nLab strict epimorphism

Definition

A strict epimorphism in a category is a morphism which is the joint coequalizer of all pairs of parallel morphisms that it coequalizes. In other words, $f \colon B\to C$ is a strict epimorphism if it is the colimit of the (possibly large) diagram consisting of all parallel pairs $g,h \colon A \;\rightrightarrows\; B$ such that $f g = f h$.

Although this definition does not include this explicitly, it follows that $f$ is an epimorphism.

A strict monomorphism is a morphism such that its dual is strict epimorphism in the dual category.

Relation to other epimorphism classes

If $f$ has a kernel pair $r,s \,\colon\,ker(f) \;\rightrightarrows\; B$ (such as if the ambient category has pullbacks), then any such pair $g,h$ factor uniquely through the kernel pair, which is itself such a pair (that is, $f r = f s$). Thus, for any $k \colon B\to D$, we have $k g = k h$ for all $g,h$ with $f g = f h$ if and only if $k r = k s$. Therefore, $f$ is strict epi if and only if it is the coequalizer of its kernel pair, hence if and only if it is an effective epimorphism and therefore a regular epimorphism.

For this reason, some sources define “regular epimorphism” in a category without pullbacks to mean what we have called a “strict epimorphism.”

It is easy to see that in any category, any regular epimorphism is strict. In a category without pullbacks, it seems that not every strict epimorphism need be regular. However, every strict epimorphism is strong, and hence extremal, for the same reason that any regular epimorphism is.

Properties

If for an epimorphism $f$, the composition $g \circ f$ is a strict epimorphism then $g$ is a strict epimorphism.

References

• Alexander Grothendieck, Definition 2.2, p. 304 in: Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats (FGA), Séminaire Bourbaki, no. 5 (1960) [numdam:SB_1958-1960__5__299_0]

See also Definition 2.1 in

• Eduardo J. Dubuc, From Yoneda to Topoi morphisms, arXiv.

Textbook accounts:

• Ion Bucur?, Aristide Deleanu?, Introduction to the theory of categories and functors, Wiley, 1968.

• Masaki Kashiwara, Pierre Schapira, p. 115-116 in: Categories and Sheaves