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

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

Remarks:

1. such a factorization does not necessarily exist.
2. $e$ is unique by definition of $m$ monic.
3. $m^{\prime }{e}^{\prime }=f=me=m^{\prime }ve$, therefore $e^{\prime }=ve$ by $m^{\prime }$ monic.
4. $v$ is monic.
5. $m=m^{\prime }v$ 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 $f=me$ 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 $(Y{\bigsqcup }_{X}Y,i_1,i_2)$, which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms $i_1,i_2:Y\to Y{\bigsqcup }_{X}Y$, on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.

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 morphisms. (Recall also that an equalizer is automatically a monomorphism).
3. In an abelian category, the cokernel pair property can be written $i_{1}f=i_{2}f\iff (i_1-i_2)f=0=0f$ and the equalizer condition $i_{1}m=i_{2}m\iff (i_1-i_2)m=0m$. Moreover, all monomorphisms are regular.

Examples

In the category of sets the image of a morphism $f:X\to Y$ 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.

Essential Image

A related notion to image is essential image.

A subcategory $C\subset B$ of a (strict) category is said to be replete if for every $x\in C$, and for every isomorphism $\iota :x\to y$, both $\iota$ and $y$ belong to C.

Given a functor $F:A\to B$ between categories, the smallest replete subcategory of the target n-category B containing the image of A under F.

See also

• Subobject
• Coimage
• Image (mathematics)