As will become clear, this is in some sense a follow up on my earlier question Why should I prefer bundles to (surjective) submersions?. As with that one, I hope that it's not too open-ended or discussion-y. If y'all feel it is too discussion-y, I will happily close it.
Let $\rm Man$ be the category of smooth (finite-dimensional) manifolds. I can think of (at least) two natural "smooth structures" on $\rm Man$, which I will outline. My question is whether one of these is the "right" one, or if there is a better one.
I should mention first of all that there many subtly different definitions of "smooth structure" — see e.g. n-Lab: smooth space and n-Lab: generalized smooth space and the many references therein — and I don't know enough to know which to prefer. Moreover, I haven't checked that my proposals match any of those definitions. In any case, the definition of "smooth structure" that I'm happiest with is one where I only have to tell you what all the smooth curves are (and these should satisfy some compatibility condition). So that's what I'll do, but I'm not sure if they do satisfy the compatibility conditions. Without further ado, here are two proposals:
- A smooth curve in $\rm Man$ is a fiber bundle $P \to \mathbb R$.
- A smooth curve in $\rm Man$ is a submersion $Y \to \mathbb R$.
Then given a manifold $M$, we can make it into a category by declaring that it has only identity morphisms. Then I believe that the smooth functors $M \to {\rm Man}$ under definition 1 are precisely the fiber bundles over $M$, whereas in definition 2 they are precisely the submersions over $M$.
(Each of these claims requires checking. In the first case, it's clear that bundles pull back, so all bundles are smooth functors, and so it suffices to check that if a surjective submersion to the disk is trivializable over any curve, then it is trivializable. In the second case, it's clear that if a smooth map restricts to a submersion over each curve, then it is a submersion, so any smooth functor in a submersion, and so one must check that submersions pull back along curves.)
I can see arguments in support of either of these. On the one hand, bundles are cool, so it would be nice if they were simply "smooth functors". On the other hand, we should not ask for smooth functions (i.e. 0-functors) to be necessarily "locally trivializable", as then they'd necessarily be constant. Maybe the correct answer is definition 2, and that bundles are "locally constant smooth functors", or something.
Anyway, thoughts? Or am I missing some other good definition?
Addendum
In the comments, folks have asked for applications, which is very reasonable. The answer is that I would really like to have a good grasp of words like "smooth functor", at least in the special case of "smooth functor to $\rm Man$". Of course, Waldorf and Shreiber have explained these words in certain cases in terms of local gluing data (charts), but I expect that a more universal definition would come directly from a good notion of "smooth structure" on a category directly.
Here's an example. Once we have a smooth structure on $\rm Man$, we can presumably talk about smooth structures on subcategories, like the category of $G$-torsors for $G$ your favorite group. Indeed, for the two definitions above, I think the natural smooth structure on $G\text{-tor}$ coincide: either we want fiber bundles where all the fibers are $G$-torsors, or submersions where all the fibers are $G$-torsors, and in either case we should expect that the $G$ action is smooth. So then we could say something like: "A principle $G$-bundle on $M$ is (i.e. there is a natural equivalence of categories) a smooth functor $M \to G\text{-tor}$", where $M \rightrightarrows M$ is the (smooth) category whose objects are $M$ and with only trivial morphisms. (Any category object internal to $\rm Man$ automatically has a smooth structure.) And if I understood the path groupoid mod thin homotopy $\mathcal P^1(M) \rightrightarrows M$ as a smooth category, then I would hope that the smooth functors $\mathcal P^1(M) \to G\text{-tor}$ would be the same as principle $G$-bundles on $M$ with connection. Functors from the groupoid of paths mod "thick" homotopy should of course be bundles with flat connections. Again, Schreiber and Waldorf have already defined these things categorically, but their definition is reasonably long, because they don't have smooth structures on $\rm Man$ that are strong enough to let them take advantage of general smooth-space yoga.
Here's another example. When I draw a bordism between manifolds, what am I actually drawing? I would like to say that I'm drawing something close to a "smooth map $[0,1] \to \rm Man$". I'm not quite, by my definitions — if you look at the pair of pants, for instance, at the "crotch" it is not a submersion to the interval. So I guess there's at least one more possible definition of "smooth curve in $\rm Man$":
- A smooth curve in $\rm Man$ is a smooth map $X \to \mathbb R$.
But this, I think, won't be as friendly a definition as those above: I bet that it does not satisfy the compatibility axioms that your favorite notion of "smooth space" demands.
