# 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 ${\displaystyle C}$ and a morphism ${\displaystyle f\colon X\to Y}$ in ${\displaystyle C}$, the image [1] of ${\displaystyle f}$ is a monomorphism ${\displaystyle m\colon I\to Y}$ satisfying the following universal property:

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

Remarks:

1. such a factorization does not necessarily exist
2. ${\displaystyle e}$ is unique by definition of ${\displaystyle m}$ monic
3. ${\displaystyle v}$ is monic.
4. ${\displaystyle m=m'\,v}$ already implies that ${\displaystyle v}$ is unique.

The image of ${\displaystyle f}$ is often denoted by ${\displaystyle {\text{Im}}f}$ or ${\displaystyle {\text{Im}}(f)}$.

Proposition: If ${\displaystyle C}$ has all equalizers then the ${\displaystyle e}$ in the factorization ${\displaystyle f=m\,e}$ of (1) is an epimorphism. [2]

## Second definition

In a category ${\displaystyle C}$ with all finite limits and colimits, the image is defined as the equalizer ${\displaystyle (Im,m)}$ of the so-called cokernel pair ${\displaystyle (Y\sqcup _{X}Y,i_{1},i_{2})}$ .[3]

Remarks:

1. Finite bicompleteness of the category ensures that pushouts and equalizers exist.
2. ${\displaystyle (Im,m)}$ can be called regular image as ${\displaystyle m}$ 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 ${\displaystyle i_{1}\,f=i_{2}\,f\ \Leftrightarrow \ (i_{1}-i_{2})\,f=0=0\,f}$ and the equalizer condition ${\displaystyle i_{1}\,m=i_{2}\,m\ \Leftrightarrow \ (i_{1}-i_{2})\,m=0\,m}$. Moreover, all monomorphisms are regular.

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

## Examples

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