The n-Category Café

I like the word ‘smooth’ better than ‘diffeological’ because it’s one syllable instead of three, English instead of Greek, and generally less scary. However, I’m not sure what the best category of smooth spaces is — or if there even is one. In addition to the Chen approach being used here (equipping a space with maps from nice test spaces), there’s also the Mostow approach (equipping a space with a sheaf of maps from it to nice test spaces). There are also approaches that combine the two ideas.

If there turns out to be a best approach, ultimately I would like the term ‘smooth spaces’ to be used for that.

I haven’t read Paul Feit’s paper, but smooth spaces as Chen defined them are really just special sheaves on the category of convex sets equipped with a certain Grothendieck topology. Sheaves are very much a ‘local-to-global’ kind of construction.

Why are these sheaves ‘special’? Because of axiom 3, which says they have enough global points to make the global points functor faithful. This axiom is thrown in to make sure that smooth spaces act like sets with extra structure! Keeping it is probably just a sign of my reluctance to fully accept the tao of mathematics. If I dropped it — and also dropped the insistence that a smooth space ‘be a set’ — we’d have a better-behaved category of smooth spaces, namely a topos. As it is, we just have a ‘concrete quasitopos’ — or so I’ve been told.

Michael wrote:

I actually like diffeological terminology better than “smooth space”, but regardless of what one calls the notion, I hope your notes increase general awareness of it.

Alas, the definition of diffeological space used by Patrick Iglesias-Zemmour is a bit different from K. T. Chen’s 1986 definition of ‘differentiable space’, which is what I call a ‘smooth space’. And, this difference of definitions seems to make it harder to treat manifolds with boundary using diffeological spaces.

Chen’s 1986 definition allows the domain $C$ of a plot $\phi: C \to X$ to be any convex set of Euclidean space. So, we can take $C$ to be a half-space:

$$\{(x_1,\dots,x_n) \in \mathbb{R}^n : x_1 \ge 0 \}$$

Plots where $C$ is a half-space are useful for manifolds with boundary.

Iglesias-Zemmour instead uses a definition that requires $C$ be an open subset of Euclidean space. This rules out half-spaces.

This makes a correct treatment of manifolds with boundary a bit tricky. In fact, I’m not sure it’s possible. I’m very curious.

Iglesias-Zemmour says in the second sentence of his book that the definition of ‘diffeological space’ is equivalent to Chen’s definition. However, he cites Chen’s 1977 paper, which actually gives a different definition than the 1986 paper! I think Chen changed his definition for a good reason.

I believe the issue boils down to this. Suppose

$$f: [0,1] \to \mathbb{R}$$

is a function such that for every smooth function

$$\phi: \mathbb{R} \to [0,1]$$

the composite $f \circ \phi$ is smooth. Is $f$ smooth?

This is obvious away from the points $0$ and $1$. But is it true at $0$ and $1$? More precisely, does $f$ have continuous one-sided derivatives of all orders at $0$ and $1$?

Note the derivative $\phi^'(x)$ vanishes if $\phi(x)$ is $0$ or $1$. That’s what makes the question annoyingly tricky. If we were allowed to use plots defined on convex sets, like the identity function

$$\phi: [0,1] \to [0,1],$$

the question would be easy — in fact trivial!

I should figure this out, but I’m feeling lazy. Maybe someone else would enjoy taking a crack at it.

A smooth point is isomorphic to another smooth point if and only if both have the same collection of plots on them.

But can we have two smooth points with differing collection of plots? I’d think not, since by one of the axioms, every constant map is a plot and all maps to the point are constant, so all maps to the point are necessarily plots.

Maybe if one were to relax that axiom, things would be different.

But I don’t think that the point of diffeology is to work like synthetic differential geometry does. Instead of providing us with a concrete notion of ‘infinitesimal’ it rather provides us with a way to rephrase every potential question about infinitesimals in terms of ordinary differential calculus.

Urs is right; there’s only one smooth structure on a point in this setup. With this smooth structure, every function to or from a point is smooth.

As Urs notes, this uses the 3rd axiom in the definition of ‘smooth space’. This the axiom that makes smooth spaces be merely sets equipped with extra structure. In other words, this axiom guarantees that the forgetful functor from smooth spaces to sets is faithful.

This is deliberately different from algebraic geometry or synthetic differential geometry, where we have many nonisomorphic spaces that have only one point. (Or more precisely, one ‘global point’.)

The difference is that we are not trying to give our smooth spaces ‘infinitesimal fuzz’… if you know what I mean.

It’s possible that in the long run this will be regarded as a mistake, but it’s definitely worth exploring.

Thanks, Urs and John. What then are the advantages of Chen’s point of view to that of synthetic differential geometry?

Also, suppose we look at unions instead of intersections in my original question. Is the union $X\cup P$ of the $x$-axis and the parabola isomorphic to the union $X\cup Y$ of the $x$-axis and the $y$-axis?

The difference is that we are not trying to give our smooth spaces ‘infinitesimal fuzz’… if you know what I mean.

It’s possible that in the long run this will be regarded as a mistake, but it’s definitely worth exploring.

Currently I tend to think that it is a virtue.

As you know, for quite a while, under the influence of Kock/Breen-Messing, I was trying to do what we need to in synthetic language (this or that).

Probably it’s all just a matter of taste, but currently I have come back to the point where I tend to like the less fancy smooth world better.

For the reason encoded in the following slogan:

infinitesimal is the same as smooth and functorial.

It seems to me that, when properly combined with categorical notions, Chen-like smoothnes neatly does for us all that we would like to hope for.

As an example: one of the motivations for using synthetic reasoning (and also super-reasoning, by the way) is that we want to be able to say that $$T$$ is the “infinitesimal interval”, that $$\mathrm{Hom}(T,X) \simeq T X$$ is the tangent space to $X$ and that $$\mathrm{Hom}(T X,\mathbb{R}) \simeq \Omega^1(X)$$ is the space of 1-forms on $X$.

The axioms of synthetic differential geometry are geared such that this works.

But consider this: the “infinitesimal interval” should be a special case of a path between two points.

But a path between two points is begging to be regarded not as an object, but as a morphism.

So I’d do this: I take the ordinary point $\{\bullet\}$. Regarded as a 0-category, this is just a point. But now regard it as a 1-category (with only the identity morphism on its single object).

What’s the difference? The difference is of course that now maps from the point into something may have paths between them. And in a moment we will see that, albeit all these paths are finite, only their infinitesimal part really matters.

So, I set $$\mathrm{Hom}(\{\bullet\},X) := \mathrm{Hom}_{\mathrm{Cat}}( \{\bullet\},P(X)) \,,$$ where $P(X)$ is the strict path $\omega$-groupoid of $X$.

Then this “space” of maps is $$\mathrm{Hom}(\{\bullet\},X) \simeq P_1(X)$$ the path 1-groupoid of $X$.

We know, indeed, that this behaves like the tangent bundle $T X$ to $X$ in the following sense:

smooth morphisms $$\mathrm{Hom}_{\mathrm{Cat}_{C^\infty}} (P_1(X), \Sigma \mathbb{R})$$ are in bijection with ordinary 1-forms on $X$ $$\mathrm{Hom}_{\mathrm{Cat}_{C^\infty}} (P_1(X), \Sigma \mathbb{R}) \simeq \Omega^1(X) \,.$$

1-Forms appear here automatically, even though I never used infinitesimals. Everything is finite.

The magic happens when we combine smoothness with functoriality.

Functoriality implies that maps on paths are determined by maps on sub-paths, which are determined by maps of sub-paths, etc. Magically, if the functor is smooth, this automatically knows about “infinitesimal subpaths”, in the sense that we have the theorem that $\mathrm{Hom}_{\mathrm{Cat}_{C^\infty}} (P_1(X), \Sigma \mathbb{R}) \simeq \Omega^1(X) \,.$

And that’s nice. For one, the theorem goes even further: we may replace $\Sigma \mathbb{R}$ with any Lie groupoid and get “Lie groupoid-valued 1-forms” this way. For single-object Lie groupoids this coincides with ordinary Lie-algebra valued 1-forms.

Also, we might want to modify the domain. The domain here just encodes the structure of connectivity in our space $X$. There are lots of other (smooth) categories which we may want to regard as encoding the information of “points and paths between them”.

Is every subset of $\mathbb{R}^n$ a smooth space? Are nonmeasurable sets smooth spaces? That would disturb me.

A smooth set is a set equipped with an extra structure.

Every set admits at least one (boring) choice of that extra structure: the discrete smooth structure where the only smooth maps from (convex if you like, open if you like) subsets of some $\mathbb{R}^n$ into your set are the constant maps. (see also slide number 5 in John’s lecture notes).

So the question is what kind of “interesting” smooth structures a given set admits.

If you have any smooth space, then any of its subsets naturally inherits a smooth structure: that defined in terms of the collection of all plots into the former that factor through the latter.

In this sense every subset of any manifold (every manifold is a smooth space) naturally inherits a smooth structure.

But potentially this will coincide with the boring discrete smooth structure.

For instance the subset $\{0,1\} \subset \mathbb{R}$ of two disjoint points on the line inherits a smooth structure from that of the line, but it is boring.

I’d guess that for generic non-measurable subsets of $\mathbb{R}^n$ the induced smooth structure will turn out to be the boring discrete one, too.

That depends on how many ordinary smooth maps into $\mathbb{R}^n$ factor through your chosen non-measurable subset. I guess generically only the constant smooth maps do.

Walt wrote:

Is every subset of $\mathbb{R}^n$ a smooth space? Are nonmeasurable sets smooth spaces?

Yes.

That would disturb me.

Perhaps this is an argument for calling them ‘diffeological spaces’ or ‘Chen spaces’ instead of ‘smooth spaces’. The point about these spaces is not that they’re smooth sort of like a manifold. The point is that we can recognize when a map between them is smooth.

Suppose $X$ is any smooth space. Even if $S \subseteq \mathbb{R}$ is some horrible nonmeasurable set, we have no trouble recognizing when a function

$$f : X \to S$$

is smooth: it’s smooth precisely when its composite with the inclusion

$$i: S \to \mathbb{R}$$

is smooth. In other words: it’s a smooth real-valued function that just happens to take values in $S$.

Similarly, a function

$$f : S \to X$$

deserves to be called smooth if its composite with every map

$$g: Y \to S$$

is smooth, for all smooth spaces $Y$. Note that here we are using the fact that we can tell when a function to $S$ is smooth, to decide when a function from $S$ is smooth.

Finally, there’s nothing special about subsets $S \subset \mathbb{R}$ in what I just said. The same stuff is true for any subset of any smooth space!

I have heard that one problem with Chen’s spaces is that one gives up the inverse function theorem. Is this correct?

I would think so. In fact, I would not even know how to state the inverse function theorem for smooth spaces, since it is not obvious what the rank of the differential of a smooth map should be.

understanding loop spaces of orbifolds as stacks.

What I would tend to do is:

think of the orbifold as a Lie groupoid (the action groupoid of the orbifold group action) and then consider paths in that Lie groupoid (or loops, if desired).

These would be generated from ordinary paths in the space of objects combined with morphisms of the groupoid itself. There is an obvious Chen-smooth structure on this, where a plot into such orbifold-paths is any map that arises as a collection of plots into paths in the space of objects and into the space of morphisms, postcomposed with the formal composition of these.

Speaking as an ignorant categorist, I’ll hazard some guesses as to what else one might give up. John Baez said that rumor has it that the category of smooth spaces is a “concrete quasitopos”, which certainly sounds right to me, given the strong family resemblance between the axioms for smooth spaces and the axioms for quasitopological spaces, which I think is the ur-example of quasitopos [and even where ‘quasitopos’ gets its name].

In a typical quasitopos, one has a lot of the Girard exactness conditions for a topos, with the notable exception of one, that every epi is a coequalizer (of its kernel pair). This allows one to prove for example that a morphism which is both mono and epi is an isomorphism. This corollary typically fails in a quasitopos. I’m guessing it fails for smooth spaces in particular. So while smooth spaces are sets with extra structure, they do not behave much like sets in some sense.

If there is some advantage to synthetic differential geometry over smooth spaces, it may be just in the fact that you can reason with the objects in a smooth topos “synthetically”, i.e., pretty much as you reason with sets, provided that you do so “constructively”. (But then eventually, you come back down to earth and ask, “just what are these objects, analytically?” – there’s a rub.)

Eugene wrote:

I have heard that one problem with Chen’s spaces is that one gives up the inverse function theorem. Is this correct?

That’s probably true — as Urs points out, it’s not even clear how one would state such a theorem.

Here are two ‘weird’ smooth spaces to expand your intuition:

1. Take any manifold $X$ and let $D$ be any sub-vector-bundle of the tangent bundle of $X$. Say that a map $f: A \to X$ is smooth (where $A$ is any smooth space) iff it’s smooth in the usual sense and $d f$ applied to any tangent vector gives a tangent vector in $D$.
   
   
2. Take any topological space $X$ and say that a map $f: A \to X$ is smooth iff it’s continuous with respect to the natural topology on the smooth space $A$.

By the way: if you don’t feel comfortable enough with smooth spaces to enjoy my use of a general smooth space $A$ in the above definitions, just pretend $A$ is a smooth manifold! Then the ‘natural topology’ is just the one you’re used to.

John sort of addressed my other question, but I still don’t quite understand. If one ‘improves’ the category of manifolds by embedding it into the category of sheaves of sets, why not then consider all sheaves and not just the ones represented by sets?

Because some backwards people still think mathematical gadgets should be sets with extra structure! That’s the main reason so far. Urs and I have certain concrete applications of smooth spaces to problems in mathematical physics. These applications don’t seem to require the more general sort of smooth space that’s not a set. So, why bother? It would only make it harder to market our results.

See, it already takes some work to convince physicists that they should be interested in ‘2-category objects in the category of smooth spaces’. I don’t think it’ll help to say ‘oh yeah, and smooth spaces are just sheaves on a certain site’.

I have ideas about some other reasons for restricting to ‘concrete’ smooth spaces, but I haven’t worked out enough details yet to make a good case.