In a recent MO question I asked about the relation between surjective submersions (in the category of smooth or otherwise manifolds) and maps that admit local sections. The latter, it turns out, are more general than the former, as surjective submersions $f:X\to Y$ admit local sections *through each point of the domain*. This means that for each $y\in Y$ there is an $x\in X$ such that $f(x)=y$ and there is an open neighbourhood $U\ni y$ and a map $\sigma:U\to X$ such that $f\circ \sigma$ is the inclusion of $U$ into $Y$ and $x=\sigma(y)$.

Now there is a class of local-section-admitting maps in $Top$ which have a similar property: every point in the codomain has a local section through it; these are called (surjective) topological submersions. In some respects these are better than just maps which have local sections, because a very complicated map could have very few local sections, none of which pass through regions of interest. Having many local sections seems to force the map to be rather nice. For example, a topological submersion with discrete fibres is a local homeomorphism, whereas a map with local sections with discrete fibres can be pretty wild. Ditto submersions in Diff: such a thing with discrete fibres is a local diffeomorphism.

Now for a general Grothendieck pretopology $J$ on a concrete category $C$ one could ask formally for this sort of map. Define a surjective $C$-submersion to be a map $f:X\to Y$ in $C$ such that for each point $x$ of $X$ there is a covering family $(U_i \to Y)_{i\in I}$ from $J$ such that $x$ is in the image of some family of maps $\sigma_i:U_i \to X$ which are local sections of $f$ (or equivalently, some map $U_i \to X$ for some $i$ which is a local section of $f$).

I can imagine an extension beyond concrete categories, and so here's the question(s) (and I hope it justifies the alg.geom. tag):

(1)Is this sort of setup seen in other categories? (most notably in Schemes (or some subcategory) or a topos)

(2) Since in Schemes (or a subcategory) one has $R$-points for arbitrary (nice enough) rings, one can consider how the collection of local sections changes under extension/restriction of scalars. Is this sort of thing considered?

(3) Do we recover some sort of characterisation of etale maps (or similar) by analogy with the result for Diff- and Top-submersions with discrete fibres?

somethinglike this idea has merit? $\endgroup$3more comments