nLab
strict epimorphism

A strict epimorphism in a category $C$ is a morphism which is the joint coequalizer of all parallel pairs that it coequalizes. In other words, $f:B\to C$ is a strict epimorphism if it is the colimit of the (possibly large) diagram consisting of all parallel pairs $g,h:A\rightrightarrows B$ such that $fg=fh$. Although the definition does not include this explicitly, it follows that $f$ is an epimorphism.

If $C$ has pullbacks, then any such $g,h$ factor uniquely through the kernel pair $r,s:\ker (f)\rightrightarrows B$ of $f$, which is itself such a pair (that is, $fr=fs$). Thus, for any $k:B\to D$, we have $kg=kh$ for all $g,h$ with $fg=fh$ if and only if $kr=ks$. 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 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.

David Roberts: I’m interested in a bicategorical version of this. You haven’t happened to have done this Mike?

Mike Shulman: Not more than can be extracted from 2-congruence and regular 2-category. What is there called an “eso” is the bicategorical version of a strong epi (which agrees with an extremal epi in the presence of pullbacks), and what is there called “the quotient of a 2-congruence” is the bicategorical version of a regular epi. I’ve never thought about the bicategorical version of a strict epi; since strict epis agree with regular epis in the presence of finite limits I’ve never really had occasion to care about them independently.

References

• Categories and Sheaves, p115-116.