TITLE: What abstract nonsense is necessary to say the word "submersion"?
QUESTION [15 upvotes]: 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.

REPLY [2 votes]: Given two manifolds $M$and $N$ and a differentiable map $f:M\to N$, pull back the tangent bundle of $N$. The derivative arrow $Df: TM \to f^*TN$ is a morphism of vector bundles over $M$ and a regular epimorphism iff $f$ is a submersion. So the extra structure we need is something like the tangent bundle on every object of the category.