Is composition of regular epimorphisms always regular? In a finitely complete and cocomplete category. Does it always hold that the composition of two regular epimorphisms is regular? And if it's not the case, what kind of additional constraints can make it true (say, a pre-abelian category)?
What I already knew is it holds for categories where regular epimorphisms and strong epimorphisms conincide.
 A: My standard example of a category where regular epimorphisms are not closed under composition is the category $\mathbf{Cat}$ of small categories.
Let $\mathbb{2}=\{0\to 1\}$ be the category with two objects and one non-identity morphism between them, and let $F:\mathbb{2}\to\mathbb{N}$ be the functor sending this morphism to $1$, where $\mathbb{N}$ is the additive monoid of natural numbers, viewed as 1-object category.
Let $G:\mathbb{N}\to\mathbb{Z}$ be the inclusion of additive monoids, viewed as functor between the associated 1-object categories, and let $H: \mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}$ be the quotient map, again viewed as functor between 1-object categories.
Then $F$ and $H\circ G$ are regular epis in $\mathbf{Cat}$, but $H\circ G\circ F$ is not.
A: In a category with all kernel pairs and coequalisers of kernel pairs, the following conditions are equivalent:


*

*regular epimorphisms are stable under composition;

*regular epimorphisms coincide with strong epimorphisms;

*for any morphism $f$, if $m_f \circ e_f$ is its factorisation through the coequaliser of its kernel pair, $m_f$ is a monomorphism;

*regular epimorphisms and monomorphisms form a factorisation system.


This is proved in Monomorphisms, Epimorphisms, and Pull-backs by Kelly (Propositions 2.7 and 3.8).  Note: Kelly takes as definition of regular epimorphism what is called elsewhere strict epimorphism, but these notions coincide when kernel pairs exist.  (And Kelly doesn’t give the “factorisation system” version.)
Kelly gives an example of a pre-abelian category in which regular monomorphisms do not compose: the category of abelian groups with no elements of order 4 (last paragraph of p. 126).  The dual category is a pre-abelian category in which regular epimorphisms do not compose.
