So Dan Christensen, who used to be my supervisor while I was a postdoc at the University of Western Ontario, came to Lisbon last week and gave a talk about a topic I remember hearing about while I was there. This is the category of diffeological spaces as a setting for homotopy theory. Just to make things scan more nicely, I’m going to say “smooth space” for “diffeological space” here, although this term is in fact ambiguous (see Andrew Stacey’s “Comparative Smootheology” for lots of details about options). There’s a lot of information about in Patrick Iglesias-Zimmour’s draft-of-a-book.

**Motivation**

The point of the category , initially, is that it extends the category of manifolds while having some nicer properties. Thus, while all manifolds are smooth spaces, there are others, which allow to be closed under various operations. These would include taking limits and colimits: for instance, any subset of a smooth space becomes a smooth space, and any quotient of a smooth space by an equivalence relation is a smooth space. Then too, has exponentials (that is, if and are smooth spaces, so is ).

So, for instance, this is a good context for constructing loop spaces: a manifold is a smooth space, and so is its loop space , the space of all maps of the circle into . This becomes important for talking about things like higher cohomology, gerbes, etc. When starting with the category of manifolds, doing this requires you to go off and define infinite dimensional manifolds before can even be defined. Likewise, the irrational torus is hard to talk about as a manifold: you take a torus, thought of as . Then take a direction in with irrational slope, and identify any two points which are translates of each other in along the direction of this line. The orbit of any point is then dense in the torus, so this is a very nasty space, certainly not a manifold. But it’s a perfectly good smooth space.

Well, these examples motivate the kinds of things these nice categorical properties allow us to do, but wouldn’t deserve to be called a category of “smooth spaces” (Souriau’s original name for them) if they didn’t allow a notion of smooth maps, which is the basis for most of what we do with manifolds: smooth paths, derivatives of curves, vector fields, differential forms, smooth cohomology, smooth bundles, and the rest of the apparatus of differential geometry. As with manifolds, this notion of smooth map ought to get along with the usual notion for in some sense.

**Smooth Spaces**

Thus, a smooth (i.e. diffeological) space consists of:

- A set (of “points”)
- A set (of “plots”) for every n and open such that:

- All constant maps are plots
- If is a plot, and is a smooth map, is a plot
- If is an open cover of , and is a map, whose restrictions are all plots, so is

A smooth map between smooth spaces is one that gets along with all this structure (i.e. the composite with every plot is also a plot).

These conditions mean that smooth maps agree with the usual notion in , and we can glue together smooth spaces to produce new ones. A manifold becomes a smooth space by taking all the usual smooth maps to be plots: it’s a full subcategory (we introduce new objects which aren’t manifolds, but no new morphisms between manifolds). A choice of a set of plots for some space is a “diffeology”: there can, of course, be many different diffeologies on a given space.

So, in particular, diffeologies can encode a little more than the charts of a manifold. Just for one example, a diffeology can have “stop signs”, as Dan put it – points with the property that any smooth map from which passes through them must stop at that point (have derivative zero – or higher derivatives, if you like). Along the same lines, there’s a nonstandard diffeology on itself with the property that any smooth map from this into a manifold must have all derivatives zero at the endpoints. This is a better object for defining smooth fundamental groups: you can concatenate these paths at will and they’re guaranteed to be smooth.

**As a Quasitopos**

An important fact about these smooth spaces is that they are *concrete sheaves* (i.e. sheaves with underlying sets) on the concrete site (i.e. a Grothendieck site where objects have underlying sets) whose objects are the . This implies many nice things about the category . One is that it’s a quasitopos. This is almost the same as a topos (in particular, it has limits, colimits, etc. as described above), but where a topos has a “subobject classifier”, a quasitopos has a weak subobject classifier (which, perhaps confusingly, is “weak” because it only classifies the *strong subobjects*).

So remember that a subobject classifier is an object with a map from the terminal object, so that any monomorphism (subobject) is the pullback of along some map (the classifying map). In the topos of sets, this is just the inclusion of a one-element set into a two-element set : the classifying map for a subset sends everything in (i.e. in the image of the inclusion map) to , and everything else to . (That is, it’s the characteristic function.) So pulling back

Any topos has one of these – in particular the topos of sheaves on the diffeological site has one. But consists of the concrete sheaves, not all sheaves. The subobject classifier of the topos won’t be concrete – but it does have a “concretification”, which turns out to be the weak subobject classifier. The subobjects of a smooth space which it classifies (i.e. for which there’s a classifying map as above) are exactly the subsets *equipped with the subspace diffeology*. (Which is defined in the obvious way: the plots are the plots of which land in ).

We’ll come back to this quasitopos shortly. The main point is that Dan and his graduate student, Enxin Wu, have been trying to define a different kind of structure on . We know it’s good for doing differential geometry. The hope is that it’s also good for doing homotopy theory.

**As a Model Category**

The basic idea here is pretty well supported: naively, one can do a lot of the things done in homotopy theory in : to start with, one can define the “smooth homotopy groups” of a pointed space. It’s a theorem by Dan and Enxin that several possible ways of doing this are equivalent. But, for example, Iglesias-Zimmour defines them inductively, so that is the set of path-components of , and is defined recursively using loop spaces, mentioned above. The point is that this all works in much as for topological spaces.

In particular, there are analogs for the for standard theorems like the long exact sequence of homotopy groups for a bundle. Of course, you have to define “bundle” in – it’s a smooth surjective map , but saying a diffeological bundle is “locally trivial” doesn’t mean “over open neighborhoods”, but “under pullback along any plot”. (Either of these converts a bundle over a whole space into a bundle over part of , where things are easy to define).

Less naively, the kind of category where homotopy theory works is a model category (see also here). So the project Dan and Enxin have been working on is to give this sort of structure. While there are technicalities behind those links, the essential point is that this means you have a closed category (i.e. with all limits and colimits, which does), on which you’ve defined three classes of morphisms: fibrations, cofibrations, and weak equivalences. These are supposed to abstract the properties of maps in the homotopy theory of topological spaces – in that case weak equivalences being maps that induce isomorphisms of homotopy groups, the other two being defined by having some lifting properties (i.e. you can lift a homotopy, such as a path, along a fibration).

So to abstract the situation in , these classes have to satisfy some axioms (including an abstract form of the lifting properties). There are slightly different formulations, but for instance, the “2 of 3″ axiom says that if two of , latex $g$ and are weak equivalences, so is the third. Or, again, there should be a factorization for any morphism into a fibration and an acyclic cofibration (i.e. one which is also a weak equivalence), and also vice versa (that is, moving the adjective “acyclic” to the fibration). Defining some classes of maps isn’t hard, but it tends to be that proving they satisfy all the axioms IS hard.

Supposing you could do it, though, you have things like the homotopy category (where you formally allow all weak equivalences to have inverses), derived functors(which come from a situation where homotopy theory is “modelled” by categories of chain complexes), and various other fairly powerful tools. Doing this in would make it possible to use these things in a setting that supports *differential* geometry. In particular, you’d have a lot of high-powered machinery that you could apply to prove things about manifolds, even though it doesn’t work in the category itself – only in the larger setting .

Dan and Enxin are still working on nailing down some of the proofs, but it appears to be working. Their strategy is based on the principle that, for purposes of homotopy, topological spaces act like simplicial complexes. So they define an affine “simplex”, . These aren’t literally simplexes: they’re affine planes, which we understand as smooth spaces – with the subspace diffeology from . But they behave like simplexes: there are face and degeneracy maps for them, and the like. They form a “cosimplicial object”, which we can think of as a functor , where is the simplex category).

Then the point is one can look at, for a smooth space , the smooth singular simplicial set : it’s a simplicial set where the sets are sets of smooth maps from the affine simplex into . Likewise, for a simplicial set , there’s a smooth space, the “geometric realization” . These give two functors and , which are adjoints ( is the left adjoint). And then, weak equivalences and fibrations being defined in simplicial sets (w.e. are homotopy equivalences of the realization in , and fibrations are “Kan fibrations”), you can just pull the definition back to : a smooth map is a w.e. if its image under is one. The cofibrations get indirectly defined via the lifting properties they need to have relative to the other two classes.

So it’s still not completely settled that this definition actually gives a model category structure, but it’s pretty close. Certainly, some things are known. For instance, Enxin Wu showed that if you have a fibrant object (i.e. one where the unique map to the terminal object is a fibration – these are generally the “good” objects to define homotopy groups on), then the smooth homotopy groups agree with the simplicial ones for . This implies that for these objects, the weak equivalences are exactly the smooth maps that give isomorphisms for homotopy groups. And so forth. But notice that even some fairly nice objects aren’t fibrant: two lines glued together at a point isn’t, for instance.

There are various further results. One, a consquences of a result Enxin proved, is that all manifolds are fibrant objects, where these nice properties apply. It’s interesting that this comes from the fact that, in , every (connected) manifold is a homogeneous space. These are quotients of smooth groups, – the space is a space of cosets, and is understood to be the stabilizer of the point. Usually one thinks of homogenous spaces as fairly rigid things: the Euclidean plane, say, where is the whole Euclidean group, and the rotations; or a sphere, where is all n-dimensional rotations, and the ones that fix some point on the sphere. (Actually, this gives a projective plane, since opposite points on the sphere get identified. But you get the idea). But that’s for Lie groups. The point is that , the space of diffeomorphisms from to itself, is a perfectly good smooth group. Then the subgroup of diffeomorphisms that fix any point is a fine smooth subgroup, and is a homogeneous space in . But that’s just , with acting transitively on it – any point can be taken anywhere on .

**Cohesive Infinity-Toposes**

One further thing I’d mention here is related to a related but more abstract approach to the question of how to incorporate homotopy-theoretic tools with a setting that supports differential geometry. This is the notion of a cohesive topos, and more generally of a cohesive infinity-topos. Urs Schreiber has advocated for this approach, for instance. It doesn’t really conflict with the kind of thing Dan was talking about, but it gives a setting for it with lot of abstract machinery. I won’t try to explain the details (which anyway I’m not familiar with), but just enough to suggest how the two seem to me to fit together, after discussing it a bit with Dan.

The idea of a cohesive topos seems to start with Bill Lawvere, and it’s supposed to characterize something about those categories which are really “categories of spaces” the way is. Intuitively, spaces consist of “points”, which are held together in lumps we could call “pieces”. Hence “cohesion”: the points of a typical space cohere together, rather than being a dust of separate elements. When that happens, in a discrete space, we just say that each piece happens to have just one point in it – but a priori we distinguish the two ideas. So we might normally say that has an “underlying set” functor , and its left adjoint, the “discrete space” functor (left adjoint since set maps from are the same as continuous maps from – it’s easy for maps out of to be continuous, since every subset is open).

In fact, any topos of sheaves on some site has a pair of functors like this (where becomes , the “set of global sections” functor), essentially because is the topos of sheaves on a single point, and there’s a terminal map from any site into the point. So this adjoint pair is the “terminal geometric morphism” into .

But this omits there are a couple of other things that apply to : has a right adjoint, , where has only and as its open sets. In , all the points are “stuck together” in one piece. On the other hand, itself has a left adjoint, , which gives the set of connected components of a space. is another kind of “underlying set” of a space. So we call a topos “cohesive” when the terminal geometric morphism extends to a chain of four adjoint functors in just this way, which satisfy a few properties that characterize what’s happening here. (We can talk about “cohesive sites”, where this happens.)

Now isn’t exactly a category of sheaves on a site: it’s the category of *concrete* sheaves on a (concrete) site. There is a cohesive topos of all sheaves on the diffeological site. (What’s more, it’s known to have a model category structure). But now, it’s a fact that any cohesive topos has a subcategory of concrete objects (ones where the canonical unit map is mono: roughly, we can characterize the morphisms of by what they do to its points). This category is always a quasitopos (and it’s a reflective subcategory of : see the previous post for some comments about reflective subcategories if interested…) This is where fits in here. Diffeologies define a “cohesion” just as topologies do: points are in the same “piece” if there’s some plot from a connected part of that lands on both. Why is only a quasitopos? Because in general, the subobject classifier in isn’t concrete – but it will have a “concretification”, which is the weak subobject classifier I mentioned above.

Where the “infinity” part of “infinity-topos” comes in is the connection to homotopy theory. Here, we replace the topos with the infinity-topos of infinity-groupoids. Then the “underlying” functor captures not just the set of points of a space , but its whole fundamental infinity-groupoid. Its objects are points of , its morphisms are paths, 2-morphisms are homotopies of paths, and so on. All the homotopy groups of live here. So a cohesive inifinity-topos is defined much like above, but with playing the role of , and with that functor replaced by , something which, implicitly, gives all the homotopy groups of . We might look for cohesive infinity-toposes to be given by the (infinity)-categories of simplicial sheaves on cohesive sites.

This raises a point Dan made in his talk over the diffeological site , we can talk about a cube of different structures that live over it, starting with presheaves: . We can add different modifiers to this: the sheaf condition; the adjective “concrete”; the adjective “simplicial”. Various combinations of these adjectives (e.g. simplicial presheaves) are known to have a model structure. is the case where we have *concrete sheaves* on . So far, it hasn’t been proved, but it looks like it shortly will be, that this has a model structure. This is a particularly nice one, because these things really do seem a lot like spaces: they’re just sets with some easy-to-define and well-behaved (that’s what the sheaf condition does) structure on them, and they include all the examples a differential geometer requires, the manifolds.

June 15, 2011 at 12:35 pm

Hi Jeffrey, the approach that Enxin and Dan are taking, using affine spaces for simplices, is similar to that Finnur Larusson takes in his paper “Affine simplices in Oka manifolds” (available on the arXiv), and the model category aspects from another paper of his, “Model Structures and the Oka Principle” (also arXiv). This is set in the holomorphic category, so more rigid than manifolds, but perhaps some of the ideas are similar, in that he also uses simplicial presheaves.

June 17, 2011 at 3:06 pm

Thanks for that reference, David. Smplicial presheaves are a very powerful tool which I probably ought to learn more about. I picked up a little of it from being at UWO, where Rick Jardine and his students and postdocs have done a lot of work to develop and use that machinery for homotopy theory etc. It’s interesting to see it in a rigid context like complex manifolds…

August 23, 2011 at 9:52 am

I have only now come across this post of yours.

One comment:

that cosimplicial object of smooth “affine” simplices looks familiar: this has been described here a while ago

http://ncatlab.org/nlab/show/interval+object#FundGeomInftCat

(for the interval object being the real line with two point inclusions). I used to give talks about how one obtains smooth path oo-groupoids using this … until I realized that there is a more powerful way, using that left adjoint on the cohesive oo-topos.

August 23, 2011 at 9:54 am

Another comment: as far as I understand, this model structure on Diff that you mention — if it can be made to exist — looks like it will be close to being equivalent to the standard model structure on Top: it will model simplices and simplicial homotopies not just in topological spaces, but in topological space equipped with smooth structure. But the smooth structure will not appear much in the Quillen equivalence class.

This means that it is quite different from the model structure on simplicial pressheaves over smooth spaces. This contains genuinely smooth homotopical structures such as Lie groupoids.

Not that you have claimed otherwise, but maybe this deserves to be emphasized.

August 23, 2011 at 1:52 pm

Hi Urs: thanks for the remarks.

Yes, it seems that the eight different sorts of structures I mention at the end have rather different properties. I’m not entirely clear why the smooth structure wouldn’t affect the Quillen equivalence class of this model category, but neither is it particularly surprising.

There might be inequivalent model structures which could be put on the same category, and conceivably one on which more resembles the one on the category of simplicial presheaves. That would be interesting to know.

However, from what Dan has said, they’ve tried several different approaches to get a model structure on , which encounter various technical obstacles. This one is the most promising candidate, but it’s not clear if it’s the only one, or the best one – it’s just one that seems (so far) that it will work.

December 29, 2011 at 9:13 pm

“There is a cohesive topos of all sheaves on the diffeological site. (What’s more, it’s known to have a model category structure).”

What is this model category structure on the category of all sheaves for the diffeological site? I can’t seem to find a reference, but I’m not very familiar with model categories.