I'm reviewing some notes of mine on regular categories to try and get a better feel for regular, strong, and extremal epimorphisms, and this leads me to ask: is a category $\mathsf C$ regular if and only if $(\mathrm{ExtEpi},\mathrm{Mono})$ is a strong/orthogonal factorization system?


If anyone cares, here's my motivation.


In the category of sets, an image factorization of a function $f:A\to B$ is a composite $A\twoheadrightarrow \operatorname{Im}f\rightarrowtail B$ where $\operatorname{Im}f$ is in bijection with the set of fibers of $f$, the left arrow collapses fibers, and the right arrow takes a fiber to the value of $f$ on it. Thus image factorizations can be constructed functorially by taking for each function the coequalizer of its kernel pair - this collapses the fibers. Travelling further down this path leads to regular epis and the classical approach to regular categories.


Instead, we can define the image of an arrow $f$ (when it exists) to be the smallest subobject of its codomain through which $f$ factors. Often enough the first arrow in the factorization is an epi, so we define an *extremal epi* to be an *arrow* which doesn't factor through any proper subobjects of its codomain. Using this as a primitive notion, we can almost *define* the image of an arrow $f$ to be a subobject of its codomain which factors $f$ through an extremal epi. The only obstruction is the fact extremal epis are in general red herrings - they may not be epimorphisms. However, if $\mathsf C$ has equalizers, they are, and all is well.


Thus, in order for a category with equalizers $\mathsf C$ to have "image factorizations", it seems all we want is for extremal epimorphisms and monomorphisms to constitute a strong factorization system. The [nlab](https://ncatlab.org/nlab/show/regular+category) writes:



> 
> A regular category is a finitely complete category which admits a good notion of image factorization
> 
> 
> 


so tentatively I'm guessing that regular categories are precisely the finitely complete categories in which $(\mathrm{ExtEpi},\mathrm{Mono})$ is a strong/orthogonal factorization system. Is this correct?


**Added.** Just spilling some more thoughts. In general, if $\mathsf C$ admits intersections and pullbacks of monomorphisms, the prefactorization system $(\mathrm{StrEpi},\mathrm{Mono})$ is a factorization system. Moreover, in complete generality, strong epi implies extremal epi. Thus, if $(\mathrm{StrEpi},\mathrm{Mono})$ is already a strong factorization system, asking for $(\mathrm{StrEpi},\mathrm{Mono})$ to also be a strong factorization system amounts to asking every extremal epi to also be strong. What I'm having trouble seeing is how the notion of regular epi pops out here...



Is a category regular iff extremal epis and monomorphisms are a strong factorization system?
It is an old result of Joyal that a category is regular if and only if it is finitely complete and if the classes (strong epi, mono) form a **stable** (orthogonal) factorisation system, that is, if every morphism can be factorised as a strong epi followed by a mono, and the pullback of a strong epi is a strong epi. For one can show that under these hypotheses, every strong epi is the coequaliser of its kernel pair, and hence a regular epi. For proofs, see 



> 
> Kelly, G. M. A note on relations relative to a factorization system. Category theory (Como, 1990), 249--261, Lecture Notes in Math., 1488, Springer, Berlin, 1991. 
> 
> 
> 


or Proposition A1.3.4 of Johnstone's *Sketches of an elephant*.


Also note that in a category with finite limits, the classes of strong epis and extremal epis coincide. A good reference for such matters is



> 
> Kelly, G. M. Monomorphisms, epimorphisms, and pull-backs. J. Austral. Math. Soc. 9 1969 124--142.
> 
> 
> 


