### sheaves

(Note: WordPress seems to be having some intermittent technical problem parsing my math markup in this post, so please bear with me until it, hopefully, goes away…)

As August is the month in which Portugal goes on vacation, and we had several family visitors toward the end of the summer, I haven’t posted in a while, but the term has now started up at IST, and seminars are underway, so there should be some interesting stuff coming up to talk about.

New Blog

First, I’ll point out that that Derek Wise has started a new blog, called simply “Simplicity“, which is (I imagine) what it aims to contain: things which seem complex explained so as to reveal their simplicity.  Unless I’m reading too much into the title.  As of this writing, he’s posted only one entry, but a lengthy one that gives a nice explanation of a program for categorified Klein geometries which he’s been thinking a bunch about.  Klein’s program for describing the geometry of homogeneous spaces (such as spherical, Euclidean, and hyperbolic spaces with constant curvature, for example) was developed at Erlangen, and goes by the name “The Erlangen Program”.  Since Derek is now doing a postdoc at Erlangen, and this is supposed to be a categorification of Klein’s approach, he’s referred to it the “2-Erlangen Program”.  There’s more discussion about it in a (somewhat) recent post by John Baez at the n-Category Cafe.  Both of them note the recent draft paper they did relating a higher gauge theory based on the Poincare 2-group to a theory known as teleparallel gravity.  I don’t know this theory so well, except that it’s some almost-equivalent way of formulating General Relativity

I’ll refer you to Derek’s own post for full details of what’s going on in this approach, but the basic motivation isn’t too hard to set out.  The Erlangen program takes the view that a homogeneous space is a space $X$ (let’s say we mean by this a topological space) which “looks the same everywhere”.  More precisely, there’s a group action by some $G$, which we understand to be “symmetries” of the space, which is transitive.  Since every point is taken to every other point by some symmetry, the space is “homogeneous”.  Some symmetries leave certain points $x \in X$ where they are – they form the stabilizer subgroup $H = Stab(x)$.  When the space is homogeneous, it is isomorphic to the coset space, $X \cong G / H$.  So Klein’s idea is to say that any time you have a Lie group $G$ and a closed subgroup $H < G$, this quotient will be called a “homogeneous space”.  A familiar example would be Euclidean space, $\mathbb{R}^n \cong E(n) / O(n)$, where $E$ is the Euclidean group and $O$ is the orthogonal group, but there are plenty of others.

This example indicates what Cartan geometry is all about, though – this is the next natural step after Klein geometry (Edit:  Derek’s blog now has a visual explanation of Cartan geometry, a.k.a. “generalized hamsterology”, new since I originally posted this).  We can say that Cartan is to Klein as Riemann is to Euclid.  (Or that Cartan is to Riemann as Klein is to Euclid – or if you want to get maybe too-precisely metaphorical, Cartan is the pushout of Klein and Riemann over Euclid).  The point is that Riemannian geometry studies manifolds – spaces which are not homogeneous, but look like Euclidean space locally.  Cartan geometry studies spaces which aren’t homogeneous, but can be locally modelled by Klein geometries.  Now, a Riemannian geometry is essentially a manifold with a metric, describing how it locally looks like Euclidean space.  An equivalent way to talk about it is a manifold with a bundle of Euclidean spaces (the tangent spaces) with a connection (the Levi-Civita connection associated to the metric).  A Cartan geometry can likewise be described as a $G$-bundle with fibre $X$ with a connection

Then the point of the “2-Erlangen program” is to develop similar geometric machinery for 2-groups (a.k.a. categorical groups).  This is, as usual, a bit more complicated since actions of 2-groups are trickier than group-actions.  In their paper, though, the point is to look at spaces which are locally modelled by some sort of 2-Klein geometry which derives from the Poincare 2-group.  By analogy with Cartan geometry, one can talk about such Poincare 2-group connections on a space – that is, some kind of “higher gauge theory”.  This is the sort of framework where John and Derek’s draft paper formulates teleparallel gravity.  It turns out that the 2-group connection ends up looking like a regular connection with torsion, and this plays a role in that theory.  Their draft will give you a lot more detail.

Talk on Manifold Calculus

On a different note, one of the first talks I went to so far this semester was one by Pedro Brito about “Manifold Calculus and Operads” (though he ran out of time in the seminar before getting to talk about the connection to operads).  This was about motivating and introducing the Goodwillie Calculus for functors between categories of spaces.  (There are various references on this, but see for instance these notes by Hal Sadofsky). In some sense this is a generalization of calculus from functions to functors, and one of the main results Goodwillie introduced with this subject, is a functorial analog of Taylor’s theorem.  I’d seen some of this before, but this talk was a nice and accessible intro to the topic.

So the starting point for this “Manifold Calculus” is that we’d like to study functors from spaces to spaces (in fact this all applies to spectra, which are more general, but Pedro Brito’s talk was focused on spaces).  The sort of thing we’re talking about is a functor which, given a space $M$, gives a moduli space of some sort of geometric structures we can put on $M$, or of mappings from $M$.  The main motivating example he gave was the functor

$Imm(-,N) : [Spaces] \rightarrow [Spaces]$

for some fixed manifold $N$. Given a manifold $M$, this gives the mapping space of all immersions of $M$ into $N$.

(Recalling some terminology: immersions are maps of manifolds where the differential is nondegenerate – the induced map of tangent spaces is everywhere injective, meaning essentially that there are no points, cusps, or kinks in the image, but there might be self-intersections. Embeddings are, in addition, local homeomorphisms.)

Studying this functor $Imm(-,N)$ means, among other things, looking at the various spaces $Imm(M,N)$ of immersions of each $M$ into $N$. We might first ask: can $M$ be immersed in $N$ at all – in other words, is $\pi_0(Imm(M,N))$ nonempty?

So, for example, the Whitney Embedding Theorem says that if $dim(N)$ is at least $2 dim(M)$, then there is an embedding of $M$ into $N$ (which is therefore also an immersion).

In more detail, we might want to know what $\pi_0(Imm(M,N))$ is, which tells how many connected components of immersions there are: in other words, distinct classes of immersions which can’t be deformed into one another by a family of immersions. Or, indeed, we might ask about all the homotopy groups of $Imm(M,N)$, not just the zeroth: what’s the homotopy type of $Imm(M,N)$? (Once we have a handle on this, we would then want to vary $M$).

It turns out this question is manageable, party due to a theorem of Smale and Hirsch, which is a generalization of Gromov’s h-principle – the original principle applies to solutions of certain kinds of PDE’s, saying that any solution can be deformed to a holomorphic one, so if you want to study the space of solutions up to homotopy, you may as well just study the holomorphic solutions.

The Smale-Hirsch theorem likewise gives a homotopy equivalence of two spaces, one of which is $Imm(M,N)$. The other is the space of “formal immersions”, called $Imm^f(M,N)$. It consists of all $(f,F)$, where $f : M \rightarrow N$ is smooth, and $F : TM \rightarrow TN$ is a map of tangent spaces which restricts to $f$, and is injective. These are “formally” like immersions, and indeed $Imm(M,N)$ has an inclusion into $Imm^f(M,N)$, which happens to be a homotopy equivalence: it induces isomorphisms of all the homotopy groups. These come from homotopies taking each “formal immersion” to some actual immersion. So we’ve approximated $Imm(-,N)$, up to homotopy, by $Imm^f(-,N)$. (This “homotopy” of functors makes sense because we’re talking about an enriched functor – the source and target categories are enriched in spaces, where the concepts of homotopy theory are all available).

We still haven’t got to manifold calculus, but it will be all about approximating one functor by another – or rather, by a chain of functors which are supposed to be like the Taylor series for a function. The way to get this series has to do with sheafification, so first it’s handy to re-describe what the Smale-Hirsch theorem says in terms of sheaves. This means we want to talk about some category of spaces with a Grothendieck topology.

So lets let $\mathcal{E}$ be the category whose objects are $d$-dimensional manifolds and whose morphisms are embeddings (which, of course, are necessarily codimension 0). Now, the point here is that if $f : M \rightarrow M'$ is an embedding in $\mathcal{E}$, and $M'$ has an immersion into $N$, this induces an immersion of $M$ into $N$. This amounst to saying $Imm(-,N)$ is a contravariant functor:

$Imm(-,N) : \mathcal{E}^{op} \rightarrow [Spaces]$

That makes $Imm(-,N)$ a presheaf. What the Smale-Hirsch theorem tells us is that this presheaf is a homotopy sheaf – but to understand that, we need a few things first.

First, what’s a homotopy sheaf? Well, the condition for a sheaf says that if we have an open cover of $M$, then

So to say how $Imm(-,N) : \mathcal{E}^{op} \rightarrow [Spaces]$ is a homotopy sheaf, we have to give $\mathcal{E}$ a topology, which means defining a “cover”, which we do in the obvious way – a cover is a collection of morphisms $f_i : U_i \rightarrow M$ such that the union of all the images $\cup f_i(U_i)$ is just $M$. The topology where this is the definition of a cover can be called $J_1$, because it has the property that given any open cover and choice of 1 point in $M$, that point will be in some $U_i$ of the cover.

This is part of a family of topologies, where $J_k$ only allows those covers with the property that given any choice of $k$ points in $M$, some open set of the cover contains them all. These conditions, clearly, get increasingly restrictive, so we have a sequence of inclusions (a “filtration”):

$J_1 \leftarrow J_2 \leftarrow J_3 \leftarrow \dots$

Now, with respect to any given one of these topologies $J_k$, we have the usual situation relating sheaves and presheaves.  Sheaves are defined relative to a given topology (i.e. a notion of cover).  A presheaf on $\mathcal{E}$ is just a contravariant functor from $\mathcal{E}$ (in this case valued in spaces); a sheaf is one which satisfies a descent condition (I’ve discussed this before, for instance here, when I was running the Stacks Seminar at UWO).  The point of a descent condition, for a given topology is that if we can take the values of a functor $F$ “locally” – on the various objects of a cover for $M$ – and “glue” them to find the value for $M$ itself.  In particular, given a cover for $M \in \mathcal{E}$, and a cover, there’s a diagram consisting of the inclusions of all the double-overlaps of sets in the cover into the original sets.  Then the descent condition for sheaves of spaces is that

The general fact is that there’s a reflective inclusion of sheaves into presheaves (see some discussion about reflective inclusions, also in an earlier post).  Any sheaf is a contravariant functor – this is the inclusion of $Sh( \mathcal{E} )$ into $latex PSh( \mathcal{E} )$.  The reflection has a left adjoint, sheafification, which takes any presheaf in $PSh( \mathcal{E} )$ to a sheaf which is the “best approximation” to it.  It’s the fact this is an adjoint which makes the inclusion “reflective”, and provides the sense in which the sheafification is an approximation to the original functor.

The way sheafification works can be worked out from the fact that it’s an adjoint to the inclusion, but it also has a fairly concrete description.  Given any one of the topologies $J_k$,  we have a whole collection of special diagrams, such as:

$U_i \leftarrow U_{ij} \rightarrow U_j$

(using the usual notation where $U_{ij} = U_i \cap U_j$ is the intersection of two sets in a cover, and the maps here are the inclusions of that intersection).  This and the various other diagrams involving these inclusions are special, given the topology $J_k$.  The descent condition for a sheaf $F$ says that if we take the image of this diagram:

$F(U_i) \rightarrow F(U_{ij}) \leftarrow F(U_j)$

then we can “glue together” the objects $F(U_i)$ and $F(U_j)$ on the overlap to get one on the union.  That is, $F$ is a sheaf if $F(U_i \cup U_j)$ is a colimit of the diagram above (intuitively, by “gluing on the overlap”).  In a presheaf, it would come equipped with some maps into the $F(U_i)$ and $F(U_j)$: in a sheaf, this object and the maps satisfy some universal property.  Sheafification takes a presheaf $F$ to a sheaf $F^{(k)}$ which does this, essentially by taking all these colimits.  More accurately, since these sheaves are valued in spaces, what we really want are homotopy sheaves, where we can replace “colimit” with “homotopy colimit” in the above – which satisfies a universal property only up to homotopy, and which has a slightly weaker notion of “gluing”.   This (homotopy) sheaf is called $F^{(k)}$ because it depends on the topology $J_k$ which we were using to get the class of special diagrams.

One way to think about $F^{(k)}$ is that we take the restriction to manifolds which are made by pasting together at most $k$ open balls.  Then, knowing only this part of the functor $F$, we extend it back to all manifolds by a Kan extension (this is the technical sense in which it’s a “best approximation”).

Now the point of all this is that we’re building a tower of functors that are “approximately” like $F$, agreeing on ever-more-complicated manifolds, which in our motivating example is $F = Imm(-,N)$.  Whichever functor we use, we get a tower of functors connected by natural transformations:

$F^{(1)} \leftarrow F^{(2)} \leftarrow F^{(3)} \leftarrow \dots$

This happens because we had that chain of inclusions of the topologies $J_k$.  Now the idea is that if we start with a reasonably nice functor (like $F = Imm(-,N)$ for example), then $F$ is just the limit of this diagram.  That is, it’s the universal thing $F$ which has a map into each $F^{(k)}$ commuting with all these connecting maps in the tower.  The tower of approximations – along with its limit (as a diagram in the category of functors) – is what Goodwillie called the “Taylor tower” for $F$.  Then we say the functor $F$ is analytic if it’s just (up to homotopy!) the limit of this tower.

By analogy, think of an inclusion of a vector space $V$ with inner product into another such space $W$ which has higher dimension.  Then there’s an orthogonal projection onto the smaller space, which is an adjoint (as a map of inner product spaces) to the inclusion – so these are like our reflective inclusions.  So the smaller space can “reflect” the bigger one, while not being able to capture anything in the orthogonal complement.  Now suppose we have a tower of inclusions $V \leftarrow V' \leftarrow V'' \dots$, where each space is of higher dimension, such that each of the $V$ is included into $W$ in a way that agrees with their maps to each other.  Then given a vector $w \in W$, we can take a sequence of approximations $(v,v',v'',\dots)$ in the $V$ spaces.  If $w$ was “nice” to begin with, this series of approximations will eventually at least converge to it – but it may be that our tower of $V$ spaces doesn’t let us approximate every $w$ in this way.

That’s precisely what one does in calculus with Taylor series: we have a big vector space $W$ of smooth functions, and a tower of spaces we use to approximate.  These are polynomial functions of different degrees: first linear, then quadratic, and so forth.  The approximations to a function $f$ are orthogonal projections onto these smaller spaces.  The sequence of approximations, or rather its limit (as a sequence in the inner product space $W$), is just what we mean by a “Taylor series for $f$“.  If $f$ is analytic in the first place, then this sequence will converge to it.

The same sort of phenomenon is happening with the Goodwillie calculus for functors: our tower of sheafifications of some functor $F$ are just “projections” onto smaller categories (of sheaves) inside the category of all contravariant functors.  (Actually, “reflections”, via the reflective inclusions of the sheaf categories for each of the topologies $J_k$).  The Taylor Tower for this functor is just like the Taylor series approximating a function.  Indeed, this analogy is fairly close, since the topologies $J_k$ will give approximations of $F$ which are in some sense based on $k$ points (so-called $k$-excisive functors, which in our terminology here are sheaves in these topologies).  Likewise, a degree-$k$ polynomial approximation approximates a smooth function, in general in a way that can be made to agree at $k$ points.

Finally, I’ll point out that I mentioned that the Goodwillie calculus is actually more general than this, and applies not only to spaces but to spectra. The point is that the functor $Imm(-,N)$ defines a kind of generalized cohomology theory – the cohomology groups for $M$ are the $\pi_i(Imm(M,N))$. So the point is, functors satisfying the axioms of a generalized cohomology theory are represented by spectra, whereas $N$ here is a special case that happens to be a space.

Lots of geometric problems can be thought of as classified by this sort of functor – if $N = BG$, the classifying space of a group, and we drop the requirement that the map be an immersion, then we’re looking at the functor that gives the moduli space of $G$-connections on each $M$.  The point is that the Goodwillie calculus gives a sense in which we can understand such functors by simpler approximations to them.

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.

Last week I spoke in Montreal at a session of the Philosophy of Science Association meeting.  Here are some notes for it.  Later on I’ll do a post about the other talks at the meeting.

Right now, though, the meeting slowed me down from describing a recent talk in the seminar here at IST.  This was Gonçalo Rodrigues’ talk on categorifying measure theory.  It was based on this paper here, which is pretty long and goes into some (but not all) of the details.  Apparently an updated version that fills in some of what’s not there is in the works.

In any case, Gonçalo takes as the starting point for categorifying ideas in analysis the paper “Measurable Categories” by David Yetter, which is the same point where I started on this topic, although he then concludes that there are problems with that approach.  Part of the reason for saying this has to do with the fact that the category of Hilbert spaces has many bad properties – or rather, fails to have many of the good ones that it should to play the role one might expect in categorifying ideas from analysis.

Yetter’s idea can be described, very roughly, as follows: we would like to categorify the concept of a function-space on a measure space $(X,\mu)$.  That is, spaces like $L^2(X,\mu)$ or $L^{\infty}(X,\mu)$.  The reason for this is that the 2-vector-spaces of Kapranov and Voevodsky are very elegant, but intrinsically finite-dimensional, categorifications of “vector space”.  An infinite-dimensional version would be important for representation theory, particularly of noncompact Lie groups or 2-groups, but even just infinite ones, since there are relatively few endomorphisms of KV 2-vector spaces.  Yetter’s paper constructs analogs to the space of measurable functions $\mathcal{M}(X)$, where “functions” take values in Hilbert spaces.

A measurable field of Hilbert spaces is, roughly, a family of Hilbert spaces indexed by points of $X$, together with a nice space of “measurable sections”.  This is supposed to be an infinite-dimensional, measure-theoretic counterpart to an object in a KV 2-vector space, which always looks like $\mathbf{Vect}^k$ for some natural number $k$, which is now being replaced by $(X,\mu)$.  One of the key tools in Yetter’s paper is the direct integral of a field of Hilbert spaces, which is similarly the counterpart to the direct sum $\bigoplus$ in the discrete world.  It just gives the space of measurable sections (taken up to almost-everywhere equivalence, as usual).  This was the main focus of Gonçalo’s talk.

The direct integral has one major problem, compared to the (finite) direct sum it is supposed to generalize – namely, the direct sum is a categorical coproduct, in $\mathbf{Vect}$ or any other KV 2-vector space.  Actually, it is both a product and a coproduct ($\mathbf{Vect}$ is abelian), so it is defined by a nice universal property.  The direct integral, on the other hand, is not.  It doesn’t have any similarly nice universal property.  (In the infinite-dimensional case, colimits and limits would be expected to become different in any case, but the direct integral is neither).  This means that many proofs in analysis will be hard to reproduce in the categorified setting – universal properties mean one doesn’t have to do nearly as much work to do this, among their other good qualities.  This is related to the issue that the category $\mathbf{Hilb}$ does not have all limits and colimits

Gonçalo’s paper and talk outline a program where one can categorify a lot of the proofs in analysis, by using a slightly different framework which uses a bigger category than $\mathbf{Hilb}$, namely $Ban_C$, whose objects are Banach spaces and whose maps are (linear) contractions.  A Banach Category is a category enriched in $Ban_C$.  Now, Banach spaces have a norm, but not necessarily an inner product, and this small weakening makes them much worse than Hilbert spaces as objects.  Many intuitions from Hilbert spaces, like the one that says any subspace has a complement, just fail: the corresponding notion for Banach spaces is the quasicomplement ($X$ and $Y$ are quasicomplements if they intersect only at zero, and their sum is dense in the whole space), and it’s quite possible to have subspaces which don’t have one.  Other unpleasant properties abound.

Yet $Ban_C$ is a much nicer category than $Hilb$.  (So we follow the general dictum that it’s better to have a nice category with bad objects than a bad category with nice objects – the same motivation behind “smooth spaces” instead of manifolds, and the like.)  It’s complete and cocomplete (i.e. has all limits and colimits), as well as monoidal closed – for Banach spaces $A$ and $B$, the space $Hom(A,B)$ is also in $Ban_C$.  None of these facts holds for $Hilb$.  On the other hand, the space of bounded maps between Hilbert spaces is a Banach space (with the operator norm), but not necessarily a Hilbert space.  So even $Hilb$ is already a Banach category.

It also turns out that, unlike in $Hilb$, limits and colimits (where those exist in $Hilb$) are not necessarily isomorphic.  In particular, in $Ban_C$, the coproduct and product $A + B$ and $A \times B$ both have the same underlying vector space $A \oplus B$, but the norms are different.  For Hilbert spaces, the inner product comes from the Pythagorean formula in either case, but for Banach spaces, the coproduct gets the sum of the two norms, and the product gets the supremum.  It turns out that coproducts are the more important concept, and this is where the direct integral comes in.

First, we can talk about Banach 2-spaces (the analogs of 2-vector spaces): these are just Banach categories which are cocomplete (have all weighted colimits).  Maps between them are cocontinuous functors – that is, colimit-preserving ones.  (Though properly, functors between Banach categories ought to be contractions on Hom-spaces).  Then there are categorified analogs of all sorts of Banach space structure in a familiar way – the direct sum (coproduct) is the analog of vector addition, the category $Ban_C$ is the analog of the base field (say, $\mathbb{R}$), and so on.

This all gives the setting for categorified measure theory.  Part of the point of choosing $Ban_C$ is that you can now reason out at least some of how it works by analogy.  To start with, one needs to fix a Boolean algebra $\Omega$ – this is to be the $\sigma$-algebra of measurable sets for some measure space, though it’s important that it needn’t have any actual points (this is a notion of measure space akin to the notion of locale in the topological world).  This part of the theory isn’t categorified (arguably a limitation of this approach, but not one that’s any different from Yetter’s).  Instead, we categorify the definition of measure itself.

A measure is a function $\mu : \Omega \mapsto \mathbb{R}$ – it assigns a number to each measurable set.  The pair $(\Omega,\mu)$ is a measure algebra, and relates to a measure space the way a locale relates to a topological space.  So a categorified measure $\nu$ should be a functor from $\Omega$ (seen now as a category) into $Ban_C$.  (We can generalize this: the measure could be valued in some vector space over $\mathbb{R}$, and a categorified measure could be a functor into some other Banach 2-space.)  Since we’re thinking of $\Omega$ as a lattice of subsets, it makes some sense to call $\nu$ a presheaf, or rather co-presheaf.  What’s more, just as a measure is additive ($\mu(A + B) = \mu(A) + \mu(B)$, for disjoint sets, where $+$ is the union), so also the categorical measure $\nu$ should be (finitely) additive up to isomorphism.  So we’re assigning Banach spaces to all the measurable sets.  This is a “co”-presheaf – which is to say, a covariant functor, so the spaces “nest”: when for measurable sets, we have $A \subset B$, then $\nu(A) \leq \nu(B)$ also.

An intuition for how this works comes from a special case (not at all exhaustive), where we start with an actual, uncategorified, measure space $(X,\mu)$.  Then one categorified measure will arise by taking $\nu(E) = L_1(E,\mu)$: the Banach space associated to a measurable set $E$ is the space of integrable functions.  We can take any “scalar” multiple of this, too: given a fixed Banach space $B$, let $\nu(E) = L_1(E,\mu) \otimes B$.  But there are lots of examples that aren’t like this.

All this is fine, but the point here is to define integration.  The usual way to go about this when you learn analysis is to start with characteristic functions of measurable sets, then define a sequence through simple functions, measurable functions, and so forth.  Eventually one can define $L^p$ spaces based on the convergence of various integrals.  Something similar happens here.

The analog of a function here is a sheaf: a (compatible) assignment of Banach spaces to measurable sets.  (Technically, to get to sheaves, we need an idea of “cover” by measurable sets, but it’s pretty much the obvious one, modulo the subtlety that we should only allow countable covers.) The idea will be to start with characteristic sheaves for measurable sets, then take some kind of completion of the category of all of these as a definition of “measurable sheaf”.  Then the point will be that we can extend the measure from characteristic sheaves to all measurable sheaves using a limit (actually, a colimit), analogous to the way we define a Lebesgue integral as a limit of simple functions approximating a measurable one.

A characteristic sheaf $\chi(E)$ for a measurable set $E \in \Omega$ might be easiest to visualize in terms of a characteristic bundle, which just puts a copy of the base field (we’ve been assuming it’s $\mathbb{R}$) at each point of $E$, and the zero vector space everywhere else.  (This is a bundle in the measurable sense, not the topological one – assuming $X$ has a topology other than $\Omega$ itself.)  Very intuitively, to turn this into a sheaf, one can just use brute force and take a set $A$ the product of all the spaces lying in $A$.  A little less crudely, one should take a space of sections with decent properties – so that $\chi(E)$ assigns to $A$ a space of functions on $E \cap A$.  In particular, the functor $\chi : \Omega \rightarrow L_{\infty}(\Omega)$ which picks out all the (measurable) bounded sections is a universal way to do this.

Now the point is that the algebra of measurable sets, $\Omega$, thought of as a category, embeds into the category of presheaves on it by $\chi : \Omega \rightarrow \mathbf{PShv}(\Omega)$, taking a set to its characteristic sheaf.  Given a measure valued in some Banach category, $\nu : \Omega \rightarrow \mathcal{B}$, we can find the left Kan extension $\int_X d\nu : \mathbf{PShv}(\Omega) \rightarrow \mathcal{B}$, such that $\nu = \int_X d\nu \circ \chi$.  The Kan extension is a universal way to extend $\nu$ to all of $\mathbf{PShv}(\Omega)$ so that this is true, and it can be calculated as a colimit.

The essential fact here is that the characteristic sheaves are dense in $\mathbf{PShv}(\Omega)$: any presheaf can be found as a colimit of the characteristic ones.  This is analogous to how any function can be approximated by linear combinations of characteristic functions.  This means that the integral defined above will actually give interesting results for all the sheaves one might expect.

I’m glossing over some points here, of course – for example, the distinction between sheaves and presheaves, the role of sheafification, etc.  If you want to get a more accurate picture, check out the paper I linked to up above.

All of this granted, however, many of the classical theorems of measure theory have analogs that are proved in essentially the same way as the standard versions.  One can see the presheaf category as a categorified analog of $L_1(X,\nu)$, and get the Fubini theorem, for instance: there is a canonical equivalence (no longer isomorphism) between (a suitable) tensor product of $\mathbf{PShv}(X)$ and $\mathbf{PShv}(Y)$ on one hand, and on the other $\mathbf{PShv}(X \times Y)$.  Doing integration, one can then do all the usual things – exchange order of integration between $X$ and $Y$, say – in analogous conditions.  The use of universal properties to define integrals etc. means that one doesn’t need to fuss about too much with coherence laws, and so the proofs of the categorified facts are much the same as the original proofs.