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 $Diff$ 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 $Diff$ in Patrick Iglesias-Zimmour’s draft-of-a-book.

Motivation

The point of the category $Diff$, 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 $Diff$ 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, $Diff$ has exponentials (that is, if $A$ and $B$ are smooth spaces, so is $A^B = Hom(B,A)$).

So, for instance, this is a good context for constructing loop spaces: a manifold $M$ is a smooth space, and so is its loop space $LM = M^{S^1} = Hom(S^1,M)$, the space of all maps of the circle into $M$.  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 $LM$ can even be defined.  Likewise, the irrational torus is hard to talk about as a manifold: you take a torus, thought of as $\mathbb{R}^2 / \mathbb{Z}^2$.  Then take a direction in $\mathbb{R}^2$ with irrational slope, and identify any two points which are translates of each other in $\mathbb{R}^2$ 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 $Diff$ 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 $\mathbb{R}^n$ in some sense.

Smooth Spaces

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

• A set $X$ (of “points”)
• A set $\{ f : U \rightarrow X \}$ (of “plots”) for every n and open $U \subset \mathbb{R}^n$ such that:
1. All constant maps are plots
2. If $f: U \rightarrow X$ is a plot, and $g : V \rightarrow U$ is a smooth map, $f \circ g : V \rightarrow X$ is a plot
3. If $\{ g_i : U_i \rightarrow U\}$ is an open cover of $U$, and $f : U \rightarrow X$ is a map, whose restrictions $f \circ g_i : U_i \rightarrow X$ are all plots, so is $f$

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 $\mathbb{R}^n$, 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 $X$ 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 $I= [0,1]$ 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 $I$ itself with the property that any smooth map from this $I$ into a manifold $M$ 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 $U \subset \mathbb{R}^n$.  This implies many nice things about the category $Diff$.  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 $t : 1 \rightarrow \Omega$ from the terminal object, so that any monomorphism (subobject) $A \rightarrow X$ is the pullback of $t$ along some map $X \rightarrow \Omega$ (the classifying map).  In the topos of sets, this is just the inclusion of a one-element set $\{\star\}$ into a two-element set $\{T,F\}$: the classifying map for a subset $A \subset X$ sends everything in $A$ (i.e. in the image of the inclusion map) to $T = Im(t)$, and everything else to $F$.  (That is, it’s the characteristic function.)  So pulling back $T$

Any topos has one of these – in particular the topos of sheaves on the diffeological site has one.  But $Diff$ 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 $X$ which it classifies (i.e. for which there’s a classifying map as above) are exactly the subsets $A \subset X$ equipped with the subspace diffeology.  (Which is defined in the obvious way: the plots are the plots of $X$ which land in $A$).

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 $Diff$.  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 $Diff$: to start with, one can define the “smooth homotopy groups” $\pi_n^s(X;x_0)$ 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 $\pi_0^s(X)$ is the set of path-components of $X$, and $\pi_k^s(X) = \pi_{k-1}^s(LX)$ is defined recursively using loop spaces, mentioned above.  The point is that this all works in $Diff$ much as for topological spaces.

In particular, there are analogs for the $\pi_k^s$ for standard theorems like the long exact sequence of homotopy groups for a bundle.  Of course, you have to define “bundle” in $Diff$ – it’s a smooth surjective map $X \rightarrow Y$, 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 $\mathbb{R}^n$, 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 $Diff$ 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 $Diff$ 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 $Top$, 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 $f$, latex $g$ and $f \circ g$ 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 $Diff$ 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 $Man$ itself – only in the larger setting $Diff$.

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”, $\mathbb{A}^n = \{ (x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} | \sum x_i = 1 \}$.  These aren’t literally simplexes: they’re affine planes, which we understand as smooth spaces – with the subspace diffeology from $\mathbb{R}^{n+1}$.  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 $\Delta \rightarrow Diff$, where $\Delta$ is the simplex category).

Then the point is one can look at, for a smooth space $X$, the smooth singular simplicial set $S(X)$: it’s a simplicial set where the sets are sets of smooth maps from the affine simplex into $X$.  Likewise, for a simplicial set $S$, there’s a smooth space, the “geometric realization” $|S|$.  These give two functors $|\cdot |$ and $S$, which are adjoints ($| \cdot |$ is the left adjoint).  And then, weak equivalences and fibrations being defined in simplicial sets (w.e. are homotopy equivalences of the realization in $Top$, and fibrations are “Kan fibrations”), you can just pull the definition back to $Diff$: a smooth map is a w.e. if its image under $S$ 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 $X$ (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 $S(X)$.  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 $Diff$, every (connected) manifold is a homogeneous space.  These are quotients of smooth groups, $G/H$ – the space is a space of cosets, and $H$ is understood to be the stabilizer of the point.  Usually one thinks of homogenous spaces as fairly rigid things: the Euclidean plane, say, where $G$ is the whole Euclidean group, and $H$ the rotations; or a sphere, where $G$ is all n-dimensional rotations, and $H$ 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 $G = Diff(M,M)$, the space of diffeomorphisms from $M$ to itself, is a perfectly good smooth group.  Then the subgroup $H$ of diffeomorphisms that fix any point is a fine smooth subgroup, and $G/H$ is a homogeneous space in $Diff$.  But that’s just $M$, with $G$ acting transitively on it – any point can be taken anywhere on $M$.

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 $Top$ 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 $Top$ has an “underlying set” functor $U : Top \rightarrow Set$, and its left adjoint, the “discrete space” functor $Disc: Set \rightarrow Top$ (left adjoint since set maps from $S$ are the same as continuous maps from $Disc(S)$ – it’s easy for maps out of $Disc(S)$ 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 $U$ becomes $\Gamma$, the “set of global sections” functor), essentially because $Set$ 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 $Set$.

But this omits there are a couple of other things that apply to $Top$: $U$ has a right adjoint, $Codisc: Set \rightarrow Top$, where $Codisc(S)$ has only $S$ and $\emptyset$ as its open sets.  In $Codisc(S)$, all the points are “stuck together” in one piece.  On the other hand, $Disc$ itself has a left adjoint, $\Pi_0: Top \rightarrow Set$, which gives the set of connected components of a space.  $\Pi_0(X)$ is another kind of “underlying set” of a space.  So we call a topos $\mathcal{E}$ “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 $Diff$ 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 $\mathcal{E}$ has a subcategory of concrete objects (ones where the canonical unit map $X \rightarrow Codisc(\Gamma(X))$ is mono: roughly, we can characterize the morphisms of $X$ by what they do to its points).  This category is always a quasitopos (and it’s a reflective subcategory of $\mathcal{E}$: see the previous post for some comments about reflective subcategories if interested…)  This is where $Diff$ 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 $\mathbb{R}^n$ that lands on both.  Why is $Diff$ only a quasitopos?  Because in general, the subobject classifier in $\mathcal{E}$ 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 $Sets$ with the infinity-topos of infinity-groupoids.  Then the “underlying” functor captures not just the set of points of a space $X$, but its whole fundamental infinity-groupoid.  Its objects are points of $X$, its morphisms are paths, 2-morphisms are homotopies of paths, and so on.  All the homotopy groups of $X$ live here.  So a cohesive inifinity-topos is defined much like above, but with $\infty-Gpd$ playing the role of $Set$, and with that $\Pi_0$ functor replaced by $\Pi$, something which, implicitly, gives all the homotopy groups of $X$.  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 $D$, we can talk about a cube of different structures that live over it, starting with presheaves: $PSh(D)$.  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.  $Diff$ is the case where we have concrete sheaves on $D$.  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.