Article

# Retraction and coretraction

In category theory, a retraction is understood to be a morphism

${displaystyle f}$

f

{displaystyle f}

which has a right inverse, that is, for which there is a morphism

${displaystyle g}$

g

{displaystyle g}

gives with

${displaystyle fcirc g=operatorname {id} }$

f

g
=
id

{displaystyle fcirc g=operatorname {id} }

. The dual notion of a retraction is that of a coretraction (or cut), that is, a morphism that has a left inverse. The right inverse of a retraction is a coretraction and vice versa.

One object

${displaystyle X}$

X

{displaystyle X}

one category

${displaystyle {mathcal {C}}}$

C

{displaystyle {mathcal {C}}

means the retract of an object

${displaystyle Yin {mathcal {C}}}$

Y

C

{displaystyle Yin {mathcal {C}}

,
when it’s in

${displaystyle {mathcal {C}}}$

C

{displaystyle {mathcal {C}}

a morphism

${displaystyle fcolon Xto Y}$

f
:
X

Y

{displaystyle fcolon Xto Y}

and a retraction

${displaystyle rcolon Yto X}$

r
:
Y

X

{displaystyle rcolon Yto X}

sync and corrections by n17t01

${displaystyle f}$

f

{displaystyle f}

, i.e. a morphism

${displaystyle r}$

r

{displaystyle r}

with

${displaystyle rcirc f=operatorname {id} _{X}}$

r

f
=

id

X

{displaystyle rcirc f=operatorname {id} _{X}}

, there.

Every retraction is extreme and even regular epimorphism. Likewise, every coretraction is extreme and even regular monomorphism and even difference kernel.[1]

## Special categories

### Topological spaces

The notion of retraction finds application in algebraic topology. In the category

${displaystyle mathbf {Top} }$

T
o
p

{displaystyle mathbf {top} }

of topological spaces, all extreme monomorphisms and hence all coretractions are topological embeddings.[2] This allows for a different view and definition in the case of topological spaces: a retraction is a continuous left inverse of a topological embedding. Or put more concretely: A retraction is a continuous mapping from a topological space into itself such that every element of the image set is a fixed point.[3]

This also allows a concrete definition of the retract: A subspace

${displaystyle A}$

A

{displaystyle A}

of a topological space

${displaystyle X}$

X

{displaystyle X}

is called Retract of

${displaystyle X}$

X

{displaystyle X}

if there’s a retraction

${displaystyle r}$

r

{displaystyle r}

for embedding

${displaystyle icolon Ato X}$

i
:
A

X

{displaystyle icolon Ato X}

there.

${displaystyle A}$

A

{displaystyle A}

is exactly then retract of

${displaystyle X}$

X

{displaystyle X}

if every continuous mapping

${displaystyle fcolon Ato Y}$

f
:
A

Y

{displaystyle fcolon Ato Y}

${displaystyle gcolon Xto Y}$

g
:
X

Y

{displaystyle gcolon Xto Y}

can be continued:

• Is there a retraction
• A continuation of

In a Hausdorff room, each retract is self-contained: Be

${displaystyle Asubset X}$

A

X

{displaystyle Asubset X}

Retract with retraction

${displaystyle rcolon Xto A}$

r
:
X

A

{displaystyle rcolon Xto A}

. Now consider a convergent network

${displaystyle N_{i}to a}$

N

i

a

{displaystyle N_{i}to a}

at

${displaystyle A}$

A

{displaystyle A}

. The image network

${displaystyle r(N_{i})}$

r
(

N

i

)

{displaystyle r(N_{i})}

converges towards

${displaystyle r(a)}$

r
(
a
)

{displaystyle r(a)}

(since

${displaystyle r}$

r

{displaystyle r}

continuous) and is equal to the original net. Since the boundary value of a mesh in Hausdorf spaces is unique, the following holds true

${displaystyle r(a)=ain A}$

r
(
a
)
=
a

A

{displaystyle r(a)=ain A}

and

${displaystyle A}$

A

{displaystyle A}

is closed. This is not true in non-Hausdorf spaces: in non-T₁-spaces there exist non-closed one-element sets, but they are obviously retracts. As an example of a T₁-space with non-closed retracts, consider the cofinite topology on

${displaystyle mathbb {N} }$

N

{displaystyle mathbb {N} }

${displaystyle rcolon mathbb {N} to mathbb {N} setminus {0}}$

r
:

N

N

{
0
}

{displaystyle rcolon mathbb {N} to mathbb {N} setminus {0}}

with

${displaystyle r(0)=1}$

r
(
0
)
=
1

{displaystyle r(0)=1}

and

${displaystyle r(n)=n}$

r
(
n
)
=
n

{displaystyle r(n)=n}

for

${displaystyle nneq 0}$

n

0

{displaystyle nneq 0}

is a retraction, but the image is not completed.

#### Deformation tract

${displaystyle A}$

A

{displaystyle A}

is called deformation retract of

${displaystyle X}$

X

{displaystyle X}

if

${displaystyle icirc r}$

i

r

{displaystyle icirc r}

homotopic to

${displaystyle operatorname {id} _{X}}$

id

X

{displaystyle operatorname {id} _{X}}

relatively

${displaystyle A}$

A

{displaystyle A}

is.

Deformation retractions are special homotopy equivalences that generate this equivalence relation.

#### Examples

##### Elementary example

The following figure is an illustrative example of a retraction in the real numbers:

##### Brouwer’s fixed point theorem in the one-dimensional case

Brouwer’s fixed point theorem states that every continuous mapping of a solid sphere has a fixed point in itself. A one-dimensional solid sphere corresponds topologically just to a closed interval, for instance

${displaystyle left[-1,1right]}$

[

– −
1
,
1

]

{displaystyle left[-1,1right]}

. If there were now a continuous, fixed point free mapping

${displaystyle fcolon left[-1,1right]to left[-1,1right]}$

f
:

[

– −
1
,
1

]

[

– −
1
,
1

]

{displaystyle fcolon left[-1,1right]to left[-1,1right]}

then this would result in a retraction

${displaystyle gcolon left[-1,1right]to {-1,1}}$

g
:

[

– −
1
,
1

]

{
– −
1
,
1
}

{displaystyle gcolon left[-1,1right]to {-1,1}}

by

${displaystyle g(x)=operatorname {sgn}(x-f(x))=textstyle {frac {x-f(x)}{left|x-f(x)right|}}}$

g
(
x
)
=
sgn

(
x
– −
f
(
x
)
)
=

x
– −
f
(
x
)

|

x
– −
f
(
x
)

|

{displaystyle g(x)=operatorname {sgn}(x-f(x))=textstyle {frac {x-f(x)}{left|x-f(x)right|}}

(since the denominator would never disappear), i. e

${displaystyle left{-1,1right}}$

{

– −
1
,
1

}

{displaystyle left{-1,1right}}

would have to be retract from

${displaystyle left[-1,1right]}$

[

– −
1
,
1

]

{displaystyle left[-1,1right]}

be. But such a retraction cannot exist, since coherence is preserved under continuous mappings.[3]

##### Closed sub-areas of the Baire region

In the Baire room

${displaystyle {mathcal {N}}}$

N

{displaystyle {mathcal {N}}

applies: For any closed sub-areas (these are always Polish sub-areas)

${displaystyle Xsubset Ysubset {mathcal {N}}}$

X

Y

N

{displaystyle Xsubset Ysubset {mathcal {N}}

sync and corrections by n17t01

${displaystyle X}$

X

{displaystyle X}

Retract from

${displaystyle Y}$

Y

{displaystyle Y}

. Note that the Baire space is totally disjoint, and therefore the coherence notion does not provide any constraints on retracts.

### Arrow category

Be

${displaystyle {mathcal {C}}}$

C

{displaystyle {mathcal {C}}

a category, the corresponding arrow category is then the category of functors from the category with two objects and three morphisms into the category

${displaystyle {mathcal {C}}}$

C

{displaystyle {mathcal {C}}

. These are called arrows and can be linked to the morphisms in

${displaystyle {mathcal {C}}}$

C

{displaystyle {mathcal {C}}

can be identified. An arrow

${displaystyle f}$

f

{displaystyle f}

is retract of an arrow

${displaystyle g}$

g

{displaystyle g}

if it is a natural transformation (i.e., a commutative square)

${displaystyle eta colon fto g}$

η
:
f

g

{displaystyle eta colon fto g}

and a retraction

${displaystyle rcolon gto f}$

r
:
g

f

{displaystyle rcolon gto f}

so the following diagram commutates:

### Set Theory

In the category

${displaystyle mathbf {Set} }$

S
e
t

{displaystyle mathbf {set} }

of all sets and the functions between them, a morphism (that is, a function between two sets) is a retraction exactly if it is surjective. This statement is equivalent to the axiom of choice in set theory. Correspondingly, a morphism is a coretraction exactly if it is injective and there is a morphism in the opposite direction. However, this statement does not require the axiom of choice. From these statements it follows that in any concrete category the retractions must be surjective and the coretractions injective, which in general does not hold for general epi- or monomorphisms, which in the category of sets coincide with the retractions or coretractions.

## Individual references

1. Dieter Pumplün: Elements of category theory. 1. Auflage. Spektrum Akademischer Verlag, Heidelberg 1999, ISBN 3-86025-676-9, p. 64.
2. extremal monomorphism, entry in nLab. (English)
3. a b William Fulton: Algebraic Topology. 1. Edition. Springer, New York 1995. section 4b, ISBN 0-387-94327-7