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This is a simple question about terminology and a request for any related references. Specifically, what would you call a functor $F:\mathbf{D}\rightarrow\mathbf{C}$ with the following property?

$(*)$ For all $f:C\rightarrow F(D)$, there exists epic $e:C\rightarrow F(D')$ and monic $m:D'\rightarrow D$ such that $f=F(m)\circ e$, i.e. the following diagram commutes.

$\begin{array}[c]{ccc} C&\stackrel{e}{\rightarrow}& F(D')\\ &\stackrel{\searrow}{\scriptstyle{f}}&\downarrow\scriptstyle{F(m)}\\ && F(D) \end{array}$

Have these kind of functors been examined before somewhere in the literature?

As an example, you can of course consider the identity functor on any category where the epis and monos yield a factorization system in the usual sense. A slightly more interesting example, more in line with what I had in mind, is the following.

Let $\mathbf{C}$ be the category of C*-algebras with the usual algebraic homomorphisms as morphisms, and let $\oplus:\mathbf{C}\times\mathbf{C}\rightarrow\mathbf{C}$ be the usual functor taking pairs of C*-algebras (and their morphisms) to their direct sum (which is also the categorical product in $\mathbf{C}$). As it stands, $\oplus$ does not have property $(*)$ on $\mathbf{C}$, but it does when we restrict to the wide subcategory $\mathbf{HC}$ of $\mathbf{C}$ (i.e. the object class of $\mathbf{HC}$ still consists of all C*-algebras) where a morphism $\pi:C\rightarrow B$ in $\mathbf{C}$ is in $\mathbf{HC}$ if and only if $\pi[C]$ is a hereditary C*-subalgebra of $B$, i.e. for all $c,c'\in C$ and $b\in B$ we have $\pi(c)b\pi(c')=\pi(a)$, for some $a\in C$ (note, however, that $B\oplus C$ is no longer a categorical product of $B$ and $C$ in $\mathbf{HC}$).

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  • $\begingroup$ There are simpler categories where (*) holds, for example $\bf Set$ with its cartesian product bifunctor $\times\colon \bf Set\times Set\to Set$: just factor $a\colon A\to X\times Y$ wrt the canonical (Epi, Mono) FS. I think that reasonable hypotheses on a cartesian category $\bf C$ with the (Extremal Epi, Mono) FS should give a similar result. $\endgroup$
    – fosco
    Commented Dec 27, 2013 at 18:56
  • $\begingroup$ I don't think $(*)$ holds for $\times:\mathbf{Set}\times\mathbf{Set}\rightarrow\mathbf{Set}$ - consider the function $f:\{0,1\}\rightarrow\{0,1\}\times\{0,1\}$ defined by $f(0)=(0,0)$ and $f(1)=(1,1)$. Then $f$ is not epic and does not factor as $(m_1\times m_2)\circ e$ for any monics $m_1$ and $m_2$ and epic $e$, the point being that the monic part has to come from $\mathbf{Set}\times\mathbf{Set}$, not $\mathbf{Set}$. $\endgroup$
    – Tristan Bice
    Commented Dec 27, 2013 at 19:16

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Looks like the first MO question I answer is going to be my own. I eventually found factorization structures for functors discussed in $\S$17 of "Abstract and Concrete Categories: The Joy of Cats", except that there they require factorization for arbitrary sources, i.e. collections of (F-structured) arrows with a common domain, rather than single arrows as in my question. And the distinction is important - the monos in the category $\mathbf{HC}\times\mathbf{HC}$ of my example can not be extended to a conglomerate of sources to form a factorization system with the epis for arbitrary sources (e.g. the source consisting of the two canonical injections $i_1,i_2:\mathbb{C}\rightarrow\mathbb{C}\oplus\mathbb{C}$ can not be factored - you might think to use the diagonal morphism $\delta_\mathbb{C}:\mathbb{C}\rightarrow\mathbb{C}\oplus\mathbb{C}$ in $\mathbf{C}$ but this is not epi and, in fact, not even in $\mathbf{HC}$!). Nonetheless, if you were going to use the same terminology you would simply call the $F$ of my question an $(\mathrm{Epi},\mathrm{Mono})$-functor. If anyone has any more information about these kinds of functors, I would still be very keen to know more about them.

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