Given a cat $\bf C$, the class $\mathcal{S}$ of all $\mathbf{C}$-morphisms that are neither left nor right invertible, generates a "genuine" subsemicat $\bf S$ of $\bf C$ (if necessary, see here for the relevant terminology), by which I mean that $\bf S$ is not, in general, a cat in its own right (even though, of course, it can in some cases). To my eyes, this gives another interesting example of a kind of "natural" structures encountered in the everyday practice which, on the other hand, fail to be categories in any (apparent) "natural" way. Then, I'd like to mention it at some point in my current work, all the more that it looks like "the" appropriate way to go for introducing other (and possibly more significant) notions like that of irreducible arrow (say, in the sense of Auslander and Reiten) or almost irreducible arrow (say, in the sense of Margolis and Steinberg). Note that similar considerations can be repeated for the class $\mathcal P$ of all $\bf C$-morphisms that are neither left nor right cancellative (which in turn don't ever generate a category). Then my questions are:
Q1. Is there any standard name for the class $\mathcal{S}$ and its members? Q2. Is there any standard name for the class $\mathcal{P}$ and its members? Q3. Is anybody aware of any paper, book, or whatever else taking a similar point of view for laying out (the rudiments of) an abstract theory of factorizations subsuming aspects of the factorization theory, say, of monoids (as presented, for instance, by Geroldinger and Halter-Kock in the first chapters of their book, though only in the commutative case)?
For the record, I'm referring to the elements of $\mathcal S$ as singular arrows and to the elements of $\mathcal P$ as promiscuous arrows, but I'm not very happy with either of these...
