Nice application of generalized smooth spaces

Question

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.

Answer 1

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.

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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?

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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.

Answer 2

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.)