nLab strict morphism

Idea

A strict morphism is a morphism for which the notion of regular image and regular 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 regular 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 regular 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

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.