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I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Chen spaces, Frolicher spaces ...) form much nicer categories of spaces at the expense of having somewhat more convoluted objects. I might be interested in taking up the study of one form of generalized smooth space or another, but my conscience will not let me unless I see that they can actually buy me a more conceptual understanding of regular old manifolds.

So I would like a Big List of theorems about manifolds whose proof can be made significantly shorter or more conceptual by making use of generalized smooth spaces and maps between them. Something like a standard construction in the manifold setting becoming representable in the new setting, and this makes short work of some (previously) complicated theorem.

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Frechet manifolds aren't generalised smooth spaces. They are regular smooth spaces modeled on Frechet TVSs, which for most applications of interest are infinite-dimensional. Perhaps you are thinking of Chen spaces, Frohlicher space, diffeological spaces and so on. –  David Roberts Feb 11 '11 at 20:36
Steven, have you already looked at the book by Moerdijk and Reyes? Because there are a number of synthetic differential geometry applications to classical theorems in differential geometry mentioned in the introduction. –  Todd Trimble Feb 11 '11 at 20:46
@Todd - No I have not looked at the book. Thanks for the reference! I have looked at Kock's synthetic differential geometry book - it concentrates more on working internally in the topos and less on building models it seems. Is the tangent space functor representable in any of the other categories of smooth spaces? For some reason I thought not. I don't know, but I feel for some reason that SDG is in a slightly different world from diffeological spaces, chen spaces, and the like. –  Steven Gubkin Feb 11 '11 at 21:06
This doesn't quite answer your question, but, I think one of the points of generalized smooth spaces is to avoid the troubles of infinite-dimensional manifolds. We have that Frolicher spaces embed fully faithfully into diffeological spaces which in turn embed into Chen spaces. In fact, all infinite dimensional manifolds in the sense of Kriegl and Michor embed fully faithfully into Frolicher spaces. So, you can construct a space and not have to worry about whether or not it is a manifold, if it is, the generalized space you construct will agree with the manifold structure. –  David Carchedi Feb 11 '11 at 21:34
Steven, it's true that those other categories are concrete categories for which the underlying functor to sets is faithful, and for this reason the tangent space functor is not representable in such categories (because it has just one point). It wasn't clear to me that you meant to restrict to such concrete categories. It's true that Moerdijk-Reyes spend some time on model-building (as is clear from the title), but there's also some material on classical applications to connections and sprays, to illustrate the directness of synthetic reasoning. –  Todd Trimble Feb 11 '11 at 21:43
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2 Answers

I think there are some theorems which are easier to prove in the diffeological framework, or as you say: for which the proof reveals more conceptual reasons. For example this one ?

Proposition Let $X$ be a connected diffeological space, let $\omega$ be a closed 2-form on $X$. Let $P_\omega \subset {\bf R}$ be its group of periods. If the group of periods $P_\omega$ is (diffeologically) discrete (that is, is a strict subgroup of $\bf R$) then there exists a family of non-equivalent principal fiber bundles $\pi : Y \to X$, with structure group the torus of periods $T_\omega = {\rm R} / P_\omega$, equipped with a connexion form $\lambda$ of curvature $\omega$. This family is indexed by the extension group ${\rm Ext}({\rm Ab}(\pi_1(X)), P_\omega)$.

This theorem is a generalization of the classical construction of the prequantization bundle of an integral symplectic (or pre-symplectic) manifold, that is the ones for which $P_\omega = a {\bf Z}$. Why such a generalization is interesting? Well, here are some comments:

1) The only condition for the existence of such "integration structures" is that the group of periods is diffeologically discrete, which is hidden in the classical construction by some technical hypothesis (countable at infinity or analog statements).

2) The space $Y$ is a quotient of the space ${\rm Paths}(X)$ on which the form $\omega$ is lifted modulo the action of a "Chain-Homotopy" operator (actually what is built by quotient is a groupoid and the bundle $Y$ is just the "half-groupoid"). So, the diffeological space ${\rm Paths}(X)$ is a master piece of this construction (but almost everywhere in diffeology), and the fact that diffeological spaces support differential forms (in particular ${\rm Paths}(X)$) with the whole tools of Cartan calculus is fundamental.

3) The generality of this theorem involve essentially "irrational tori", since in general the quotient $T_\omega$ is of course not a Lie group.

The last point illustrates why irrational tori are important in diffeology: or you accept these objects or you give up this (kind of) theorems. You may note that such a theorem doesn't exists in the restricted category of Frölicher spaces since irrational tori are trivial there. You may be happy with just the integral case, but in my opinion you miss a lot by not taking the whole generality of the construction, and putting fences where they do not exist.

I may give some other examples where diffeology give a shortcut for known classical theorems, and by the way extend them to objects which do not belong to the category of manifolds.

Here is an example of a more conventional theorem: the homotopic invariance of De Rham cohomology. Differential forms and De Rham cohomology are well defined concept in diffeology, they apply in particular on space of paths of diffeological spaces, spaces of smooth maps, quotients etc.

We use here the Chain-Homotopy operator $$ K : \Omega^p(X) \to \Omega^{p-1}({\rm Paths}(X)) \quad \mbox{which satisfies} \quad K \circ d + d \circ K = \hat 1^* - \hat 0^*, $$ where $\hat 1$ and $\hat 0$ are the maps defined from ${\rm Paths}(X)$ to $X$ by $\hat 1(\gamma) = \gamma(1)$ and $\hat 0(\gamma) = \gamma(0)$.

Proposition Let $X$ and $X'$ be two diffeological spaces, let $f_0$ and $f_1$ be two homotopic smooth maps from $X$ to $X'$, let $\alpha$ be a closed $p$-form on $X'$. The pullbacks $f_0^*(\alpha)$ and $f_1^*(\alpha)$ are cohomologous.

Proof Let $\varphi : X \to {\rm Paths}(X')$ be the map defined by $\varphi(x) = [t \mapsto f_t(x)]$. The pullback by $\varphi$ of the identity $K(d\alpha) + d(K\alpha) = {\hat 1^*}(\alpha) - {\hat 0^*}(\alpha)$ gives $d(\varphi^*(K\alpha)) = f_1^*(\alpha) - f_0^*(\alpha)$. $\square$

This is an example of simplification/generalization of a classical theorem by short-cuting the proof through diffeology. Here also the space of paths of a diffeological space, and the Chain-Homotopy operator, are crucial. May be something more fundamental is hidden behind that. Enxin Wu a Dan Christensen student is working on a possible Quillen model based on diffeology, it will give maybe some lighting on this question?

BTW Frölicher spaces is equivalent to the full subcategory of what we call "reflexive diffeological spaces" (a work in progress with Y. Karshon and al), the ones whose diffeology is completely defined by the real smooth maps. They are the "intersection" of the category {Diffeology} and the category {Sikorski}. There are nice examples and counter-examples which illustrate the difference between these categories.

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Thank you for the answer, and for the great service to mathematics of writing your book! I am not familiar with the classical version of this, so I will have to do some reading before I can say more. This certainly looks promising though! –  Steven Gubkin Feb 12 '11 at 21:53
You welcome :-) –  Patrick I-Z Feb 12 '11 at 22:54
Ah, your second example hits the spot! I will wait for more answers to role in, but that is a very nice example. –  Steven Gubkin Feb 12 '11 at 23:39
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I see that this hasn't garnered an answer yet (though there's several leads in the comments that I endorse following) so I'll post a sort of answer. The following is from the start of Section 2 of Comparative Smootheology (which I - obviously - recommend reading).

... these categories [of generalised smooth spaces] were initially introduced to correct some defect in the category of smooth manifolds. For several of the categories, in particular Chen's and Souriau's, the motivation for the definition was to apply the tools of differential topology to some space that does not quite fit the definition of a (finite dimensional) smooth manifold. Examples of such spaces include loop spaces and diffeomorphism groups. Note that whilst loop spaces can be treated as infinite dimensional manifolds, the closely associated path space (with domain $\mathbb{R}$) cannot be so described. A closely related idea is that of seeing how far it is possible to push a particular concept in differential topology. Smith's category of smooth spaces was introduced to see how far the de Rham theorem can be extended. Another motivation is to apply the tools of another category, for example the category of rings, to smooth manifolds. To do this, one wants to associate a ring to every smooth manifold and characterise those rings that can be obtained in this fashion. Inevitably in this situation one has to balance precision with usability and allow for things that are sufficiently similar to the type of ring that comes from a smooth manifold. This was the motivation for the category of Sikorski.

My favourite motivating example (as hinted in the above) is the path space $C^\infty(\mathbb{R},M)$. There is absolutely no local model space for this whatsoever unless you are prepared to seriously mess with the topology. Nonetheless, as a smooth space it is very well behaved and easy to study since $C^\infty(X,C^\infty(\mathbb{R},M)) \cong C^\infty(X \times \mathbb{R},M)$. Moreover, it is a very natural space to consider when doing all sorts of things in ordinary differential topology.

(Patrick IZ has a nice example of the "irrational torus", but I'll let him say something about that, or you could read it in his book.)

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@Andrew : This doesn't really answer the question, does it? It looked to me like the OP wanted results (about things that people would care about even if they had never heard about generalized smooth spaces) that could be proved using generalized smooth spaces. Being a bit allergic to "generalization for generalization's sake", I'd be very interested in hearing about such results too. –  Andy Putman Feb 12 '11 at 19:33
@Andy: True, which is why I held off for a while before posting. But as no-one posted anything I thought I'd at least contribute something. I actually disagree with the OP's stance: that one studies generalised spaces in order to cast more light on regular manifolds (I think that it does do that, but as a by-product) or even to simplify proofs (again, this can be a by-product). I think that one studies generalised spaces because one has something that requires it. I have a similar allergy to yours, but I have to put up with it because the spaces that I naturally come up against (ctd) –  Andrew Stacey Feb 12 '11 at 20:41
(ctd) require me to treat spaces that aren't manifolds as if they were. Such as loop spaces and path spaces. I don't know of anyone who studies generalised smooth spaces with the aim of simplifying proofs, but everyone does it because they want to extend theorems to new spaces. One can say that by seeing how far a standard theorem extends, one gains insight in to what that theorem actually depends on. So I don't think that the OPs list is going to materialise, whereupon I wanted to at least put something about why one might want to study these spaces. –  Andrew Stacey Feb 12 '11 at 20:45
@Andrew : I think that maybe an answer would be to list some theorems about loop spaces or path spaces that don't involve the notion of "generalized smooth spaces" but which use generalized smooth spaces in their proofs (nb : the downvote wasn't mine -- I was pretty sure that if anyone could motivate these spaces in the way the OP wanted, then you could!). –  Andy Putman Feb 12 '11 at 20:50
+1 Thanks for the answer! A big reason I posted this question was someone on nLab analogizing Hadamard's "The shortest path between two truths in the real domain passes through the complex domain" to "The shortest path between two truths about manifolds passes through a category of generalized smooth spaces" (I can't track down where I read this though...). I know complex analysis is awesome in it's own right, but I am not sure if people would have studied them historically unless they helped to solve real cubic equations, or understand the radii of convergence of real power series. –  Steven Gubkin Feb 12 '11 at 21:11
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