image

General definition

Given a category

C

{\displaystyle C}

and a morphism

f

:

X

→

Y

{\displaystyle f\colon X\to Y}

in

C

{\displaystyle C}

, the image

of

f

{\displaystyle f}

is a monomorphism

m

:

I

→

Y

{\displaystyle m\colon I\to Y}

satisfying the following universal property:

There exists a morphism

e

:

X

→

I

{\displaystyle e\colon X\to I}

such that

f

=

m

e

{\displaystyle f=m\,e}

.

For any object

I

′

{\displaystyle I'}

with a morphism

e

′

:

X

→

I

′

{\displaystyle e'\colon X\to I'}

and a monomorphism

m

′

:

I

′

→

Y

{\displaystyle m'\colon I'\to Y}

such that

f

=

m

′

e

′

{\displaystyle f=m'\,e'}

, there exists a unique morphism

v

:

I

→

I

′

{\displaystyle v\colon I\to I'}

such that

m

=

m

′

v

{\displaystyle m=m'\,v}

.

Remarks:

such a factorization does not necessarily exist.

e

{\displaystyle e}

is unique by definition of

m

{\displaystyle m}

monic.

m

′

e

′

=

f

=

m

e

=

m

′

v

e

{\displaystyle m'e'=f=me=m've}

, therefore

e

′

=

v

e

{\displaystyle e'=ve}

by

m

′

{\displaystyle m'}

monic.

v

{\displaystyle v}

is monic.

m

=

m

′

v

{\displaystyle m=m'\,v}

already implies that

v

{\displaystyle v}

is unique.

The image of

f

{\displaystyle f}

is often denoted by

Im

f

{\displaystyle {\text{Im}}f}

or

Im

(

f

)

{\displaystyle {\text{Im}}(f)}

.

Proposition: If

C

{\displaystyle C}

has all equalizers then the

e

{\displaystyle e}

in the factorization

f

=

m

e

{\displaystyle f=m\,e}

of (1) is an epimorphism.

Second definition

In a category

C

{\displaystyle C}

with all finite limits and colimits, the image is defined as the equalizer

(

I

m

,

m

)

{\displaystyle (Im,m)}

of the so-called cokernel pair

(

Y

⊔

X

Y

,

i

1

,

i

2

)

{\displaystyle (Y\sqcup _{X}Y,i_{1},i_{2})}

, which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms

i

1

,

i

2

Examples

In the category of sets the image of a morphism

f

:

X

→

Y

{\displaystyle f\colon X\to Y}

is the inclusion from the ordinary image

{

f

(

x

)

|

x

∈

X

}

{\displaystyle \{f(x)~|~x\in X\}}

to

Y

{\displaystyle Y}

. In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.

In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism

f

{\displaystyle f}

can be expressed as follows:

im f = ker coker f

Essential Image

A related notion to image is essential image.

A subcategory

C

⊂

B

{\displaystyle C\subset B}

of a (strict) category is said to be replete if for every

x

∈

C

{\displaystyle x\in C}

, and for every isomorphism

ι

:

x

→

y

{\displaystyle \iota :x\to y}

, both

ι

{\displaystyle \iota }

and

y

{\displaystyle y}

belong to C.

Let

F

:

A

→

B

{\displaystyle F\colon A\to B}