X ⟶ Z}
{g : Y ⟶ Z}
{c : Limits.PullbackCone f g}
(h : Limits.IsLimit c)
:
IsPullback c.fst c.snd f g
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}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(w : CommSq fst snd f g)
(h : Limits.IsLimit w.cone)
:
IsPullback fst snd f g
A variant of of_isLimit that is more useful with apply.
theorem
CategoryTheory.IsPullback.of_isLimit_cone
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Functor Limits.WalkingCospan C}
{c : Limits.Cone D}
(hc : Limits.IsLimit c)
:
Variant of of_isLimit for an arbitrary cone on a diagram WalkingCospan ⥤ C.
theorem
CategoryTheory.IsPullback.hasPullback
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
:
theorem
CategoryTheory.IsPullback.of_hasPullback
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y Z : C}
(f : X ⟶ Z)
(g : Y ⟶ Z)
[Limits.HasPullback f g]
:
IsPullback (Limits.pullback.fst f g) (Limits.pullback.snd f g) f g
The pullback provided by HasPullback f g fits into an IsPullback.
theorem
CategoryTheory.IsPullback.of_is_product
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y Z : C}
{c : Limits.BinaryFan X Y}
(h : Limits.IsLimit c)
(t : Limits.IsTerminal Z)
:
IsPullback c.fst c.snd (t.from ((Limits.pair X Y).obj { as := Limits.WalkingPair.left }))
(t.from ((Limits.pair X Y).obj { as := Limits.WalkingPair.right }))
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}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
(h : Limits.IsLimit (Limits.BinaryFan.mk fst snd))
(t : Limits.IsTerminal Z)
:
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}
[Category.{v₁, u₁} C]
(X Y : C)
[Limits.HasBinaryProduct X Y]
[Limits.HasTerminal C]
:
theorem
CategoryTheory.IsPullback.of_hasBinaryProduct
{C : Type u₁}
[Category.{v₁, u₁} C]
(X Y : C)
[Limits.HasBinaryProduct X Y]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
:
noncomputable def
CategoryTheory.IsPullback.isoIsPullback
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
:
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
Instances For
@[simp]
theorem
CategoryTheory.IsPullback.isoIsPullback_hom_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
:
@[simp]
theorem
CategoryTheory.IsPullback.isoIsPullback_hom_fst_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
{Z✝ : C}
(h✝ : X ⟶ Z✝)
:
CategoryStruct.comp (isoIsPullback X Y h h').hom (CategoryStruct.comp fst' h✝) = CategoryStruct.comp fst h✝
@[simp]
theorem
CategoryTheory.IsPullback.isoIsPullback_hom_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
:
@[simp]
theorem
CategoryTheory.IsPullback.isoIsPullback_hom_snd_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
{Z✝ : C}
(h✝ : Y ⟶ Z✝)
:
CategoryStruct.comp (isoIsPullback X Y h h').hom (CategoryStruct.comp snd' h✝) = CategoryStruct.comp snd h✝
@[simp]
theorem
CategoryTheory.IsPullback.isoIsPullback_inv_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
:
@[simp]
theorem
CategoryTheory.IsPullback.isoIsPullback_inv_fst_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
{Z✝ : C}
(h✝ : X ⟶ Z✝)
:
CategoryStruct.comp (isoIsPullback X Y h h').inv (CategoryStruct.comp fst h✝) = CategoryStruct.comp fst' h✝
@[simp]
theorem
CategoryTheory.IsPullback.isoIsPullback_inv_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
:
@[simp]
theorem
CategoryTheory.IsPullback.isoIsPullback_inv_snd_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P : C}
(X 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 : IsPullback fst snd f g)
(h' : IsPullback fst' snd' f g)
{Z✝ : C}
(h✝ : Y ⟶ Z✝)
:
CategoryStruct.comp (isoIsPullback X Y h h').inv (CategoryStruct.comp snd h✝) = CategoryStruct.comp snd' h✝
noncomputable def
CategoryTheory.IsPullback.isoPullback
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
:
Any object at the top left of a pullback square is
isomorphic to the pullback provided by the HasLimit API.
Equations
- h.isoPullback = (CategoryTheory.Limits.limit.isoLimitCone { cone := h.cone, isLimit := h.isLimit }).symm
Instances For
@[simp]
theorem
CategoryTheory.IsPullback.isoPullback_hom_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
:
@[simp]
theorem
CategoryTheory.IsPullback.isoPullback_hom_fst_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
{Z✝ : C}
(h✝ : X ⟶ Z✝)
:
CategoryStruct.comp h.isoPullback.hom (CategoryStruct.comp (Limits.pullback.fst f g) h✝) = CategoryStruct.comp fst h✝
@[simp]
theorem
CategoryTheory.IsPullback.isoPullback_hom_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
:
@[simp]
theorem
CategoryTheory.IsPullback.isoPullback_hom_snd_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
{Z✝ : C}
(h✝ : Y ⟶ Z✝)
:
CategoryStruct.comp h.isoPullback.hom (CategoryStruct.comp (Limits.pullback.snd f g) h✝) = CategoryStruct.comp snd h✝
@[simp]
theorem
CategoryTheory.IsPullback.isoPullback_inv_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
:
@[simp]
theorem
CategoryTheory.IsPullback.isoPullback_inv_fst_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
{Z✝ : C}
(h✝ : X ⟶ Z✝)
:
CategoryStruct.comp h.isoPullback.inv (CategoryStruct.comp fst h✝) = CategoryStruct.comp (Limits.pullback.fst f g) h✝
@[simp]
theorem
CategoryTheory.IsPullback.isoPullback_inv_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
:
@[simp]
theorem
CategoryTheory.IsPullback.isoPullback_inv_snd_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[Limits.HasPullback f g]
{Z✝ : C}
(h✝ : Y ⟶ Z✝)
:
CategoryStruct.comp h.isoPullback.inv (CategoryStruct.comp snd h✝) = CategoryStruct.comp (Limits.pullback.snd f g) h✝
theorem
CategoryTheory.IsPullback.of_iso_pullback
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : CommSq fst snd f g)
[Limits.HasPullback f g]
(i : P ≅ Limits.pullback f g)
(w₁ : CategoryStruct.comp i.hom (Limits.pullback.fst f g) = fst)
(w₂ : CategoryStruct.comp i.hom (Limits.pullback.snd f g) = snd)
:
IsPullback fst snd f g
theorem
CategoryTheory.IsPullback.of_horiz_isIso_mono
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
[IsIso fst]
[Mono g]
(sq : CommSq fst snd f g)
:
IsPullback fst snd f g
theorem
CategoryTheory.IsPullback.of_horiz_isIso
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
[IsIso fst]
[IsIso g]
(sq : CommSq fst snd f g)
:
IsPullback fst snd f g
theorem
CategoryTheory.IsPullback.of_iso
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
{P' X' Y' 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 : CategoryStruct.comp fst e₂.hom = CategoryStruct.comp e₁.hom fst')
(commsnd : CategoryStruct.comp snd e₃.hom = CategoryStruct.comp e₁.hom snd')
(commf : CategoryStruct.comp f e₄.hom = CategoryStruct.comp e₂.hom f')
(commg : CategoryStruct.comp g e₄.hom = CategoryStruct.comp e₃.hom g')
:
IsPullback fst' snd' f' g'
theorem
CategoryTheory.IsPullback.isIso_fst_of_mono
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y : C}
{fst snd : P ⟶ X}
{f : X ⟶ Y}
[Mono f]
(h : IsPullback fst snd f f)
:
IsIso fst
theorem
CategoryTheory.IsPullback.isIso_snd_iso_of_mono
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y : C}
{fst snd : P ⟶ X}
{f : X ⟶ Y}
[Mono f]
(h : IsPullback fst snd f f)
:
IsIso snd
theorem
CategoryTheory.IsPullback.isIso_fst_of_isIso
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[IsIso g]
:
IsIso fst
theorem
CategoryTheory.IsPullback.isIso_snd_of_isIso
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
[IsIso f]
:
IsIso snd
def
CategoryTheory.IsPushout.cocone
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
:
The (colimiting) PushoutCocone f g implicit in the statement
that we have an IsPushout f g inl inr.
Instances For
noncomputable def
CategoryTheory.IsPushout.isColimit
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
:
The cocone obtained from IsPushout f g inl inr is a colimit cocone.
Instances For
noncomputable def
CategoryTheory.IsPushout.desc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(hP : IsPushout f g inl inr)
{W : C}
(h : X ⟶ W)
(k : Y ⟶ W)
(w : CategoryStruct.comp f h = CategoryStruct.comp g k)
:
API for PushoutCocone.IsColimit.lift for IsPushout
Equations
- hP.desc h k w = CategoryTheory.Limits.PushoutCocone.IsColimit.desc hP.isColimit h k w
Instances For
@[simp]
theorem
CategoryTheory.IsPushout.inl_desc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(hP : IsPushout f g inl inr)
{W : C}
(h : X ⟶ W)
(k : Y ⟶ W)
(w : CategoryStruct.comp f h = CategoryStruct.comp g k)
:
@[simp]
theorem
CategoryTheory.IsPushout.inl_desc_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(hP : IsPushout f g inl inr)
{W : C}
(h : X ⟶ W)
(k : Y ⟶ W)
(w : CategoryStruct.comp f h = CategoryStruct.comp g k)
{Z✝ : C}
(h✝ : W ⟶ Z✝)
:
@[simp]
theorem
CategoryTheory.IsPushout.inr_desc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(hP : IsPushout f g inl inr)
{W : C}
(h : X ⟶ W)
(k : Y ⟶ W)
(w : CategoryStruct.comp f h = CategoryStruct.comp g k)
:
@[simp]
theorem
CategoryTheory.IsPushout.inr_desc_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(hP : IsPushout f g inl inr)
{W : C}
(h : X ⟶ W)
(k : Y ⟶ W)
(w : CategoryStruct.comp f h = CategoryStruct.comp g k)
{Z✝ : C}
(h✝ : W ⟶ Z✝)
:
theorem
CategoryTheory.IsPushout.hom_ext
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(hP : IsPushout f g inl inr)
{W : C}
{k l : P ⟶ W}
(h₀ : CategoryStruct.comp inl k = CategoryStruct.comp inl l)
(h₁ : CategoryStruct.comp inr k = CategoryStruct.comp inr l)
:
theorem
CategoryTheory.IsPushout.of_isColimit
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{c : Limits.PushoutCocone f g}
(h : Limits.IsColimit c)
:
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}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(w : CommSq f g inl inr)
(h : Limits.IsColimit w.cocone)
:
IsPushout f g inl inr
A variant of of_isColimit that is more useful with apply.
theorem
CategoryTheory.IsPushout.of_isColimit_cocone
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Functor Limits.WalkingSpan C}
{c : Limits.Cocone D}
(hc : Limits.IsColimit c)
:
Variant of of_isColimit for an arbitrary cocone on a diagram WalkingSpan ⥤ C.
theorem
CategoryTheory.IsPushout.hasPushout
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
:
theorem
CategoryTheory.IsPushout.of_hasPushout
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y : C}
(f : Z ⟶ X)
(g : Z ⟶ Y)
[Limits.HasPushout f g]
:
IsPushout f g (Limits.pushout.inl f g) (Limits.pushout.inr f g)
The pushout provided by HasPushout f g fits into an IsPushout.
theorem
CategoryTheory.IsPushout.of_is_coproduct
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y : C}
{c : Limits.BinaryCofan X Y}
(h : Limits.IsColimit c)
(t : Limits.IsInitial Z)
:
IsPushout (t.to ((Limits.pair X Y).obj { as := Limits.WalkingPair.left }))
(t.to ((Limits.pair X Y).obj { as := Limits.WalkingPair.right })) 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}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : Limits.IsColimit (Limits.BinaryCofan.mk inl inr))
(t : Limits.IsInitial Z)
:
A variant of of_is_coproduct that is more useful with apply.
theorem
CategoryTheory.IsPushout.of_hasBinaryCoproduct'
{C : Type u₁}
[Category.{v₁, u₁} C]
(X Y : C)
[Limits.HasBinaryCoproduct X Y]
[Limits.HasInitial C]
:
theorem
CategoryTheory.IsPushout.of_hasBinaryCoproduct
{C : Type u₁}
[Category.{v₁, u₁} C]
(X Y : C)
[Limits.HasBinaryCoproduct X Y]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
:
noncomputable def
CategoryTheory.IsPushout.isoIsPushout
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z : C}
(X 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 : IsPushout f g inl inr)
(h' : IsPushout f g inl' inr')
:
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
Instances For
@[simp]
theorem
CategoryTheory.IsPushout.inl_isoIsPushout_hom_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z : C}
(X 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 : IsPushout f g inl inr)
(h' : IsPushout f g inl' inr')
{Z✝ : C}
(h✝ : P' ⟶ Z✝)
:
CategoryStruct.comp inl (CategoryStruct.comp (isoIsPushout X Y h h').hom h✝) = CategoryStruct.comp inl' h✝
@[simp]
theorem
CategoryTheory.IsPushout.inr_isoIsPushout_hom_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z : C}
(X 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 : IsPushout f g inl inr)
(h' : IsPushout f g inl' inr')
{Z✝ : C}
(h✝ : P' ⟶ Z✝)
:
CategoryStruct.comp inr (CategoryStruct.comp (isoIsPushout X Y h h').hom h✝) = CategoryStruct.comp inr' h✝
@[simp]
theorem
CategoryTheory.IsPushout.inl_isoIsPushout_inv_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z : C}
(X 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 : IsPushout f g inl inr)
(h' : IsPushout f g inl' inr')
{Z✝ : C}
(h✝ : P ⟶ Z✝)
:
CategoryStruct.comp inl' (CategoryStruct.comp (isoIsPushout X Y h h').inv h✝) = CategoryStruct.comp inl h✝
@[simp]
theorem
CategoryTheory.IsPushout.inr_isoIsPushout_inv_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z : C}
(X 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 : IsPushout f g inl inr)
(h' : IsPushout f g inl' inr')
{Z✝ : C}
(h✝ : P ⟶ Z✝)
:
CategoryStruct.comp inr' (CategoryStruct.comp (isoIsPushout X Y h h').inv h✝) = CategoryStruct.comp inr h✝
noncomputable def
CategoryTheory.IsPushout.isoPushout
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
:
Any object at the top left of a pullback square is
isomorphic to the pullback provided by the HasLimit API.
Equations
- h.isoPushout = (CategoryTheory.Limits.colimit.isoColimitCocone { cocone := h.cocone, isColimit := h.isColimit }).symm
Instances For
@[simp]
theorem
CategoryTheory.IsPushout.inl_isoPushout_inv
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
:
@[simp]
theorem
CategoryTheory.IsPushout.inl_isoPushout_inv_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
{Z✝ : C}
(h✝ : P ⟶ Z✝)
:
CategoryStruct.comp (Limits.pushout.inl f g) (CategoryStruct.comp h.isoPushout.inv h✝) = CategoryStruct.comp inl h✝
@[simp]
theorem
CategoryTheory.IsPushout.inr_isoPushout_inv
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
:
@[simp]
theorem
CategoryTheory.IsPushout.inr_isoPushout_inv_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
{Z✝ : C}
(h✝ : P ⟶ Z✝)
:
CategoryStruct.comp (Limits.pushout.inr f g) (CategoryStruct.comp h.isoPushout.inv h✝) = CategoryStruct.comp inr h✝
@[simp]
theorem
CategoryTheory.IsPushout.inl_isoPushout_hom
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
:
@[simp]
theorem
CategoryTheory.IsPushout.inl_isoPushout_hom_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
{Z✝ : C}
(h✝ : Limits.pushout f g ⟶ Z✝)
:
CategoryStruct.comp inl (CategoryStruct.comp h.isoPushout.hom h✝) = CategoryStruct.comp (Limits.pushout.inl f g) h✝
@[simp]
theorem
CategoryTheory.IsPushout.inr_isoPushout_hom
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
:
@[simp]
theorem
CategoryTheory.IsPushout.inr_isoPushout_hom_assoc
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
[Limits.HasPushout f g]
{Z✝ : C}
(h✝ : Limits.pushout f g ⟶ Z✝)
:
CategoryStruct.comp inr (CategoryStruct.comp h.isoPushout.hom h✝) = CategoryStruct.comp (Limits.pushout.inr f g) h✝
theorem
CategoryTheory.IsPushout.of_iso_pushout
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : CommSq f g inl inr)
[Limits.HasPushout f g]
(i : P ≅ Limits.pushout f g)
(w₁ : CategoryStruct.comp inl i.hom = Limits.pushout.inl f g)
(w₂ : CategoryStruct.comp inr i.hom = Limits.pushout.inr f g)
:
IsPushout f g inl inr
theorem
CategoryTheory.IsPushout.of_iso
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
{Z' X' Y' 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 : CategoryStruct.comp f e₂.hom = CategoryStruct.comp e₁.hom f')
(commg : CategoryStruct.comp g e₃.hom = CategoryStruct.comp e₁.hom g')
(comminl : CategoryStruct.comp inl e₄.hom = CategoryStruct.comp e₂.hom inl')
(comminr : CategoryStruct.comp inr e₄.hom = CategoryStruct.comp e₃.hom inr')
:
IsPushout f' g' inl' inr'
theorem
CategoryTheory.IsPushout.isIso_inl_iso_of_epi
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y : C}
{inl inr : X ⟶ P}
{f : Y ⟶ X}
[Epi f]
(h : IsPushout f f inl inr)
:
IsIso inl
theorem
CategoryTheory.IsPushout.isIso_inr_iso_of_epi
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y : C}
{inl inr : X ⟶ P}
{f : Y ⟶ X}
[Epi f]
(h : IsPushout f f inl inr)
:
IsIso inr
theorem
CategoryTheory.IsPullback.flip
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
:
IsPullback snd fst g f
theorem
CategoryTheory.IsPullback.flip_iff
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
:
@[simp]
theorem
CategoryTheory.IsPullback.zero_left
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X : C)
:
IsPullback 0 0 (CategoryStruct.id X) 0
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}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X : C)
:
IsPullback 0 0 0 (CategoryStruct.id X)
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}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X : C)
:
IsPullback 0 (CategoryStruct.id X) 0 0
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}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X : C)
:
IsPullback (CategoryStruct.id X) 0 0 0
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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₂₁ : X₂₁ ⟶ X₃₁}
{v₂₂ : X₂₂ ⟶ X₃₂}
(s : IsPullback h₁₁ v₁₁ v₁₂ h₂₁)
(t : IsPullback h₂₁ v₂₁ v₂₂ h₃₁)
:
IsPullback h₁₁ (CategoryStruct.comp v₁₁ v₂₁) (CategoryStruct.comp v₁₂ 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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₁₂ : X₁₂ ⟶ X₁₃}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₃ ⟶ X₂₃}
(s : IsPullback h₁₁ v₁₁ v₁₂ h₂₁)
(t : IsPullback h₁₂ v₁₂ v₁₃ h₂₂)
:
IsPullback (CategoryStruct.comp h₁₁ h₁₂) v₁₁ v₁₃ (CategoryStruct.comp h₂₁ h₂₂)
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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₂₁ : X₂₁ ⟶ X₃₁}
{v₂₂ : X₂₂ ⟶ X₃₂}
(s : IsPullback h₁₁ (CategoryStruct.comp v₁₁ v₂₁) (CategoryStruct.comp v₁₂ v₂₂) h₃₁)
(p : CategoryStruct.comp h₁₁ v₁₂ = CategoryStruct.comp v₁₁ h₂₁)
(t : IsPullback h₂₁ v₂₁ v₂₂ h₃₁)
:
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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₁₂ : X₁₂ ⟶ X₁₃}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₃ ⟶ X₂₃}
(s : IsPullback (CategoryStruct.comp h₁₁ h₁₂) v₁₁ v₁₃ (CategoryStruct.comp h₂₁ h₂₂))
(p : CategoryStruct.comp h₁₁ v₁₂ = CategoryStruct.comp v₁₁ h₂₁)
(t : IsPullback h₁₂ v₁₂ v₁₃ h₂₂)
:
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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₂₁ : X₂₁ ⟶ X₃₁}
{v₂₂ : X₂₂ ⟶ X₃₂}
(s : IsPullback h₂₁ v₂₁ v₂₂ h₃₁)
(e : CategoryStruct.comp h₁₁ v₁₂ = CategoryStruct.comp v₁₁ h₂₁)
:
IsPullback h₁₁ (CategoryStruct.comp v₁₁ v₂₁) (CategoryStruct.comp v₁₂ v₂₂) h₃₁ ↔ IsPullback h₁₁ v₁₁ v₁₂ h₂₁
theorem
CategoryTheory.IsPullback.paste_horiz_iff
{C : Type u₁}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₁₂ : X₁₂ ⟶ X₁₃}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₃ ⟶ X₂₃}
(s : IsPullback h₁₂ v₁₂ v₁₃ h₂₂)
(e : CategoryStruct.comp h₁₁ v₁₂ = CategoryStruct.comp v₁₁ h₂₁)
:
IsPullback (CategoryStruct.comp h₁₁ h₁₂) v₁₁ v₁₃ (CategoryStruct.comp h₂₁ h₂₂) ↔ IsPullback h₁₁ v₁₁ v₁₂ h₂₁
theorem
CategoryTheory.IsPullback.of_right'
{C : Type u₁}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₂ : X₁₂ ⟶ X₁₃}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₂₂ : X₂₂ ⟶ X₂₃}
{h₁₃ : X₁₁ ⟶ X₁₃}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₃ ⟶ X₂₃}
(s : IsPullback h₁₃ v₁₁ v₁₃ (CategoryStruct.comp h₂₁ h₂₂))
(t : IsPullback h₁₂ v₁₂ v₁₃ h₂₂)
:
IsPullback (t.lift h₁₃ (CategoryStruct.comp v₁₁ h₂₁) ⋯) 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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₃₁ : X₃₁ ⟶ X₃₂}
{v₃₁ : X₁₁ ⟶ X₃₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₂₁ : X₂₁ ⟶ X₃₁}
{v₂₂ : X₂₂ ⟶ X₃₂}
(s : IsPullback h₁₁ v₃₁ (CategoryStruct.comp v₁₂ v₂₂) h₃₁)
(t : IsPullback h₂₁ v₂₁ v₂₂ h₃₁)
:
IsPullback h₁₁ (t.lift (CategoryStruct.comp h₁₁ v₁₂) 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}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
IsPullback b.fst b.snd 0 0
@[simp]
theorem
CategoryTheory.IsPullback.of_has_biproduct
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X Y : C)
[Limits.HasBinaryBiproduct X Y]
:
theorem
CategoryTheory.IsPullback.inl_snd'
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
IsPullback b.inl 0 b.snd 0
@[simp]
theorem
CategoryTheory.IsPullback.inl_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X Y : C)
[Limits.HasBinaryBiproduct X Y]
:
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}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
IsPullback b.inr 0 b.fst 0
@[simp]
theorem
CategoryTheory.IsPullback.inr_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X Y : C)
[Limits.HasBinaryBiproduct X Y]
:
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}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
IsPullback 0 0 b.inl b.inr
theorem
CategoryTheory.IsPullback.of_hasBinaryBiproduct
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X Y : C)
[Limits.HasBinaryBiproduct X Y]
:
instance
CategoryTheory.IsPullback.hasPullback_biprod_fst_biprod_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
[Limits.HasBinaryBiproduct X Y]
:
def
CategoryTheory.IsPullback.pullbackBiprodInlBiprodInr
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
[Limits.HasBinaryBiproduct X Y]
:
The pullback of biprod.inl and biprod.inr is the zero object.
Equations
- CategoryTheory.IsPullback.pullbackBiprodInlBiprodInr = CategoryTheory.Limits.limit.isoLimitCone { cone := ⋯.cone, isLimit := ⋯.isLimit }
Instances For
theorem
CategoryTheory.IsPullback.op
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
:
theorem
CategoryTheory.IsPullback.unop
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : Cᵒᵖ}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
(h : IsPullback fst snd f g)
:
theorem
CategoryTheory.IsPullback.of_vert_isIso_mono
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
[IsIso snd]
[Mono f]
(sq : CommSq fst snd f g)
:
IsPullback fst snd f g
theorem
CategoryTheory.IsPullback.of_vert_isIso
{C : Type u₁}
[Category.{v₁, u₁} C]
{P X Y Z : C}
{fst : P ⟶ X}
{snd : P ⟶ Y}
{f : X ⟶ Z}
{g : Y ⟶ Z}
[IsIso snd]
[IsIso f]
(sq : CommSq fst snd f g)
:
IsPullback fst snd f g
theorem
CategoryTheory.IsPullback.of_id_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Z : C}
{f : X ⟶ Z}
:
IsPullback (CategoryStruct.id X) f f (CategoryStruct.id Z)
theorem
CategoryTheory.IsPullback.of_id_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Z : C}
{f : X ⟶ Z}
:
IsPullback f (CategoryStruct.id X) (CategoryStruct.id Z) f
theorem
CategoryTheory.IsPullback.id_vert
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Z : C}
(f : X ⟶ Z)
:
IsPullback f (CategoryStruct.id X) (CategoryStruct.id Z) f
The following diagram is a pullback
X --f--> Z
| |
id id
v v
X --f--> Z
theorem
CategoryTheory.IsPullback.id_horiz
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Z : C}
(f : X ⟶ Z)
:
IsPullback (CategoryStruct.id X) f f (CategoryStruct.id Z)
The following diagram is a pullback
X --id--> X
| |
f f
v v
Z --id--> Z
theorem
CategoryTheory.IsPullback.of_prod_fst_with_id
{C : Type u₁}
[Category.{v₁, u₁} C]
{A B : C}
(f : A ⟶ B)
(X : C)
[Limits.HasBinaryProduct A X]
[Limits.HasBinaryProduct B X]
:
In a category, given a morphism f : A ⟶ B and an object X,
this is the obvious pullback diagram:
A ⨯ X ⟶ A
| |
v v
B ⨯ X ⟶ B
theorem
CategoryTheory.IsPullback.of_isLimit_binaryFan_of_isTerminal
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
{c : Limits.BinaryFan X Y}
(hc : Limits.IsLimit c)
{T : C}
(hT : Limits.IsTerminal T)
:
IsPullback c.fst c.snd (hT.from ((Limits.pair X Y).obj { as := Limits.WalkingPair.left }))
(hT.from ((Limits.pair X Y).obj { as := Limits.WalkingPair.right }))
theorem
CategoryTheory.IsPushout.flip
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
:
IsPushout g f inr inl
@[simp]
theorem
CategoryTheory.IsPushout.zero_right
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X : C)
:
IsPushout 0 (CategoryStruct.id X) 0 0
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}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X : C)
:
IsPushout (CategoryStruct.id X) 0 0 0
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}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X : C)
:
IsPushout 0 0 (CategoryStruct.id X) 0
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}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X : C)
:
IsPushout 0 0 0 (CategoryStruct.id X)
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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₂₁ : X₂₁ ⟶ X₃₁}
{v₂₂ : X₂₂ ⟶ X₃₂}
(s : IsPushout h₁₁ v₁₁ v₁₂ h₂₁)
(t : IsPushout h₂₁ v₂₁ v₂₂ h₃₁)
:
IsPushout h₁₁ (CategoryStruct.comp v₁₁ v₂₁) (CategoryStruct.comp v₁₂ v₂₂) 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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₁₂ : X₁₂ ⟶ X₁₃}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₃ ⟶ X₂₃}
(s : IsPushout h₁₁ v₁₁ v₁₂ h₂₁)
(t : IsPushout h₁₂ v₁₂ v₁₃ h₂₂)
:
IsPushout (CategoryStruct.comp h₁₁ h₁₂) v₁₁ v₁₃ (CategoryStruct.comp h₂₁ h₂₂)
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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₂₁ : X₂₁ ⟶ X₃₁}
{v₂₂ : X₂₂ ⟶ X₃₂}
(s : IsPushout h₁₁ (CategoryStruct.comp v₁₁ v₂₁) (CategoryStruct.comp v₁₂ v₂₂) h₃₁)
(p : CategoryStruct.comp h₂₁ v₂₂ = CategoryStruct.comp v₂₁ h₃₁)
(t : IsPushout h₁₁ v₁₁ v₁₂ h₂₁)
:
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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₁₂ : X₁₂ ⟶ X₁₃}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₃ ⟶ X₂₃}
(s : IsPushout (CategoryStruct.comp h₁₁ h₁₂) v₁₁ v₁₃ (CategoryStruct.comp h₂₁ h₂₂))
(p : CategoryStruct.comp h₁₂ v₁₃ = CategoryStruct.comp v₁₂ h₂₂)
(t : IsPushout h₁₁ v₁₁ v₁₂ h₂₁)
:
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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₂₁ : X₂₁ ⟶ X₃₁}
{v₂₂ : X₂₂ ⟶ X₃₂}
(s : IsPushout h₁₁ v₁₁ v₁₂ h₂₁)
(e : CategoryStruct.comp h₂₁ v₂₂ = CategoryStruct.comp v₂₁ h₃₁)
:
IsPushout h₁₁ (CategoryStruct.comp v₁₁ v₂₁) (CategoryStruct.comp v₁₂ v₂₂) h₃₁ ↔ IsPushout h₂₁ v₂₁ v₂₂ h₃₁
theorem
CategoryTheory.IsPushout.paste_horiz_iff
{C : Type u₁}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₁₂ : X₁₂ ⟶ X₁₃}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₃ ⟶ X₂₃}
(s : IsPushout h₁₁ v₁₁ v₁₂ h₂₁)
(e : CategoryStruct.comp h₁₂ v₁₃ = CategoryStruct.comp v₁₂ h₂₂)
:
IsPushout (CategoryStruct.comp h₁₁ h₁₂) v₁₁ v₁₃ (CategoryStruct.comp h₂₁ h₂₂) ↔ IsPushout h₁₂ v₁₂ v₁₃ h₂₂
theorem
CategoryTheory.IsPushout.of_top'
{C : Type u₁}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₂ ⟶ X₃₂}
{v₂₁ : X₂₁ ⟶ X₃₁}
(s : IsPushout h₁₁ (CategoryStruct.comp v₁₁ v₂₁) v₁₃ h₃₁)
(t : IsPushout h₁₁ v₁₁ v₁₂ h₂₁)
:
IsPushout h₂₁ v₂₁ (t.desc v₁₃ (CategoryStruct.comp v₂₁ h₃₁) ⋯) 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}
[Category.{v₁, u₁} C]
{X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂}
{h₁₂ : X₁₂ ⟶ X₁₃}
{h₂₁ : X₂₁ ⟶ X₂₂}
{h₂₃ : X₂₁ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁}
{v₁₂ : X₁₂ ⟶ X₂₂}
{v₁₃ : X₁₃ ⟶ X₂₃}
(s : IsPushout (CategoryStruct.comp h₁₁ h₁₂) v₁₁ v₁₃ h₂₃)
(t : IsPushout h₁₁ v₁₁ v₁₂ h₂₁)
:
IsPushout h₁₂ v₁₂ v₁₃ (t.desc (CategoryStruct.comp h₁₂ v₁₃) 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}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
@[simp]
theorem
CategoryTheory.IsPushout.of_has_biproduct
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X Y : C)
[Limits.HasBinaryBiproduct X Y]
:
theorem
CategoryTheory.IsPushout.inl_snd'
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
theorem
CategoryTheory.IsPushout.inl_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X Y : C)
[Limits.HasBinaryBiproduct X Y]
:
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}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
theorem
CategoryTheory.IsPushout.inr_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X Y : C)
[Limits.HasBinaryBiproduct X Y]
:
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}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
theorem
CategoryTheory.IsPushout.of_hasBinaryBiproduct
{C : Type u₁}
[Category.{v₁, u₁} C]
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
(X Y : C)
[Limits.HasBinaryBiproduct X Y]
:
instance
CategoryTheory.IsPushout.hasPushout_biprod_fst_biprod_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
[Limits.HasBinaryBiproduct X Y]
:
def
CategoryTheory.IsPushout.pushoutBiprodFstBiprodSnd
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
[Limits.HasBinaryBiproduct X Y]
:
The pushout of biprod.fst and biprod.snd is the zero object.
Equations
- CategoryTheory.IsPushout.pushoutBiprodFstBiprodSnd = CategoryTheory.Limits.colimit.isoColimitCocone { cocone := ⋯.cocone, isColimit := ⋯.isColimit }
Instances For
theorem
CategoryTheory.IsPushout.op
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : C}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
:
IsPullback inr.op inl.op g.op f.op
theorem
CategoryTheory.IsPushout.unop
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X Y P : Cᵒᵖ}
{f : Z ⟶ X}
{g : Z ⟶ Y}
{inl : X ⟶ P}
{inr : Y ⟶ P}
(h : IsPushout f g inl inr)
:
IsPullback inr.unop inl.unop g.unop f.unop
theorem
CategoryTheory.IsPushout.of_id_fst
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X : C}
{f : Z ⟶ X}
:
IsPushout (CategoryStruct.id Z) f f (CategoryStruct.id X)
theorem
CategoryTheory.IsPushout.of_id_snd
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X : C}
{f : Z ⟶ X}
:
IsPushout f (CategoryStruct.id Z) (CategoryStruct.id X) f
theorem
CategoryTheory.IsPushout.id_vert
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X : C}
(f : X ⟶ Z)
:
IsPushout f (CategoryStruct.id X) (CategoryStruct.id Z) f
The following diagram is a pullback
X --f--> Z
| |
id id
v v
X --f--> Z
theorem
CategoryTheory.IsPushout.id_horiz
{C : Type u₁}
[Category.{v₁, u₁} C]
{Z X : C}
(f : X ⟶ Z)
:
IsPushout (CategoryStruct.id X) f f (CategoryStruct.id Z)
The following diagram is a pullback
X --id--> X
| |
f f
v v
Z --id--> Z
theorem
CategoryTheory.IsPushout.of_coprod_inl_with_id
{C : Type u₁}
[Category.{v₁, u₁} C]
{A B : C}
(f : A ⟶ B)
(X : C)
[Limits.HasBinaryCoproduct A X]
[Limits.HasBinaryCoproduct B X]
:
In a category, given a morphism f : A ⟶ B and an object X,
this is the obvious pushout diagram:
A ⟶ A ⨿ X
| |
v v
B ⟶ B ⨿ X
theorem
CategoryTheory.IsPushout.of_isColimit_binaryCofan_of_isInitial
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
{c : Limits.BinaryCofan X Y}
(hc : Limits.IsColimit c)
{I : C}
(hI : Limits.IsInitial I)
:
IsPushout (hI.to ((Limits.pair X Y).obj { as := Limits.WalkingPair.right }))
(hI.to ((Limits.pair X Y).obj { as := Limits.WalkingPair.left })) c.inr c.inl
noncomputable def
CategoryTheory.IsPullback.isLimitFork
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y Z : C}
{f : X ⟶ Y}
{g g' : Y ⟶ Z}
(H : IsPullback f f g g')
:
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}
[Category.{v₁, u₁} C]
{X Y Z : C}
{f f' : X ⟶ Y}
{g : Y ⟶ Z}
(H : IsPushout f f' g g)
:
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}
[Category.{v₁, u₁} C]
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
(p₁ : IsPullback f g h i)
(p₂ : IsPushout f g h i)
:
BicartesianSq f g h i
theorem
CategoryTheory.BicartesianSq.flip
{C : Type u₁}
[Category.{v₁, u₁} C]
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
(p : BicartesianSq f g h i)
:
BicartesianSq g f i h
theorem
CategoryTheory.BicartesianSq.of_is_biproduct₁
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
BicartesianSq b.fst b.snd 0 0
X ⊞ Y --fst--> X
| |
snd 0
| |
v v
Y -----0---> 0
is a bicartesian square.
theorem
CategoryTheory.BicartesianSq.of_is_biproduct₂
{C : Type u₁}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
{b : Limits.BinaryBicone X Y}
(h : b.IsBilimit)
:
BicartesianSq 0 0 b.inl b.inr
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}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
[Limits.HasBinaryBiproduct X Y]
:
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}
[Category.{v₁, u₁} C]
{X Y : C}
[Limits.HasZeroObject C]
[Limits.HasZeroMorphisms C]
[Limits.HasBinaryBiproduct X Y]
:
0 -----0---> X
| |
0 inl
| |
v v
Y --inr--> X ⊞ Y
is a bicartesian square.
theorem
CategoryTheory.Functor.map_isPullback
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Type u₂}
[Category.{v₂, u₂} D]
(F : Functor C D)
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
[Limits.PreservesLimit (Limits.cospan h i) F]
(s : IsPullback f g h i)
:
IsPullback (F.map f) (F.map g) (F.map h) (F.map i)
theorem
CategoryTheory.Functor.map_isPushout
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Type u₂}
[Category.{v₂, u₂} D]
(F : Functor C D)
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
[Limits.PreservesColimit (Limits.span f g) F]
(s : IsPushout f g h i)
:
theorem
CategoryTheory.IsPullback.map
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Type u₂}
[Category.{v₂, u₂} D]
(F : Functor C D)
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
[Limits.PreservesLimit (Limits.cospan h i) F]
(s : IsPullback f g h i)
:
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}
[Category.{v₁, u₁} C]
{D : Type u₂}
[Category.{v₂, u₂} D]
(F : Functor C D)
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
[Limits.PreservesColimit (Limits.span f g) F]
(s : IsPushout f g h i)
:
Alias of CategoryTheory.Functor.map_isPushout.
theorem
CategoryTheory.IsPullback.of_map
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Type u₂}
[Category.{v₂, u₂} D]
(F : Functor C D)
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
[Limits.ReflectsLimit (Limits.cospan h i) F]
(e : CategoryStruct.comp f h = CategoryStruct.comp g i)
(H : IsPullback (F.map f) (F.map g) (F.map h) (F.map i))
:
IsPullback f g h i
theorem
CategoryTheory.IsPullback.of_map_of_faithful
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Type u₂}
[Category.{v₂, u₂} D]
(F : Functor C D)
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
[Limits.ReflectsLimit (Limits.cospan h i) F]
[F.Faithful]
(H : IsPullback (F.map f) (F.map g) (F.map h) (F.map i))
:
IsPullback f g h i
theorem
CategoryTheory.IsPullback.map_iff
{C : Type u₁}
[Category.{v₁, u₁} C]
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
{D : Type u_1}
[Category.{u_2, u_1} D]
(F : Functor C D)
[Limits.PreservesLimit (Limits.cospan h i) F]
[Limits.ReflectsLimit (Limits.cospan h i) F]
(e : CategoryStruct.comp f h = CategoryStruct.comp g i)
:
theorem
CategoryTheory.IsPushout.of_map
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Type u₂}
[Category.{v₂, u₂} D]
(F : Functor C D)
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
[Limits.ReflectsColimit (Limits.span f g) F]
(e : CategoryStruct.comp f h = CategoryStruct.comp g i)
(H : IsPushout (F.map f) (F.map g) (F.map h) (F.map i))
:
IsPushout f g h i
theorem
CategoryTheory.IsPushout.of_map_of_faithful
{C : Type u₁}
[Category.{v₁, u₁} C]
{D : Type u₂}
[Category.{v₂, u₂} D]
(F : Functor C D)
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
[Limits.ReflectsColimit (Limits.span f g) F]
[F.Faithful]
(H : IsPushout (F.map f) (F.map g) (F.map h) (F.map i))
:
IsPushout f g h i
theorem
CategoryTheory.IsPushout.map_iff
{C : Type u₁}
[Category.{v₁, u₁} C]
{W X Y Z : C}
{f : W ⟶ X}
{g : W ⟶ Y}
{h : X ⟶ Z}
{i : Y ⟶ Z}
{D : Type u_1}
[Category.{u_2, u_1} D]
(F : Functor C D)
[Limits.PreservesColimit (Limits.span f g) F]
[Limits.ReflectsColimit (Limits.span f g) F]
(e : CategoryStruct.comp f h = CategoryStruct.comp g i)
: