# Contents

## Idea

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

Compare with strict epimorphism.

## Definition

### In a category with limits and colimits

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

$\mathrm{Im}f\simeq \mathrm{lim}\left(d⇉d{\bigsqcup }_{c}d\right)\phantom{\rule{thinmathspace}{0ex}},$Im f \simeq lim( d \rightrightarrows d \sqcup_c d ) \,,

i.e. the equalizer of $d⇉d{\bigsqcup }_{c}d$,

while the coimage is the colimit

$\mathrm{Coim}f\simeq \mathrm{colim}\left(c{×}_{d}c⇉c\right)\phantom{\rule{thinmathspace}{0ex}}.$Coim f \simeq colim( c \times_d c \rightrightarrows c) \,.

By the various universal properties, there is a unique morphism

$u:\mathrm{Coim}f\to \mathrm{Im}f$u : Coim f \to Im f

such that

$\begin{array}{ccc}c& \stackrel{f}{\to }& d\\ ↓& & ↑\\ \mathrm{Coim}f& \stackrel{u}{\to }& \mathrm{Im}f\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

Examples of categories in which every morphism is strict include

• Set;
• the category $\mathrm{Mod}\left(R\right)$ of modules over a ring $R$;
• the category $\mathrm{PSh}\left(C\right)=\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ 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)