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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeMay 15th 2011

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By the way, here is an archived version of the old query from strict epimorphism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

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeMay 17th 2011

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.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeMay 17th 2011

Oh thanks, I like the argument!

• CommentRowNumber4.
• CommentAuthoreparejatobes
• CommentTimeMar 6th 2012

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

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeMar 7th 2012

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