nLab
image

Idea

The familiar notion of the image of a map of sets may be formalized to yield a notion of image for morphisms in an arbitrary category.

Definitions

There are several definitions that are equivalent when they jointly apply.

In terms of subobjects

Let $C$ be a category, and $f:c\to d$ be a morphism. The image of $f$ is the smallest subobject $s\subseteq d$ through which $f$ factors (if it exists). There is a dual notion of co-image: the largest quotient (in the co-subobject sense) of $c$ through which $f$ factors.

Remarks

• If $C$ admits equalizers, and if $i:k\to d$ represents the image of $f:c\to d$, then the unique map $q:c\to k$ such that $f=iq$ is an epimorphism. Thus, in a finitely complete category in which every morphism admits an image, one obtains in this way an epi-mono factorization, but the factorization may not have particularly good properties (in particular, the factorization through the image might not be stable with respect to pullback).

• The notion of regular category formalizes a sense in which image factorizations do behave well: factorizations into a regular epimorphism followed by a mono which are stable under pullback.

• In Cat, the image of a functor $F:A\to B$ is the smallest subcategory of $B$ which contains images through $F$ of all morphisms in $A$. Some of the morphisms in the image may not be images of any morphism in $A$; all morphisms in the image of $F$ are compositions in $B$ of $B$-composable sequences of images of morphisms in $A$ which themselves do not necessarily form $A$-composable sequences of morphisms in $A$. Sometimes the notion of essential image is more appropriate; as the essential image is only equivalent to the image, this is somewhat $2$-category-theoretic point of view.

as a left adjoint functor

Alternatively, let $C/d$ be the slice category over $d$, and let $\mathrm{Mono}(C)/d$ be the full subcategory whose objects are monos into $d$. Assuming images exist in $C$, taking the image of a map $f:c\to d$ provides a left adjoint

$$C/d\to \mathrm{Mono}(C)/d$$C/d \to Mono(C)/d

to the inclusion $\mathrm{Mono}(C)/d\hookrightarrow C/d$.

as an equalizer

If the category $C$ admits finite limits and colimits, then the image $\mathrm{Im}f$ of a morphism $f:c\to d$ my be expressed as

$$\mathrm{Im}f\simeq \lim (d\overrightarrow{\to }d{\bigsqcup }_{c}d),$$Im f \simeq lim (d \stackrel{\to}{\to} d \sqcup_c d) \,,

where $d{\bigsqcup }_{c}d$ denotes the pushout

$$\begin{array}{ccc}c& \stackrel{f}{\to }& d\\ {\downarrow }^f& & \downarrow \\ d& \to & d{\bigsqcup }_{c}d\end{array}.$$\array{ c &\stackrel{f}{\to}& d \\ \downarrow^{f} && \downarrow \\ d &\to& d \sqcup_c d } \,.

In other words, the image is the equalizer of the cokernel pair?.

This is isomorphic to the pullback $d{\times }_{d{\bigsqcup }_{c}d}d$

$$\mathrm{im}f\simeq d{\times }_{d{\bigsqcup }_{c}d}d.$$im f \simeq d \times_{d \sqcup_c d} d \,.

So in the diagram

$$\begin{array}{ccccc}c& & \stackrel{f}{\to }& & d\\ & \searrow & & \nearrow \\ {\downarrow }^f& & \mathrm{im}f& & \downarrow \\ & \swarrow \\ d& & \to & & d{\bigsqcup }_{c}d\end{array}$$\array{ c &&\stackrel{f}{\to}&& d \\ & \searrow && \nearrow \\ \downarrow^f && im f&& \downarrow \\ & \swarrow \\ d &&\to&& d \sqcup_c d }

the outer square is a pushout and the inner one is a pullback.

Since $\mathrm{im}f$ is an equalizer, the morphism

$$\mathrm{im}f\to d$$im f \to d

is a monomorphism.

Lemma

There is a unique morphism

$$u:\mathrm{coim}f\to \mathrm{im}f$$u : coim f \to im f

from the coimage to the image of $f$ such that

$$f=(c\to \mathrm{coim}f\stackrel{u}{\to }\mathrm{im}f\to d).$$f = (c \to coim f \stackrel{u}{\to} im f \to d) \,.

Proof

Because $f$ coequalizes $c{\times }_{d}c\overrightarrow{\to }c$, a morphism $h$ in

$$\begin{array}{ccccccc}c& {\times }_{d}c\overrightarrow{\to }& c& \stackrel{f}{\to }& d& \overrightarrow{\to }& d{\bigsqcup }_{c}d\\ & & {}^{}{\downarrow }^{\mathrm{epi}}& {}^h\nearrow & {}^{}{\uparrow }^{\mathrm{mono}}\\ & & \mathrm{coim}f& & \mathrm{im}f\end{array}$$\array{ c &\times_d c \stackrel{\to}{\to}& c &\stackrel{f}{\to}& d &\stackrel{\to}{\to}& d \sqcup_{c} d \\ && {}^{\;}\downarrow^{epi} &{}^{h}\nearrow& {}^{\;}\uparrow^{mono} \\ && coim f && im f }

exists uniquely.

Because $c\to \mathrm{coim}f$ is epi it follows that $h$ equalizes $d\overrightarrow{\to }d{\bigsqcup }_{c}d$ and hence $u$ in the diagram

$$\begin{array}{ccccccc}c& {\times }_{d}c\overrightarrow{\to }& c& \stackrel{f}{\to }& d& \overrightarrow{\to }& d{\bigsqcup }_{c}d\\ & & {}^{}{\downarrow }^{\mathrm{epi}}& {}^h\nearrow & {}^{}{\uparrow }^{\mathrm{mono}}\\ & & \mathrm{coim}f& \stackrel{u}{\to }& \mathrm{im}f\end{array}$$\array{ c &\times_d c \stackrel{\to}{\to}& c &\stackrel{f}{\to}& d &\stackrel{\to}{\to}& d \sqcup_{c} d \\ && {}^{\;}\downarrow^{epi} &{}^{h}\nearrow& {}^{\;}\uparrow^{mono} \\ && coim f &\stackrel{u}{\to}& im f }

exists uniquely.

Remarks

• If $u$ is an isomorphism then $f$ is called a strict morphism.

• So if $C$ has finite limits and colimits and every morphism is a strict morphism we get an epi-mono factorization of every morphism $f:c\to d$ through its image $\simeq$ coimage.

• there are various generalizations of the notion of image to higher categorical contexts

  • in Cat there is the notion of essential image of a functor

  • in a presentable (infinity,1)-category or model category there is the notion of homotopy image.