nLab
coimage

Idea

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

Definition

The coimage of a morphism f:cd in a category C is the image of the corresponding morphism in the opposite category C op.

In terms of colimits

If C has finite limits and colimits, then the coimage of a morphism f:cd is

Coimfcolim(c× dcc),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 c× dcc

coimfc c× dcc.coim f \simeq c \sqcup_{c\times_d c} c \,.

So in

c× dc c f coimf f C f d\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

ccoimfc \to coim f

is an epimorphism.

Remarks

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