strict morphism

A *strict morphism* is a morphism for which the notion of image and coimage coincide.

Compare with strict epimorphism.

Let $C$ be a category with finite limits and colimits. Let $f : c \to d$ be a morphism in $C$.

Recall that the image of $f$ is the limit

$Im f \simeq lim( d \rightrightarrows d \sqcup_c d )
\,,$

i.e. the equalizer of $d \rightrightarrows d \sqcup_c d$,

while the coimage is the colimit

$Coim f \simeq colim( c \times_d c \rightrightarrows c)
\,.$

By the various universal properties, there is a unique morphism

$u : Coim f \to Im f$

such that

$\array{
c &\stackrel{f}{\to}& d
\\
\downarrow && \uparrow
\\
Coim f &\stackrel{u}{\to}& Im f
}
\,.$

The morphism $f$ is called a **strict morphism** if $u$ is an isomorphism.

Examples of categories in which *every* morphism is strict include

- Set;
- the category $Mod(R)$ of modules over a ring $R$;
- the category $PSh(C) = [C^{op},Set]$ of presheaves on any small category $C$;
- any abelian category;
- any topos.

Revised on July 9, 2010 07:23:40
by Urs Schreiber
(87.212.203.135)