### Comparative Smootheology

#### Posted by Urs Schreiber

Here in the $n$-Café we happen to talk about the various notions of generalized smooth spaces every now and then (last time starting here).

I was dreaming of having, at one point, a survey of the various definitions and their relations in our non-existent wiki. Luckily, while I was just dreaming, Andrew Stacey did it.

Andrew is an expert on the index theorem for Dirac operators on loop spaces (see his list of research articles), and for that work he needs to deal with generalized smooth structures that render loop space a smooth space.

Last time I visited Nils Baas in Trondheim I had the pleasure of talking quite a bit with Andrew. Ever since then I had planned to post something about the intriguing things about loop space Dirac operators he taught me, but never found the time (but see this comment).

Now recently he sent me a link to his new article, which gives a detailed survey of the various definitions of generalized smooth spaces, and a careful and detailed comparison between them:

A. Stacey
*Comparative Smootheology*

(pdf, arXiv)

Abstract.We compare the different definitions of “the category of smooth objects”.

Of the four or five different definitions he considers, Andrew favors *Frölicher spaces*, and he explains why.

I am reading this in particular with an eye towards our recent discussion in Transgression of $n$-Transport and $n$-Connections, where I am falling in love with a general definition that Andrew does not discuss explicitly: smooth spaces as general presheaves over the site of manifolds or open subsets of $\mathbb{R} \cup \mathbb{R}^2 \cup \mathbb{R}^3 \cup \cdots$.

The set such a presheaf assigns to any object $U$ of the domain category (manifold or open subset of sorts) is to be thought of as the collection of smooth maps from $U$ *into* the smooth space thus defined.

The slogan here is

A generalized smooth space (in the sense of presheaves on manifolds) is a space which need not locally look like a manifold, but which may be

probedby manifolds.

*Chen smooth spaces* and/or *diffeological spaces* are a special case of such presheaves, namely presheaves which are *quasi-representable*: while not in general representable, these are presheaves $X$ for which there exists a *set* $X_s$ such that for $U$ any test domain, we have $X(U) \subset \mathrm{Hom}_{Set}(U,X_s)$. Moreover, morphisms of diffeological or Chen-smooth spaces are morphisms of presheaves $X \to Y$ induced by maps $X_s \to X_y$ of these sets.

On the other hand, Frölicher spaces and “differentiable spaces”, as in Mostow’s article, are defined not just by specified smooth maps *into* them, but also by specified smooth maps *out of* them.

This is extremely useful for instance for having such a standard concept as the *chain rule* available for generalized smooth spaces. The chain rule for mere presheaves, as above, is, in contrast, a headache.

Moreover, the slick thing about Frölicher’s definition is that he realized that with maps in and out, it is already sufficient to consider all maps from just $\mathbb{R}^1$ into the generalized smooth space. Hence a Frölicher smooth space is a set together with a specified collection of *smooth curves* in it, and a specified collection of smooth functions *on* it, satisfying some compatibility conditions.

I like that. But personally I haven’t quite made up my mind yet.

I am thinking that maybe this is telling us that we eventually may want to consider things that are *pairs* consisting of a presheaf *and* a co-presheaf on manifolds, compatible in some way.

Notice that for presheaves on manifolds it is natural to define all *contra*variant differential geometric constructions, notably differential forms.

While for co-presheaves on manifolds, it is natural to define all *co*variant differential geometric constructions, notably vector fields.

Then recall one of the points emphasized in Transgression of $n$-transport and $n$-connections: *every* *non-negatively* differential graded commutative algebra sits inside the dg-algebra of differential forms on *some* presheaf-like generalized smooth space.

And then recall from On BV-quantization, Part VIII that we are really looking for a way to generalize this statement to dg-algebras with *no* restriction on the grading.

So possibly one way to realize this is to consider things that are both presheaves *and* co-presheaves on manifolds. “Frölicher presheaves”, in a way.

Hm…

## Re: Comparative Smootheology

I didn’t put in the “presheaves” version because I didn’t know of it. If someone sends me a precise definition then I’ll add it in (providing I understand it, of course!). Now that, thanks to Urs, I have a copy of Mostow’s paper I’ll add a section on that (again, subject to the proviso of comprehension).

Urs wrote (and I emphasised):

It’s the compatibility relationship that leads one to Frölicher spaces. I suspect that you would end up with a non-set-based version of Frölicher spaces which I don’t think anyone would have any qualms about.

As I see it, Frölicher’s main insight was this compatibility relation. The fact that he only used curves and functionals (rather than maps into or out of more general spaces) is more of a technicality. Due to useful results such as Boman’s, one can always reduce a more complicated definition to one using only curves and functionals without losing any information (just “bloat” as I refer to it).

The compatibility relationship can be thought of as “If it looks like a duck and quacks like a duck, then it is a duck.”. Without wishing to start a flame war, the other approaches are more in the line of “It is only a duck if it appears in my list of ‘approved ducks’.”.

Andrew

PS And without wishing to start a subthread on a completely irrelevant issue, a chemist I know (rather well) told me of the following version: “If it looks like a duck and quacks like a duck but doesn’t have the NMR of a duck, it ain’t a duck.”

In categorical terms, that would translate to: “If it looks like a duck and quacks like a duck but doesn’t transform like a duck, it ain’t a duck.” though, of course, it may be a 2-duck.