# The largest set $\bar{\mathcal{E}}$ of Epis, stable under composition, such that every mono in $\bar{\mathcal{E}}$ is iso

Let $\mathcal A$ be a category and $\bar{\mathcal{E}}$ be the largest subset of all Epis, stable under composition, such that every mono in $\bar{\mathcal{E}}$ is iso.

What is known about $\bar{\mathcal{E}}$?

Obviously every strong epi is in $\bar{\mathcal{E}}$ (assuming these are actually epi). What can be said about the converse, is there a counterexample?

If so, do the elements of $\bar{\mathcal{E}}$ have a name? Are there any other known properties or characterizations in certain kinds of categories?

• This seems vaguely related to the notion of an extremal monomorphism $Mono^\epsilon$, i.e. an $m$ such that whenever $m=kp$ with $p$ an epi, then $p$ is an iso. Finite co/completeness of $\cal A$ entails that $(Epi, Mono^\epsilon)$ is a factorization system on $\cal A$, and sometimes $Mono^\epsilon = Mono$; this is the greatest of all proper FS on $\cal A$. – Fosco Dec 10 '16 at 16:58
• Could you elaborate one that "greatest" part? In the presence of finite limits strong epis coincide with extremal ones. – Stefan Perko Dec 10 '16 at 20:17
• Greatest only means that it is the largest :) – Fosco Dec 10 '16 at 21:53
• @FoscoLoregian When is factorization system larger than another? And I also meant: I probably have no clue how to prove that. – Stefan Perko Dec 10 '16 at 23:05
• Since $g\circ f$ mono implies $f$ mono, then whenever $f$ is epic and $g\circ f$ is a a monic epic, $f$ is also a monic epic. Consequently, the class of all epics that are not monic epics without being isos is already closed under composition. Hence, your question is the complement of characterizing monic epics (sometimes called bimorphisms) among epics. – Vladimir Sotirov Dec 10 '16 at 23:45