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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.

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I've updated my answer Theo. Please take a look! –  David Carchedi Nov 8 '10 at 0:23
    
Theo, I tend to think that regular monos in manifolds are probably closed embeddings. A regular mono is an equalizer, and if we're talking about the underlying spaces being Hausdorff, then regular monos are necessarily closed inclusions. Whether that's sufficient I haven't convinced myself of, but I'm interested in this. –  Todd Trimble Nov 8 '10 at 0:41
    
@Todd: oh, good point. I was just going from some "regular monos are trying to be embeddings" claim on nLab, and clearly didn't check anything myself. In any case, one should expect the word "submersion" to need more than just the category; it should need some topology. –  Theo Johnson-Freyd Nov 8 '10 at 19:35
    
This is not really much of an answer, but in an answer to a question by Harry Gindi, I find this reference: ens.math.univ-montp2.fr/~toen/m2.html - Toen's course on stacks. In Cours 2 there is a section on what he calls 'geometric contexts', which may be what you are looking for. –  David Roberts Nov 14 '10 at 10:31
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5 Answers

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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.

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Awesome: I will certainly check out those sources. I have precisely no exposure to the words "admissible structure" nor "geometric context", so it's answers like this that I'm particular interested in. The short version of your answer to my question seems to be "lots of abstract nonsense is required". –  Theo Johnson-Freyd Nov 7 '10 at 23:38
    
Yes, that is exactly right. –  Harry Gindi Nov 7 '10 at 23:45
    
I would also suggest taking a look at Lurie's DAG V, but I must warn you, it is very focused on $\infty$-categorical versions of this stuff. –  Harry Gindi Nov 8 '10 at 0:20
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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.

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Hi Tom, I've seen both versions in the literature. But if you are not working with topological manifolds, but spaces, then the version that Theo mentions is probably more useful, because locally Euclidean is far too strong. –  David Roberts Nov 7 '10 at 22:56
    
Notice that if $f$ is a submersion in the sense Theo defines, $dg_x$ supplies a right-inverse for $df_x$, so it follows that $df_x$ is surjective. The converse is obvious, so the notions are equivalent. –  David Carchedi Nov 7 '10 at 23:21
    
As you say, there is another, much stronger condition that is also called "submersion"; the nLab page lists both. The condition I want is the one I glossed. I have also seen "looks locally like a projection" without requiring that you're projecting along something that's locally $\mathbb R^{m-n}$; this is still stronger. Then again, if you have an abstract-nonsense way of defining this stronger notion of submersion, please do tell! –  Theo Johnson-Freyd Nov 7 '10 at 23:26
    
@Theo: Can't you just say that there exists a cover of the form $(g_i:V_i \times W_i \to C)$ and a cover $(\tilde g_i:V_i \to d)$ such that $f \circ g_i=\tilde g_i \circ pr_1$? –  David Carchedi Nov 8 '10 at 0:59
    
@David: yes, in the smooth case the condition on df is also equivalent to the condition Theo is concerned with. So there are two different generalizations to the topological case. I just hadn't seen the word "submersion" used for this particular generalization. –  Tom Goodwillie Nov 8 '10 at 1:21
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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.

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Well, I'm trying to get away from requiring my category to be concrete. For example, there are many categories which are "concrete" in the sense that there is some object X so that Hom(X,-) is faithful, but for which Hom(1,-) is not faithful, where 1 is the terminal object; I tend to think that the latter is the correct definition of "point", not the former. (E.g. if your concretization of Man is Hom(R,-), then I think you don't get sections through every point.) How abstract-nonsensy can you say "admits local sections over the pretopology of open sets"? –  Theo Johnson-Freyd Nov 7 '10 at 23:34
    
local sections is easy, it is local sections through every point that is hard. Seeing your other question on right principal bibundles between internal categories, I can guess the problem you've run up against, as I did recently. I don't think I resolved it yet (or perhaps I did, I'll have to look at my notes later tonight. –  David Roberts Nov 7 '10 at 23:53
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You can talk about a formal class of maps which plays the role of submersions relative to a Grothendieck pretopology - I use this in my thesis (and it is also called an admissible class, independently on that in Harry's answer) to talk about internal anafunctors. But I agree with him that it is a bit more complicated that it first seems. –  David Roberts Nov 7 '10 at 23:56
    
@Theo: I should add that we had a discussion of this on the nForum that may be instructive to read if you don't want to get into the nitty-gritty technical details. –  Harry Gindi Nov 8 '10 at 0:10
    
@Harry: Cool, where? I didn't find it by searching. –  Theo Johnson-Freyd Nov 8 '10 at 0:45
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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.

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This definition doesn't make sense as stated. $\bigcup_a Sec(f)_a$ is a set of maps with codomain $c$, so can't be a covering family of $d$. Did you mean, "that is all maps $\sigma_i:d_i\to c$, ranging over all covers of $d$, is a cover of $c$"? And even that way it doesn't make sense, because images of local sections are not open in the case that $C=Diff$ with the open cover pretopology. Or do I misunderstand you? –  David Roberts Nov 7 '10 at 23:15
    
Your description is something close to "at every point in $d$ there is a local section", whereas the correct notion of "submersion" is "through every point in $c$ there is a local section". So if I understand correctly, you're generalizing an important notion, but not the one I want. –  Theo Johnson-Freyd Nov 7 '10 at 23:30
    
@David: As for your first comment- yes, I made a typo, and I fixed. As for your second comment, you're totally right. (I've been playing with etale maps too much lately). I think I've fixed things now. –  David Carchedi Nov 8 '10 at 0:13
    
Calling them étale maps is a very dangerous proposition and is very likely to be misleading in this context. Johnstone's term "local homeomorphism of topoi" is much more on-point. –  Harry Gindi Nov 8 '10 at 0:16
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How does $\{\sigma_i\}$ define a sieve? A map $c'\to c$ is in the sieve if it factors through some $\sigma_i$? This can't be right, if "covers" of manifolds are required to be the same dimension. But otherwise, I think your definition is something like "$c$ has a cover $\{h_j:c_j\to c\}$ so that each $h_j$ factors as $\sigma f h_j$ for some $\sigma:d_i\to c$ for some member of a covering family $d_i\to d$." This probably can be simplified. –  Theo Johnson-Freyd Nov 8 '10 at 0:36
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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.

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@Wouter: I agree that having something like a tangent bundle would suffice, but it is clearly not necessary: Topological spaces do not have tangent bundles, and I still have a notion of "topological submersion" given by "existence of local splittings". See also the discussion in the comments after Tom Goodwillie's answer. –  Theo Johnson-Freyd Nov 12 '10 at 18:58
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