I redid the definitions and terminology in these sections, keeping what Eduardo wrote and adding more.

]]>I’ve added this to both split epimorphism and split monomorphism.

]]>Oh thanks, I like the argument!

]]>Any absolute epimorphism is split. If $e\colon X \to Y$ is absolute epi, then it is preserved by $Hom(Y,-)$ and so $Hom(Y,X)\to Hom(Y,Y)$ is surjective, hence $1_Y$ is in the image of some $s\colon Y\to X$, which must be a section of $e$. Cf. also absolute coequalizer.

]]>New entry universal epimorphism redirectinig also universal monomorphism. It is not among those variants listed in epimorphism. We also do not list **absolute epimorphism** (epimorphism which stays epimorphism after applying any functor to it). Every split epimorphism stays split after applying a functor hence it is absolute, but is there a counterexample of an absolute epimorphism which is not in fact split ?

By the way, here is an archived version of the old query from strict epimorphism

]]>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 (michaelshulman) and regular 2-category (michaelshulman). 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.