This question is closely related to these two, 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.
Best Answer
Dear Theo, I think that you're oversimplifying things a bit too much here. The notion of a submersion depends very much on an "admissibility structure" in the sense of Lurie, or a "geometric context" in the sense of Toën-Vezzosi. That is, in addition to a Grothendieck topology, you also need a "geometry" satisfying certain properties to give further structure to your category.
I was confused a while ago about a similar point, and after learning more about the subject, I realized that it's not nearly so simple as I had hoped.
For instance, the proper notion of a submersion in the algebro-geometric context is a smooth map, but there is no notion of a smooth map between sheaves on the affine étale site before first discussing what it means for a morphism to be "relatively representable". You may want to check out Toën-Vezzosi's paper "Homotopical algebraic geometry II", where they give an inductive definition of an n-geometric algebraic stack (where stack here means simplicial sheaf) (and this inductive definition holds true for sheaves of sets as well). For a map between schemes to be smooth (resp. a submersion) you need for the map to be "relatively representable" by a "scheme" (resp. a "manifold") (when you restrict to the case of sheaves of sets, $n$ really only varies between $-1$, $0$, and $1$).
This may all sound like gibberish, but if you take a look at Toën-Vezzosi's HAG II chapter 2 and ignore the homotopical stuff, the basic idea should be clear. The moral of the story is that a Grothendieck topology alone cannot characterize the geometry of the sheaves on that site.