 Image (category Theory)
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Image Category Theory

In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.

## General Definition

Given a category $C$ and a morphism $f\colon X\to Y$ in $C$ , the image  of $f$ is a monomorphism $m\colon I\to Y$ satisfying the following universal property:

1. There exists a morphism $e\colon X\to I$ such that $f=m\,e$ .
2. For any object $I'$ with a morphism $e'\colon X\to I'$ and a monomorphism $m'\colon I'\to Y$ such that $f=m'\,e'$ , there exists a unique morphism $v\colon I\to I'$ such that $m=m'\,v$ .

Remarks:

1. such a factorization does not necessarily exist.
2. $e$ is unique by definition of $m$ monic.
3. $m'e'=f=me=m've\implies e'=ve$ by $m$ monic.
4. $v$ is monic.
5. $m=m'\,v$ already implies that $v$ is unique.  The image of $f$ is often denoted by ${\text{Im}}f$ or ${\text{Im}}(f)$ .

Proposition: If $C$ has all equalizers then the $e$ in the factorization $f=m\,e$ of (1) is an epimorphism. 

## Second definition

In a category $C$ with all finite limits and colimits, the image is defined as the equalizer $(Im,m)$ of the so-called cokernel pair $(Y\sqcup _{X}Y,i_{1},i_{2})$ .

Remarks:

1. Finite bicompleteness of the category ensures that pushouts and equalizers exist.
2. $(Im,m)$ can be called regular image as $m$ is a regular monomorphism, i.e. the equalizer of a pair of morphism. (Recall also that an equalizer is automatically a monomorphism).
3. In an abelian category, the cokernel pair property can be written $i_{1}\,f=i_{2}\,f\ \Leftrightarrow \ (i_{1}-i_{2})\,f=0=0\,f$ and the equalizer condition $i_{1}\,m=i_{2}\,m\ \Leftrightarrow \ (i_{1}-i_{2})\,m=0\,m$ . Moreover, all monomorphisms are regular.

Theorem — If $f$ always factorizes through regular monomorphisms, then the two definitions coincide.

## Examples

In the category of sets the image of a morphism $f\colon X\to Y$ is the inclusion from the ordinary image $\{f(x)~|~x\in X\}$ to $Y$ . 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 $f$ can be expressed as follows:

im f = ker coker f

In an abelian category (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f.