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:
#There exists a morphism $e\colon X\to I$ such that $f = m\, e$.
#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:
# such a factorization does not necessarily exist.
# $e$ is unique by definition of $m$ monic.
# $m'e'=f=me=m've$, therefore $e'=ve$ by $m'$ monic.
# $v$ is monic.
# $m = m'\, v$ already implies that $v$ is unique.

The image of $f$ is often denoted by $\text f$ or $\text (f)$.

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 $(Y \sqcup_X Y, i_1, i_2)$, which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms $i_1,i_2:Y\to Y\sqcup_X Y$, on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.

Remarks:
# Finite bicompleteness of the category ensures that pushouts and equalizers exist.
# $(Im,m)$ 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).
# 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.

Examples

In the category of sets the image of a morphism $f\colon X \to Y$ is the inclusion from the ordinary image $\$ 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 $F \colon A \to B$ 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)

References

{{Reflist

Category theory