What abstract nonsense is necessary to say the word "submersion"?

Question

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.

Answer 1

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.

Answer 2

The definition that you "recall" in paragraph 2: is this really standard?

In the category of differentiable manifolds a submersion is a map $f:M\to N$ which when differentiated at any point $x$ in its domain yields a surjective map $T_xM\to T_{f(x)}N$ of tangent vector spaces. It is a corollary of the Inverse Function Theorem that this is equivalent to saying that with respect to suitable coordinate charts $f$ looks locally like projection from $\mathbb R^m$ to $\mathbb R^n$. The latter condition could be taken as a definition; it's a matter of taste. But I believe that the latter condition for topological charts is what people in the subject of topological manifolds would call a topological submersion. That's stronger than what you said.

Answer 3

One characterisation of submersions that is perhaps too general is that they are the largest class of maps which admits local sections over the pretopology of open sets, and of which all pullbacks exist. Working in a site which actually has all pullbacks, then this class of maps is all maps which admits local sections, and is a sort of 'saturation': given composable $f:x\to y,\ g:y\to z$ if $g\circ f$ admits local sections then $g$ admits local sections.

However this misses the idea about sections through every point. Of course, for any concrete category we can talk about sections through every point in the domain, but I'm guessing this is not what you want.

Answer 4

I believe the definition should be as follows:

Let $C$ be a Grothendieck site. Suppose $f:c \to d$ in $C$. Let $a$ be a cover $\left(a_i:d_i \to d\right)$ of $d$. Let $Sec(f)_a$ denote the set of maps $\sigma_i:d_i \to c$ is a map such that $f \circ \sigma_i=a_i$. Say that $f$ is a submersion if the set $\underset{a} \cup Sec(f)_a$, that is all maps $\sigma_i:d_i \to d$, ranging over all covers of $d$, is a cover of $c$.

It is easy to check that this gives the same definition you gave for manifolds.

EDIT: I made a mistake. What I actually should say is:

There exists a cover of $c$ such that each $\sigma_j$ is in its associated sieve.

Answer 5

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.