nLab
strict morphism

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

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

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

while the coimage is the colimit

$$\mathrm{Coim}f\simeq \mathrm{colim}(c{\times }_{d}c\rightrightarrows c).$$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

$$\begin{array}{ccc}c& \stackrel{f}{\to }& d\\ \downarrow & & \uparrow \\ \mathrm{Coim}f& \stackrel{u}{\to }& \mathrm{Im}f\end{array}.$$\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 $\mathrm{Mod}(R)$ of modules over a ring $R$;
• the category $\mathrm{PSh}(C)=[C^{\mathrm{op}},\mathrm{Set}]$ of presheaves on any small category $C$;
• any abelian category;
• any topos.