nLab
coimage

Idea

The coimage of a morphism is the notion dual to its image.

Definition

The coimage of a morphism $f:c\to d$ in a category $C$ is the image of the corresponding morphism in the opposite category $C^{\mathrm{op}}$.

In terms of colimits

If $C$ has finite limits and colimits, then the coimage of a morphism $f:c\to d$ is

$$\mathrm{Coim}f\simeq \mathrm{colim}(c{\times }_{d}c\overrightarrow{\to }c),$$Coim f \simeq colim ( c \times_d c \stackrel{\to}{\to} c) \,,

that is the coequalizer of its kernel pair.

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

$$\mathrm{coim}f\simeq c{\bigsqcup }_{c{\times }_{d}c}c.$$coim f \simeq c \sqcup_{c\times_d c} c \,.

So in

$$\begin{array}{ccccc}c{\times }_{d}c& & \to & & c\\ & & & \swarrow \\ {\downarrow }^f& & \mathrm{coim}f& & {\downarrow }^f\\ & \nearrow & & \searrow \\ C& & \stackrel{f}{\to }& & d\end{array}$$\array{ c \times_d c &&\to&& c \\ &&& \swarrow \\ \downarrow^f && coim f && \downarrow^f \\ & \nearrow && \searrow \\ C && \stackrel{f}{\to} && d }

the outer square is a pullback square while the inner is a pushout.

Notice that being a coequalizer, the morphism

$$c\to \mathrm{coim}f$$c \to coim f

is an epimorphism.

Remarks

• Morphisms for which image and coimage coincide (in a certain sense) are strict morphisms.