This question is closely related to thesethese twotwo, but the former doesn't go far enough and the latter didn't attract much attention, and anyway I want to ask the question slightly differently.
Recall that in the category of Topological spaces or in the category of Manifolds, a submersion is a (not necessarily surjective!) map $f: X \to Y$ so that for each point $x\in X$, there exists open neighborhood $f(x) \in U \subseteq Y$ and a map $g: U \to X$ splitting $f$, i.e. $f \circ g = \operatorname{id}_U$. This definition does not generalize well to other categories: it requires at least that "points" know a lot about the objects, and that we know what are "open neighborhoods".
My question is: How much extra "abstract nonsense" structure do I need to put on a category for it to have a good theory of submersions?
On the one hand, the surjective submersions of manifolds are all regular epimorphisms (does this characterize the surjective submersions?), and so I could imagine defining "submersion" to mean a map that factors as a regular epi and a regular mono (I think that the regular monos in manifolds are the open embeddings?). Then it seems that I don't need any extra structure, but I have not checked that this conditions characterizes submersions.
On the other hand, (surjective?) submersions form a Grothendieck pretopology, and hence determine a Grothendieck topology. Conversely, I would have assumed that a Grothendieck topology (which is extra structure on a category) determines which maps are submersions, although I am sufficiently new to this that I don't have a proposal for such a definition.
