### category theory

(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.

So I recently got back from a trip to the UK – most of the time was spent in Cardiff, at a workshop on TQFT and categorification at the University of Cardiff.  There were two days of talks, which had a fair amount of overlap with our workshop in Lisbon, so, being a little worn out on the topic, I’ll refrain from summarizing them all, except to mention a really nice one by Jeff Giansiracusa (who hadn’t been in Lisbon) which related (open/closed) TQFT’s and cohomology theories via a discussion of how categories of cobordisms with various kinds of structure correspond to various sorts of operads.  For example, the “little disks” operad, which describes the structure of how to compose disks with little holes, by pasting new disks into the holes of the old ones, corresponds to the usual cobordism category.

This workshop was part of a semester-long program they’ve been having, sponsored by an EU network on noncommutative geometry.  After the workshop was done, Tim Porter and I stayed on for the rest of the week to give some informal seminars and talk to the various grad students who were visiting at the time.  The seminars started off being directed by questions, but ended up talking about TQFT’s and their relations to various kinds of algebras and higher categorical structures, via classifying spaces.  We also had some interesting discussions outside these, for example with Jennifer Maier, who’s been working with Thomas Nicklaus on equivariant Dijkgraaf-Witten theory; with Grace Kennedy, about planar algebras and their relationships to TQFT‘s. I’d also like to give some credit to Makoto Yamashita, who’s interested in noncommutative geometry (viz) and pointed out to me a paper of Alain Connes which gives an account of integration on groupoids, and what corresponds to measures in that setting, which thankfully agrees with what little of it I’d been able to work out on my own.

However, what I’d like to take the time to write up was from the earlier part of my trip, where I visited with Jamie Vicary at Oxford. While I was there, I gave a little lunch seminar about the bicategory $Span(Gpd)$ (actually a tricategory), and some of the physics- and TQFT-related uses for it. That turned out to be very apropos, because they also had another visitor at the same time, namely Jean Benabou, the fellow who invented bicategories, and introduced the idea of bicategories of spans as one of the first examples.  He gave a talk while I was there which was about the relationship between spans and what he calls “distributors” (which are often called “profunctors“, but since he was anyway the one who introduced them and gave them that name in the first place, and since he has since decided that “profunctors” should refer to only a special class of these entities, I’ll follow his terminology).

(Edit: Thanks to Thomas Streicher for passing on a reference to lecture notes he prepared from lecture by Benabou on the same general topic.)

The question to answer is: what is the relation between spans of categories and distributors?

This is related to a slightly lower-grade question about the relationship between spans of sets, and relations, although the answer turns out to be more complicated.  So, remember that a span from a set $A$ to a set $B$ is just a diagram like this: $A \leftarrow X \rightarrow B$.  They can be composed together – so that given a span from $A$ to $B$, and from $B$ to $C$, we can take fibre products over $B$ and get a span from $A$ to $C$, consisting of pairs of elements from the $X$ sets which map down to the same $b \in B$.  We can do the same thing in any category with pullbacks, not just ${Sets}$.

A span $A \leftarrow S \rightarrow B$ is a relation if the pair of arrows is “jointly monic”, which is to say that as a map $S \rightarrow A \times B$, it is a monomorphism – which, since we’re talking about sets, essentially means “a subset”.  That is, up to isomorphism of spans, $S$ just picks out a bunch of pairs $(a,b) \in A \times B$, which are the “related” pairs in this relation.  So there is an inclusion ${Rel} \hookrightarrow Span({Sets})$.  What’s more  the inclusion has a left adjoin, which turns a span into a corresponding relation.  It follows from the fact that $Sets$ has an “epi-mono factorization”: namely, the map $f: S \rightarrow A \times B$ that comes from the span (and the definition of product) will factor through the image.  That is, it is the composite $S \rightarrow Im(f) \rightarrow A \times B$, where the first part is surjective, and the second part is injective.  Then the inclusion $r(f) : Im(f) \hookrightarrow A \times B$ is a relation.  So we say the inclusion of $Rel$ into $Span(Set)$ is a reflection.  (This is a slightly misleading term: there’s an adjoint to the inclusion, but it’s not an adjoint equivalence.  “Reflecting” twice may not get you back where you started, or anywhere isomorphic to it.)

(Edit: Actually, this is a bit wrong.  See the comments below.  What’s true is that the hom-categories of $Rel$ have reflective inclusions into the hom-categories of $Span(Set)$.  Here, we think of $Rel$ as a 2-category because it’s naturally enriched in posets.  Then these reflective inclusions of hom-categories can be used to build  a lax functor from $Span(Set)$ to $Rel$ – but not an actual functor.)

So a slightly more general question is: if $\mathbb{V}$ is a monoidal category, and $\mathbb{V}' \subset \mathbb{V}$ is a  “reflective subcategory“, can we make $\mathbb{V}'$ into a monoidal category just by defining $A' \otimes ' B'$ (the product in $\mathbb{V}'$) to be the reflection $r(A' \otimes B')$ of the original product?   This is the one-object version of a question about bicategories.  Namely, say that $\mathbb{S}$ is a bicategory, and $\mathbb{S}'$ is a sub-bicategory such that every pair of objects gives a reflective subcategory: $\mathbb{S}' (A,B) \subset \mathbb{S}(A,B)$ has a reflection.  Then can we “pull” the composition of morphisms in $\mathbb{S}$ back to $\mathbb{S}'$?

The answer is no: this just doesn’t work in general.  For spans of sets, and relations, it works: composing spans essentially “counts paths” which relate elements $A$ and $B$, whereas composing relations only keeps track of whether or not there is a path.  However, composing spans which come from relations, and then squashing them back down to relations again, agrees with the composite in $Rel$ (the squashing just tells whether the set of paths from $A$ to $B$ by a sequence of relations is empty or not).  But in the case of $Span(Cat)$ and some reflective subcategory – among other possible examples – associativity and unit axioms will break, unless the reflections $r_{A,B}$ are specially tuned.  This isn’t to say that we can’t make $\mathbb{V}'$ a monoidal category (or $\mathbb{S}'$ a bicategory).  It just means that pulling back $\otimes$ or $\circ$ along the reflection won’t work.  But there is a theorem that says we can always promote such an inclusion into one where this works.

So what’s an instance of all this?  A distributor (again, often called “profunctor”) $\Phi : \mathbb{A} \nrightarrow \mathbb{B}$ from a category $\mathbb{A}$ to $\mathbb{B}$ is actually a functor $\phi : \mathbb{B}^{op} \times \mathbb{A} \rightarrow Sets$.  Then there’s a bicategory $Dist$, where for each objects there’s a category $Dist(\mathbb{A},\mathbb{B})$.  Distributors represent, in some sense, a categorification of relations. (This observation follows the periodic table of category theory, in which a 1-category is a category, a 0-category is a set, and a (-1)-category is a truth value.  There’s a 1-category of relations, with hom-sets $Rel(A,B)$, and each one is a map from $B \times A$ into truth values, specifying whether a pair $(b,a)$ is related.)

The most elementary example of a distributor is the “hom-set” construction, where $\Phi (\mathbb{A},\mathbb{B}) = hom(\mathbb{A},\mathbb{B})$, which is indeed covariant in $\mathbb{A}$ and contravariant in $\mathbb{B}$.  A way to see the general case in that $\Phi$ obviously determines a functor from $\mathbb{A}$ into presheaves on $\mathbb{B}$: $\Phi : \mathbb{A} \rightarrow \hat{\mathbb{B}}$, where $\hat{\mathbb{B}} = Psh(\mathbb{B})$ is the category $hom(\mathbb{B},Sets)$.

In fact, given a functor $F : \mathbb{A} \rightarrow \mathbb{B}$, we can define two different distributors:

$\Phi^F : \mathbb{B} \nrightarrow \mathbb{A}$ with $\Phi^F(A,B) = Hom_{\mathbb{B}}(FA,B)$

and

$\Phi_F : \mathbb{A} \nrightarrow \mathbb{B}$ with $\Phi_F(B,A) = Hom_{\mathbb{B}}(B,FA)$

(Remember, these $\Phi$ are functors from the product into $Sets$: so they are just taking hom-sets here in $\mathbb{B}$ in one direction or the other.)  This much is a tautology: putting a value in $\mathbb{A}$ in leaves a free variable, but the point is that $\hat{\mathbb{B}}$ can be interpreted as a category of “big objects of $\mathbb{B}$“.  This is since the Yoneda embedding $Y : B \hookrightarrow \mathbb{B}$ embeds $\mathbb{B}$ by taking each object $b \in B$ to the presentable presheaf $hom_B(-,b)$ which assigns each object the set of morphisms into $b$, so $\hat{\mathbb{B}}$ has “extended” objects of $\mathbb{B}$.

So distributors like $\Phi$ are “generalized functors” into $\mathbb{B}$ – and the idea is that this is in roughly the same way that “distributions” are to be seen as “generalized functions”, hence the name.  (Benabou now prefers to use the name “profunctor” to refer only to those distributors which map to “pro-objects” in $\hat{\mathbb{B}}$, which are just special presheaves, namely the “flat” ones.)

Now we have an idea that there is a bicategory $Dist$, whose hom-categories $Dist(\mathbb{A},\mathbb{B})$ consist of distributors (and natural transformations), and that the usual functors (which can be seen as distributors which only happen to land in the image of $\mathbb{B}$ under the Yoneda embedding) form a sub-bicategory: that is, post-composition with $Y$ turns a functor into a distributor.

But moreover, this operation has an adjoint: functors out of $\mathbb{B}$ can be “lifted” to functors out of $\hat{\mathbb{B}}$, just by taking the Kan extension of a functor $G : \mathbb{B} \rightarrow \mathbb{X}$ along $Y$.  This will work (pointwise, even), as long as $\mathbb{X}$ is cocomplete, so that we can basically “add up” contributions from the objects of $\mathbb{B}$ by taking colimits.  In the special case where $\mathbb{X} = \hat{\mathbb{C}}$ for some other category $\mathbb{C}$, then this tells us how to get composition of distributors $Dist(\mathbb{A},\mathbb{B}) \times Dist(\mathbb{B},\mathbb{C})\rightarrow Dist(\mathbb{A},\mathbb{C})$.

Now, for a functor $F$, there are straightforward unit and counit natural transformations which makes $\Phi^F$ (the image of $F$ under the embedding of $Cat$ into $Dist$) a left adjoint for $\Phi_F$.  So we’ve embedded $Cat$ into $Dist$ in such a way that every functor has a right adjoint.  What about $Span(Cat)$?  In general, given a bicategory $B$, we can construe $Span(B)$ as a tricategory, which contains $B$, in such a way that every morphism of $B$ has an ambidextrous adjoint (both left and right adjoint).  (There’s work on this by Toby Kenney and Dorette Pronk, and Alex Hoffnung has also been looking at this recently.)  So how does $Span(Cat)$ relate to $Dist$?

One statement is that a distributor $\Phi : \mathbb{A} \nrightarrow \mathbb{B}$ can be seen as a special kind of span, namely:

$\mathbb{A} \stackrel{q}{\longleftarrow} Elt(\Phi) \stackrel{p}{\longrightarrow} \mathbb{B}$

where $Elt(\Phi)$ consists of all the “elements of $\Phi$” (in particular, pasting together all the images in $Sets$ of pairs $(A,B)$ and the set maps that come from morphisms between them in $\mathbb{B}^{op} \times \mathbb{A}$).  (As an aside: Benabou also explained how a cospan, $\mathbb{A} \rightarrow C(\Phi) \leftarrow \mathbb{B}$ can be got from a distributor.  The objects of $C(\Phi)$ are just the disjoint union of those from $\mathbb{A}$ and $\mathbb{B}$, and the hom-sets are just taken from either $\mathbb{A}$, or $\mathbb{B}$, or as the sets given by $\Phi$, depending on the situation.  Then the span we just described completes a pullback square opposite this cospan – it’s a comma category.)

These spans $(Elt(\Phi),p,q)$ end up having some special properties that result from how they’re constructed.  In particular, $p$ will be an op-fibration and $q$ will be a fibration (this, for instance, is alifting property that let one lift morphisms – since the morphisms are found as the images of the original distributor, this makes sense).  Also, the fibres of $(p,q)$ are discrete (these are by definition the images of identity morphisms, so naturally they’re discrete categories).  Finally, these properties (fibration, op-fibration, and discrete fibres) are enough to guarantee that a given span is (isomorphic to) one that comes from a distributor.  So we have an embedding $i : Dist \rightarrow Span(Cat)$.

What’s more, it’s a reflective embedding, because we can always mangle any span to get a new one where these properties hold: it’s enough to force fibres to be discrete by taking their $\pi_0$ – the connected components.  The other properties will then follow.  But notice that this is a very nontrivial thing to do: in general, the fibres of $(p,q)$ could be any sort of category, and this operation turns them into sets (of isomorphism classes).  So there’s an adjunction between $i$ and $\pi_0$, and $Dist$ is a reflective sub-bicategory of $Span(Cat)$.  But the severity of $\pi_0$ ends up meaning that this doesn’t get along well with composition – the composition of distributors (described above) is not related to composition of spans (which works by weak pullback) via this reflection in a naive way.  However, the theorem mentioned above means that there will be SOME reflecction that makes the compositions get along.  It just may not be as nice as this one.

This is kind of surprising, and the ideal punchline to go here would be to say what that reflection is like, but I don’t know the answer to that question just now.  Anyone else know?

Thanks to Bob Coecke, here are some pictures of me, a few of the people from ComLab, and Jean Benabou at dinner at the Oxford University Club, with a variety of dopey expressions as Bob snapped the pictures unexpectedly.  Thanks Bob.

Now for a more sketchy bunch of summaries of some talks presented at the HGTQGR workshop.  I’ll organize this into a few themes which appeared repeatedly and which roughly line up with the topics in the title: in this post, variations on TQFT, plus 2-group and higher forms of gauge theory; in the next post, gerbes and cohomology, plus talks on discrete models of quantum gravity and suchlike physics.

## TQFT and Variations

I start here for no better reason than the personal one that it lets me put my talk first, so I’m on familiar ground to start with, for which reason also I’ll probably give more details here than later on.  So: a TQFT is a linear representation of the category of cobordisms – that is, a (symmetric monoidal) functor $nCob \rightarrow Vect$, in the notation I mentioned in the first school post.  An Extended TQFT is a higher functor $nCob_k \rightarrow k-Vect$, representing a category of cobordisms with corners into a higher category of k-Vector spaces (for some definition of same).  The essential point of my talk is that there’s a universal construction that can be used to build one of these at $k=2$, which relies on some way of representing $nCob_2$ into $Span(Gpd)$, whose objects are groupoids, and whose morphisms in $Hom(A,B)$ are pairs of groupoid homomorphisms $A \leftarrow X \rightarrow B$.  The 2-morphisms have an analogous structure.  The point is that there’s a 2-functor $\Lambda : Span(Gpd) \rightarrow 2Vect$ which is takes representations of groupoids, at the level of objects; for morphisms, there is a “pull-push” operation that just uses the restricted and induced representation functors to move a representation across a span; the non-trivial (but still universal) bit is the 2-morphism map, which uses the fact that the restriction and induction functors are bi-ajdoint, so there are units and counits to use.  A construction using gauge theory gives groupoids of connections and gauge transformations for each manifold or cobordism.  This recovers a form of the Dijkgraaf-Witten model.  In principle, though, any way of getting a groupoid (really, a stack) associated to a space functorially will give an ETQFT this way.  I finished up by suggesting what would need to be done to extend this up to higher codimension.  To go to codimension 3, one would assign an object (codimension-3 manifold) a 3-vector space which is a representation 2-category of 2-groupoids of connections valued in 2-groups, and so on.  There are some theorems about representations of n-groupoids which would need to be proved to make this work.

The fact that different constructions can give groupoids for spaces was used by the next speaker, Thomas Nicklaus, whose talk described another construction that uses the $\Lambda$ I mentioned above.  This one produces “Equivariant Dijkgraaf-Witten Theory”.  The point is that one gets groupoids for spaces in a new way.  Before, we had, for a space $M$ a groupoid $\mathcal{A}_G(M)$ whose objects are $G$-connections (or, put another way, bundles-with-connection) and whose morphisms are gauge transformations.  Now we suppose that there’s some group $J$ which acts weakly (i.e. an action defined up to isomorphism) on $\mathcal{A}_G(M)$.  We think of this as describing “twisted bundles” over $M$.  This is described by a quotient stack $\mathcal{A}_G // J$ (which, as a groupoid, gets some extra isomorphisms showing where two objects are related by the $J$-action).  So this gives a new map $nCob \rightarrow Span(Gpd)$, and applying $\Lambda$ gives a TQFT.  The generating objects for the resulting 2-vector space are “twisted sectors” of the equivariant DW model.  There was some more to the talk, including a description of how the DW model can be further mutated using a cocycle in the group cohomology of $G$, but I’ll let you look at the slides for that.

Next up was Jamie Vicary, who was talking about “(1,2,3)-TQFT”, which is another term for what I called “Extended” TQFT above, but specifying that the objects are 1-manifolds, the morphisms 2-manifolds, and the 2-morphisms are 3-manifolds.  He was talking about a theorem that identifies oriented TQFT’s of this sort with “anomaly-free modular tensor categories” – which is widely believed, but in fact harder than commonly thought.  It’s easy enough that such a TQFT $Z$ corresponds to a MTC – it’s the category $Z(S^1)$ assigned to the circle.  What’s harder is showing that the TQFT’s are equivalent functors iff the categories are equivalent.  This boils down, historically, to the difficulty of showing the category is rigid.  Jamie was talking about a project with Bruce Bartlett and Chris Schommer-Pries, whose presentation of the cobordism category (described in the school post) was the basis of their proof.

Part of it amounts to giving a description of the TQFT in terms of certain string diagrams.  Jamie kindly credited me with describing this point of view to him: that the codimension-2 manifolds in a TQFT can be thought of as “boundaries in space” – codimension-1 manifolds are either time-evolving boundaries, or else slices of space in which the boundaries live; top-dimension cobordisms are then time-evolving slices of space-with-boundary.  (This should be only a heuristic way of thinking – certainly a generic TQFT has no literal notion of “time-evolution”, though in that (2+1) quantum gravity can be seen as a TQFT, there’s at least one case where this picture could be taken literally.)  Then part of their proof involves showing that the cobordisms can be characterized by taking vector spaces on the source and target manifolds spanned by the generating objects, and finding the functors assigned to cobordisms in terms of sums over all “string diagrams” (particle worldlines, if you like) bounded by the evolving boundaries.  Jamie described this as a “topological path integral”.  Then one has to describe the string diagram calculus – ridigidy follows from the “yanking” rule, for instance, and this follows from Morse theory as in Chris’ presentation of the cobordism category.

There was a little more discussion about what the various properties (proved in a similar way) imply.  One is “cloaking” – the fact that a 2-morphism which “creates a handle” is invisible to the string diagrams in the sense that it introduces a sum over all diagrams with a string “looped” around the new handle, but this sum gives a result that’s equal to the original map (in any “pivotal” tensor category, as here).

Chronologically before all these, one of the first talks on such a topic was by Rafael Diaz, on Homological Quantum Field Theory, or HLQFT for short, which is a rather different sort of construction.  Remember that Homotopy QFT, as described in my summary of Tim Porter’s school sessions, is about linear representations of what I’ll for now call $Cob(d,B)$, whose morphisms are $d$-dimensional cobordisms equipped with maps into a space $B$ up to homotopy.  HLQFT instead considers cobordisms equipped with maps taken up to homology.

Specifically, there’s some space $M$, say a manifold, with some distinguished submanifolds (possibly boundary components; possibly just embedded submanifolds; possibly even all of $M$ for a degenerate case).  Then we define $Cob_d^M$ to have objects which are $(d-1)$-manifolds equipped with maps into $M$ which land on the distinguished submanifolds (to make composition work nicely, we in fact assume they map to a single point).  Morphisms in $Cob_d^M$ are trickier, and look like $(N,\alpha, \xi)$: a cobordism $N$ in this category is likewise equipped with a map $\alpha$ from its boundary into $M$ which recovers the maps on its objects.  That $\xi$ is a homology class of maps from $N$ to $M$, which agrees with $\alpha$.  This forms a monoidal category as with standard cobordisms.  Then HLQFT is about representations of this category.  One simple case Rafael described is the dimension-1 case, where objects are (ordered sets of) points equipped with maps that pick out chosen submanifolds of $M$, and morphisms are just braids equipped with homology classes of “paths” joining up the source and target submanifolds.  Then a representation might, e.g., describe how to evolve a homology class on the starting manifold to one on the target by transporting along such a path-up-to-homology.  In higher dimensions, the evolution is naturally more complicated.

A slightly looser fit to this section is the talk by Thomas Krajewski, “Quasi-Quantum Groups from Strings” (see this) – he was talking about how certain algebraic structures arise from “string worldsheets”, which are another way to describe cobordisms.  This does somewhat resemble the way an algebraic structure (Frobenius algebra) is related to a 2D TQFT, but here the string worldsheets are interacting with 3-form field, $H$ (the curvature of that 2-form field $B$ of string theory) and things needn’t be topological, so the result is somewhat different.

Part of the point is that quantizing such a thing gives a higher version of what happens for quantizing a moving particle in a gauge field.  In the particle case, one comes up with a line bundle (of which sections form the Hilbert space) and in the string case one comes up with a gerbe; for the particle, this involves associated 2-cocycle, and for the string a 3-cocycle; for the particle, one ends up producing a twisted group algebra, and for the string, this is where one gets a “quasi-quantum group”.  The algebraic structures, as in the TQFT situation, come from, for instance, the “pants” cobordism which gives a multiplication and a comultiplication (by giving maps $H \otimes H \rightarrow H$ or the reverse, where $H$ is the object assigned to a circle).

There is some machinery along the way which I won’t describe in detail, except that it involves a tricomplex of forms – the gradings being form degree, the degree of a cocycle for group cohomology, and the number of overlaps.  As observed before, gerbes and their higher versions have transition functions on higher numbers of overlapping local neighborhoods than mere bundles.  (See the paper above for more)

## Higher Gauge Theory

The talks I’ll summarize here touch on various aspects of higher-categorical connections or 2-groups (though at least one I’ll put off until later).  The division between this and the section on gerbes is a little arbitrary, since of course they’re deeply connected, but I’m making some judgements about emphasis or P.O.V. here.

Apart from giving lectures in the school sessions, John Huerta also spoke on “Higher Supergroups for String Theory”, which brings “super” (i.e. $\mathbb{Z}_2$-graded) objects into higher gauge theory.  There are “super” versions of vector spaces and manifolds, which decompose into “even” and “odd” graded parts (a.k.a. “bosonic” and “fermionic” parts).  Thus there are “super” variants of Lie algebras and Lie groups, which are like the usual versions, except commutation properties have to take signs into account (e.g. a Lie superalgebra’s bracket is commutative if the product of the grades of two vectors is odd, anticommutative if it’s even).  Then there are Lie 2-algebras and 2-groups as well – categories internal to this setting.  The initial question has to do with whether one can integrate some Lie 2-algebra structures to Lie 2-group structures on a spacetime, which depends on the existence of some globally smooth cocycles.  The point is that when spacetime is of certain special dimensions, this can work, namely dimensions 3, 4, 6, and 10.  These are all 2 more than the real dimensions of the four real division algebras, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$.  It’s in these dimensions that Lie 2-superalgebras can be integrated to Lie 2-supergroups.  The essential reason is that a certain cocycle condition will hold because of the properties of a form on the Clifford algebras that are associated to the division algebras.  (John has some related material here and here, though not about the 2-group case.)

Since we’re talking about higher versions of Lie groups/algebras, an important bunch of concepts to categorify are those in representation theory.  Derek Wise spoke on “2-Group Representations and Geometry”, based on work with Baez, Baratin and Freidel, most fully developed here, but summarized here.  The point is to describe the representation theory of Lie 2-groups, in particular geometrically.  They’re to be represented on (in general, infinite-dimensional) 2-vector spaces of some sort, which is chosen to be a category of measurable fields of Hilbert spaces on some measure space, which is called $H^X$ (intended to resemble, but not exactly be the same as, $Hilb^X$, the space of “functors into $Hilb$ from the space $X$, the way Kapranov-Voevodsky 2-vector spaces can be described as $Vect^k$).  The first work on this was by Crane and Sheppeard, and also Yetter.  One point is that for 2-groups, we have not only representations and intertwiners between them, but 2-intertwiners between these.  One can describe these geometrically – part of which is a choice of that measure space $(X,\mu)$.

This done, we can say that a representation of a 2-group is a 2-functor $\mathcal{G} \rightarrow H^X$, where $\mathcal{G}$ is seen as a one-object 2-category.  Thinking about this geometrically, if we concretely describe $\mathcal{G}$ by the crossed module $(G,H,\rhd,\partial)$, defines an action of $G$ on $X$, and a map $X \rightarrow H^*$ into the character group, which thereby becomes a $G$-equivariant bundle.  One consequence of this description is that it becomes possible to distinguish not only irreducible representations (bundles over a single orbit) and indecomposible ones (where the fibres are particularly simple homogeneous spaces), but an intermediate notion called “irretractible” (though it’s not clear how much this provides).  An intertwining operator between reps over $X$ and $Y$ can be described in terms of a bundle of Hilbert spaces – which is itself defined over the pullback of $X$ and $Y$ seen as $G$-bundles over $H^*$.  A 2-intertwiner is a fibre-wise map between two such things.  This geometric picture specializes in various ways for particular examples of 2-groups.  A physically interesting one, which Crane and Sheppeard, and expanded on in that paper of [BBFW] up above, deals with the Poincaré 2-group, and where irreducible representations live over mass-shells in Minkowski space (or rather, the dual of $H \cong \mathbb{R}^{3,1}$).

Moving on from 2-group stuff, there were a few talks related to 3-groups and 3-groupoids.  There are some new complexities that enter here, because while (weak) 2-categories are all (bi)equivalent to strict 2-categories (where things like associativity and the interchange law for composing 2-cells hold exactly), this isn’t true for 3-categories.  The best strictification result is that any 3-category is (tri)equivalent to a Gray category – where all those properties hold exactly, except for the interchange law $(\alpha \circ \beta) \cdot (\alpha ' \circ \beta ') = (\alpha \cdot \alpha ') \circ (\beta \circ \beta ')$ for horizontal and vertical compositions of 2-cells, which is replaced by an “interchanger” isomorphism with some coherence properties.  John Barrett gave an introduction to this idea and spoke about “Diagrams for Gray Categories”, describing how to represent morphisms, 2-morphisms, and 3-morphisms in terms of higher versions of “string” diagrams involving (piecewise linear) surfaces satisfying some properties.  He also carefully explained how to reduce the dimensions in order to make them both clearer and easier to draw.  Bjorn Gohla spoke on “Mapping Spaces for Gray Categories”, but since it was essentially a shorter version of a talk I’ve already posted about, I’ll leave that for now, except to point out that it linked to the talk by Joao Faria Martins, “3D Holonomy” (though see also this paper with Roger Picken).

The point in Joao’s talk starts with the fact that we can describe holonomies for 3-connections on 3-bundles valued in Gray-groups (i.e. the maximally strict form of a general 3-group) in terms of Gray-functors $hol: \Pi_3(M) \rightarrow \mathcal{G}$.  Here, $\Pi_3(M)$ is the fundamental 3-groupoid of $M$, which turns points, paths, homotopies of paths, and homotopies of homotopies into a Gray groupoid (modulo some technicalities about “thin” or “laminated”  homotopies) and $\mathcal{G}$ is a gauge Gray-group.  Just as a 2-group can be represented by a crossed module, a Gray (3-)group can be represented by a “2-crossed module” (yes, the level shift in the terminology is occasionally confusing).  This is a chain of groups $L \stackrel{\delta}{\rightarrow} E \stackrel{\partial}{\rightarrow} G$, where $G$ acts on the other groups, together with some structure maps (for instance, the Peiffer commutator for a crossed module becomes a lifting $\{ ,\} : E \times E \rightarrow L$) which all fit together nicely.  Then a tri-connection can be given locally by forms valued in the Lie algebras of these groups: $(\omega , m ,\theta)$ in  $\Omega^1 (M,\mathfrak{g} ) \times \Omega^2 (M,\mathfrak{e}) \times \Omega^3(M,\mathfrak{l})$.  Relating the global description in terms of $hol$ and local description in terms of $(\omega, m, \theta)$ is a matter of integrating forms over paths, surfaces, or 3-volumes that give the various $j$-morphisms of $\Pi_3(M)$.  This sort of construction of parallel transport as functor has been developed in detail by Waldorf and Schreiber (viz. these slides, or the full paper), some time ago, which is why, thematically, they’re the next two speakers I’ll summarize.

Konrad Waldorf spoke about “Abelian Gauge Theories on Loop Spaces and their Regression”.  (For more, see two papers by Konrad on this)  The point here is that there is a relation between two kinds of theories – string theory (with $B$-field) on a manifold $M$, and ordinary $U(1)$ gauge theory on its loop space $LM$.  The relation between them goes by the name “regression” (passing from gauge theory on $LM$ to string theory on $M$), or “transgression”, going the other way.  This amounts to showing an equivalence of categories between [principal $U(1)$-bundles with connection on $LM$] and [$U(1)$-gerbes with connection on $M$].  This nicely gives a way of seeing how gerbes “categorify” bundles, since passing to the loop space – whose points are maps $S^1 \rightarrow M$ means a holonomy functor is now looking at objects (points in $LM$) which would be morphisms in the fundamental groupoid of $M$, and morphisms which are paths of loops (surfaces in $M$ which trace out homotopies).  So things are shifted by one level.  Anyway, Konrad explained how this works in more detail, and how it should be interpreted as relating connections on loop space to the $B$-field in string theory.

Urs Schreiber kicked the whole categorification program up a notch by talking about $\infty$-Connections and their Chern-Simons Functionals .  So now we’re getting up into $\infty$-categories, and particularly $\infty$-toposes (see Jacob Lurie’s paper, or even book if so inclined to find out what these are), and in particular a “cohesive topos”, where derived geometry can be developed (Urs suggested people look here, where a bunch of background is collected). The point is that $\infty$-topoi are good for talking about homotopy theory.  We want a setting which allows all that structure, but also allows us to do differential geometry and derived geometry.  So there’s a “cohesive” $\infty$-topos called $Smooth\infty Gpds$, of “sheaves” (in the $\infty$-topos sense) of $\infty$-groupoids on smooth manifolds.  This setting is the minimal common generalization of homotopy theory and differential geometry.

This is about a higher analog of this setup: since there’s a smooth classifying space (in fact, a Lie groupoid) for $G$-bundles, $BG$, there’s also an equivalence between categories $G-Bund$ of $G$-principal bundles, and $SmoothGpd(X,BG)$ (of functors into $BG$).  Moreover, there’s a similar setup with $BG_{conn}$ for bundles with connection.  This can be described topologically, or there’s also a “differential refinement” to talk about the smooth situation.  This equivalence lives within a category of (smooth) sheaves of groupoids.  For higher gauge theory, we want a higher version as in $Smooth \infty Gpds$ described above.  Then we should get an equivalence – in this cohesive topos – of $hom(X,B^n U(1))$ and a category of $U(1)$-$(n-1)$-gerbes.

Then the part about the  “Chern-Simons functionals” refers to the fact that CS theory for a manifold (which is a kind of TQFT) is built using an action functional that is found as an integral of the forms that describe some $U(1)$-connection over the manifold.  (Then one does a path-integral of this functional over all connections to find partition functions etc.)  So the idea is that for these higher $U(1)$-gerbes, whose classifying spaces we’ve just described, there should be corresponding functionals.  This is why, as Urs remarked in wrapping up, this whole picture has an explicit presentation in terms of forms.  Actually, in terms of Cech-cocycles (due to the fact we’re talking about gerbes), whose coefficients are taken in sheaves of complexes (this is the derived geometry part) of differential forms whose coefficients are in $L_\infty$-algebroids (the $\infty$-groupoid version of Lie algebras, since in general we’re talking about a theory with gauge $\infty$-groupoids now).

Whew!  Okay, that’s enough for this post.  Next time, wrapping up blogging the workshop, finally.

Continuing from the previous post, there are a few more lecture series from the school to talk about.

## Higher Gauge Theory

The next was John Huerta’s series on Higher Gauge Theory from the point of view of 2-groups.  John set this in the context of “categorification”, a slightly vague program of replacing set-based mathematical ideas with category-based mathematical ideas.  The general reason for this is to get an extra layer of “maps between things”, or “relations between relations”, etc. which tend to be expressed by natural transformations.  There are various ways to go about this, but one is internalization: given some sort of structure, the relevant example in this case being “groups”, one has a category ${Groups}$, and can define a 2-group as a “category internal to ${Groups}$“.  So a 2-group has a group of objects, a group of morphisms, and all the usual maps (source and target for morphisms, composition, etc.) which all have to be group homomorphisms.  It should be said that this all produces a “strict 2-group”, since the objects $G$ necessarily form a group here.  In particular, $m : G \times G \rightarrow G$ satisfies group axioms “on the nose” – which is the only way to satisfy them for a group made of the elements of a set, but for a group made of the elements of a category, one might require only that it commute up to isomorphism.  A weak 2-group might then be described as a “weak model” of the theory of groups in $Cat$, but this whole approach is much less well-understood than the strict version as one goes to general n-groups.

Now, as mentioned in the previous post, there is a 1-1 correspondence between 2-groups and crossed modules (up to equivalence): given a crossed module $(G,H,\partial,\rhd)$, there’s a 2-group $\mathcal{G}$ whose objects are $G$ and whose morphisms are $G \ltimes H$; given a 2-group $\mathcal{G}$ with objects $G$, there’s a crossed module $(G, Aut(1_G),1,m)$.  (The action $m$ acts on a morphism in such as way as to act by multiplication on its source and target).  Then, for instance, the Peiffer identity for crossed modules (see previous post) is a consequence of the fact that composition of morphisms is supposed to be a group homomorphism.

Looking at internal categories in [your favourite setting here] isn’t the only way to do categorification, but it does produce some interesting examples.  Baez-Crans 2-vector spaces are defined this way (in $Vect$), and built using these are Lie 2-algebras.  Looking for a way to integrate Lie 2-algebras up to Lie 2-groups (which are internal categories in Lie groups) brings us back to the current main point.  This is the use of 2-groups to do higher gauge theory.  This requires the use of “2-bundles”.  To explain these, we can say first of all that a “2-space” is an internal category in $Spaces$ (whether that be manifolds, or topological spaces, or what-have-you), and that a (locally trivial) 2-bundle should have a total 2-space $E$, a base 2-space $M$, and a (functorial) projection map $p : E \rightarrow M$, such that there’s some open cover of $M$ by neighborhoods $U_i$ where locally the bundle “looks like” $\pi_i : U_i \times F \rightarrow U_i$, where $F$ is the fibre of the bundle.  In the bundle setting, “looks like” means “is isomorphic to” by means of isomorphisms $f_i : E_{U_i} \rightarrow U_i \times F$.  With 2-bundles, it’s interpreted as “is equivalent to” in the categorical sense, likewise by maps $f_i$.

Actually making this precise is a lot of work when $M$ is a general 2-space – even defining open covers and setting up all the machinery properly is quite hard.  This has been done, by Toby Bartels in his thesis, but to keep things simple, John restricted his talk to the case where $M$ is just an ordinary manifold (thought of as a 2-space which has only identity morphisms).   Then a key point is that there’s an analog to how (principal) $G$-bundles (where $F \cong G$ as a $G$-set) are classified up to isomorphism by the first Cech cohomology of the manifold, $\check{H}^1(M,G)$.  This works because one can define transition functions on double overlaps $U_{ij} := U_i \cap U_j$, by $g_{ij} = f_i f_j^{-1}$.  Then these $g_{ij}$ will automatically satisfy the 1-cocycle condidion ($g_{ij} g_{jk} = g_{ik}$ on the triple overlap $U_{ijk}$) which means they represent a cohomology class $[g] = \in \check{H}^1(M,G)$.

A comparable thing can be said for the “transition functors” for a 2-bundle – they’re defined superficially just as above, except that being functors, we can now say there’s a natural isomorphism $h_{ijk} : g_{ij}g_{jk} \rightarrow g_{ik}$, and it’s these $h_{ijk}$, defined on triple overlaps, which satisfy a 2-cocycle condition on 4-fold intersections (essentially, the two ways to compose them to collapse $g_{ij} g_{jk} g_{kl}$ into $g_{il}$ agree).  That is, we have $g_{ij} : U_{ij} \rightarrow Ob(\mathcal{G})$ and $h_{ijk} : U_{ijk} \rightarrow Mor(\mathcal{G})$ which fit together nicely.  In particular, we have an element $[h] \in \check{H}^2(M,G)$ of the second Cech cohomology of $M$: “principal $\mathcal{G}$-bundles are classified by second Cech cohomology of $M$“.  This sort of thing ties in to an ongoing theme of the later talks, the relationship between gerbes and higher cohomology – a 2-bundle corresponds to a “gerbe”, or rather a “1-gerbe”.  (The consistent terminology would have called a bundle a “0-gerbe”, but as usual, terminology got settled before the general pattern was understood).

Finally, having defined bundles, one usually defines connections, and so we do the same with 2-bundles.  A connection on a bundle gives a parallel transport operation for paths $\gamma$ in $M$, telling how to identify the fibres at points along $\gamma$ by means of a functor $hol : P_1(M) \rightarrow G$, thinking of $G$ as a category with one object, and where $P_1(M)$ is the path groupoid whose objects are points in $M$ and whose morphisms are paths (up to “thin” homotopy). At least, it does so once we trivialize the bundle around $\gamma$, anyway, to think of it as $M \times G$ locally – in general we need to get the transition functions involved when we pass into some other local neighborhood.  A connection on a 2-bundle is similar, but tells how to parallel transport fibres not only along paths, but along homotopies of paths, by means of $hol : P_2(M) \rightarrow \mathcal{G}$, where $\mathcal{G}$ is seen as a 2-category with one object, and $P_2(M)$ now has 2-morphisms which are (essentially) homotopies of paths.

Just as connections can be described by 1-forms $A$ valued in $Lie(G)$, which give $hol$ by integrating, a similar story exists for 2-connections: now we need a 1-form $A$ valued in $Lie(G)$ and a 2-form $B$ valued in $Lie(H)$.  These need to satisfy some relations, essentially that the curvature of $A$ has to be controlled by $B$.   Moreover, that $B$ is related to the $B$-field of string theory, as I mentioned in the post on the pre-school… But really, this is telling us about the Lie 2-algebra associated to $\mathcal{G}$, and how to integrate it up to the group!

## Infinite Dimensional Lie Theory and Higher Gauge Theory

This series of talks by Christoph Wockel returns us to the question of “integrating up” to a Lie group $G$ from a Lie algebra $\mathfrak{g} = Lie(G)$, which is seen as the tangent space of $G$ at the identity.  This is a well-understood, well-behaved phenomenon when the Lie algebras happen to be finite dimensional.  Indeed the classification theorem for the classical Lie groups can be got at in just this way: a combinatorial way to characterize Lie algebras using Dynkin diagrams (which describe the structure of some weight lattice), followed by a correspondence between Lie algebras and Lie groups.  But when the Lie algebras are infinite dimensional, this just doesn’t have to work.  It may be impossible to integrate a Lie algebra up to a full Lie group: instead, one can only get a little neighborhood of the identity.  The point of such infinite-dimensional groups, and ultimately their representation theory, is to deal with string groups that have to do with motions of extended objects.  Christoph Wockel was describing a result which says that, going to 2-groups, this problem can be overcome.  (See the relevant paper here.)

The first lecture in the series presented some background on a setting for infinite dimensional manifolds.  There are various approaches, a popular one being Frechet manifolds, but in this context, the somewhat weaker notion of locally convex spaces is sufficient.  These are “locally modelled” by (infinite dimensional) locally convex vector spaces, the way finite dimensonal manifolds are locally modelled by Euclidean space.  Being locally convex is enough to allow them to support a lot of differential calculus: one can find straight-line paths, locally, to define a notion of directional derivative in the direction of a general vector.  Using this, one can build up definitions of differentiable and smooth functions, derivatives, and integrals, just by looking at the restrictions to all such directions.  Then there’s a fundamental theorem of calculus, a chain rule, and so on.

At this point, one has plenty of differential calculus, and it becomes interesting to bring in Lie theory.  A Lie group is defined as a group object in the category of manifolds and smooth maps, just as in the finite-dimensional case.  Some infinite-dimensional Lie groups of interest would include: $G = Diff(M)$, the group of diffeomorphisms of some compact manifold $M$; and the group of smooth functions $G = C^{\infty}(M,K)$ from $M$ into some (finite-dimensional) Lie group $K$ (perhaps just $\mathbb{R}$), with the usual pointwise multiplication.  These are certainly groups, and one handy fact about such groups is that, if they have a manifold structure near the identity, on some subset that generates $G$ as a group in a nice way, you can extend the manifold structure to the whole group.  And indeed, that happens in these examples.

Well, next we’d like to know if we can, given an infinite dimensional Lie algebra $X$, “integrate up” to a Lie group – that is, find a Lie group $G$ for which $X \cong T_eG$ is the “infinitesimal” version of $G$.  One way this arises is from central extensions.  A central extension of Lie group $G$ by $Z$ is an exact sequence $Z \hookrightarrow \hat{G} \twoheadrightarrow G$ where (the image of) $Z$ is in the centre of $\hat{G}$.  The point here is that $\hat{G}$ extends $G$.  This setup makes $\hat{G}$ is a principal $Z$-bundle over $G$.

Now, finding central extensions of Lie algebras is comparatively easy, and given a central extension of Lie groups, one always falls out of the induced maps.  There will be an exact sequence of Lie algebras, and now the special condition is that there must exist a continuous section of the second map.  The question is to go the other way: given one of these, get back to an extension of Lie groups.  The problem of finding extensions of $G$ by $Z$, in particular as a problem of finding a bundle with connection having specified curvature, which brings us back to gauge theory.  One type of extension is the universal cover of $G$, which appears as $\pi_1(G) \hookrightarrow \hat{G} \twoheadrightarrow G$, so that the fibre is $\pi_1(G)$.

In general, whether an extension can exist comes down to a question about a cocycle: that is, if there’s a function $f : G \times G \rightarrow Z$ which is locally smooth (i.e. in some neighborhood in $G$), and is a cocyle (so that $f(g,h) + f(gh,k) = f(g,hk) + f(h,k)$), by the same sorts of arguments we’ve already seen a bit of.  For this reason, central extensions are classified by the cohomology group $H^2(G,Z)$.  The cocycle enables a “twisting” of the multiplication associated to a nontrivial loop in $G$, and is used to construct $\hat{G}$ (by specifying how multiplication on $G$ lifts to $\hat{G}$).  Given a  2-cocycle $\omega$ at the Lie algebra level (easier to do), one would like to lift that up the Lie group.  It turns out this is possible if the period homomorphism $per_{\omega} : \Pi_2(G) \rightarrow Z$ – which takes a chain $[\sigma]$ (with $\sigma : S^2 \rightarrow G$) to the integral of the original cocycle on it, $\int_{\sigma} \omega$ – lands in a discrete subgroup of $Z$. A popular example of this is when $Z$ is just $\mathbb{R}$, and the discrete subgroup is $\mathbb{Z}$ (or, similarly, $U(1)$ and $1$ respectively).  This business of requiring a cocycle to be integral in this way is sometimes called a “prequantization” problem.

So suppose we wanted to make the “2-connected cover” $\pi_2(G) \hookrightarrow \pi_2(G) \times_{\gamma} G \twoheadrightarrow G$ as a central extension: since $\pi_2(G)$ will be abelian, this is conceivable.  If the dimension of $G$ is finite, this is trivial (since $\pi_2(G) = 0$ in finite dimensions), which is why we need some theory  of infinite-dimensional manifolds.  Moreover, though, this may not work in the context of groups: the $\gamma$ in the extension $\pi_2(G) \times_{\gamma} G$ above needs to be a “twisting” of associativity, not multiplication, being lifted from $G$.  Such twistings come from the THIRD cohomology of $G$ (see here, e.g.), and describe the structure of 2-groups (or crossed modules, whichever you like).  In fact, the solution (go read the paper for more if you like) to define a notion of central extension for 2-groups (essentially the same as the usual definition, but with maps of 2-groups, or crossed modules, everywhere).  Since a group is a trivial kind of 2-group (with only trivial automorphisms of any element), the usual notion of central extension turns out to be a special case.  Then by thinking of $\pi_2(G)$ and $G$ as crossed modules, one can find a central extension which is like the 2-connected cover we wanted – though it doesn’t work as an extension of groups because we think of $G$ as the base group of the crossed module, and $\pi_2(G)$ as the second group in the tower.

The pattern of moving to higher group-like structures, higher cohomology, and obstructions to various constructions ran all through the workshop, and carried on in the next school session…

## Higher Spin Structures in String Theory

Hisham Sati gave just one school-lecture in addition to his workshop talk, but it was packed with a lot of material.  This is essentially about cohomology and the structures on manifolds to which cohomology groups describe the obstructions.  The background part of the lecture referenced this book by Fridrich, and the newer parts were describing some of Sati’s own work, in particular a couple of papers with Schreiber and Stasheff (also see this one).

The basic point here is that, for physical reasons, we’re often interested in putting some sort of structure on a manifold, which is really best described in terms of a bundle.  For instance, a connection or spin connection on spacetime lets us transport vectors or spinors, respectively, along paths, which in turn lets us define derivatives.  These two structures really belong on vector bundles or spinor bundles.  Now, if these bundles are trivial, then one can make the connections on them trivial as well by gauge transformation.  So having nontrivial bundles really makes this all more interesting.  However, this isn’t always possible, and so one wants to the obstruction to being able to do it.  This is typically a class in one of the cohomology groups of the manifold – a characteristic class.  There are various examples: Chern classes, Pontrjagin classes, Steifel-Whitney classes, and so on, each of which comes in various degrees $i$.  Each one corresponds to a different coefficient group for the cohomology groups – in these examples, the groups $U$ and $O$ which are the limits of the unitary and orthogonal groups (such as $O := O(\infty) \supset \dots \supset O(2) \supset O(1)$)

The point is that these classes are obstructions to building certain structures on the manifold $X$ – which amounts to finding sections of a bundle.  So for instance, the first Steifel-Whitney classes, $w_1(E)$ of a bundle $E$ are related to orientations, coming from cohomology with coefficients in $O(n)$.  Orientations for the manifold $X$ can be described in terms of its tangent bundle, which is an $O(n)$-bundle (tangent spaces carry an action of the rotation group).  Consider $X = S^1$, where we have actually $O(1) \simeq \mathbb{Z}_2$.  The group $H^1(S^1, \mathbb{Z}_2)$ has two elements, and there are two types of line bundle on the circle $S^1$: ones with a nowhere-zero section, like the trivial bundle; and ones without, like the Moebius strip.  The circle is orientable, because its tangent bundle is of the first sort.

Generally, an orientation can be put on $X$ if the tangent bundle, as a map $f : X \rightarrow B(O(n))$, can be lifted to a map $\tilde{f} : X \rightarrow B(SO(n))$ – that is, it’s “secretly” an $SO(n)$-bundle – the special orthogonal group respects orientation, which is what the determinant measures.  Its two values, $\pm 1$, are what’s behind the two classes of bundles.  (In short, this story relates to the exact sequence $1 \rightarrow SO(n) \rightarrow O(n) \stackrel{det}{\rightarrow} O(1) = \mathbb{Z}_2 \rightarrow 1$; in just the same way we have big groups $SO$, $Spin$, and so forth.)

So spin structures have a story much like the above, but where the exact sequence $1 \rightarrow \mathbb{Z}_2 \rightarrow Spin(n) \rightarrow SO(n) \rightarrow 1$ plays a role – the spin groups are the universal covers (which are all double-sheeted covers) of the special rotation groups.  A spin structure on some $SO(n)$ bundle $E$, let’s say represented by $f : X \rightarrow B(SO(n))$ is thus, again, a lifting to $\tilde{f} : X \rightarrow B(Spin(n))$.  The obstruction to doing this (the thing which must be zero for the lifting to exist) is the second Stiefel-Whitney class, $w_2(E)$.  Hisham Sati also explained the example of “generalized” spin structures in these terms.  But the main theme is an analogous, but much more general, story for other cohomology groups as obstructions to liftings of some sort of structures on manifolds.  These may be bundles, for the lower-degree cohomology, or they may be gerbes or n-bundles, for higher-degree, but the setup is roughly the same.

The title’s term “higher spin structures” comes from the fact that we’ve so far had a tower of classifying spaces (or groups), $B(O) \leftarrow B(SO) \leftarrow B(Spin)$, and so on.  Then the problem of putting various sorts of structures on $X$ has been turned into the problem of lifting a map $f : X \rightarrow S(O)$ up this tower.  At each point, the obstruction to lifting is some cohomology class with coefficients in the groups ($O$, $SO$, etc.)  So when are these structures interesting?

This turns out to bring up another theme, which is that of special dimensions – it’s just not true that the same phenomena happen in every dimension.  In this case, this has to do with the homotopy groups  – of $O$ and its cousins.  So it turns out that the homotopy group $\pi_k(O)$ (which is the same as $\pi_k(O_n)$ as long as $n$ is bigger than $k$) follows a pattern, where $\pi_k(O) = \mathbb{Z}_2$ if $k = 0,1 (mod 8)$, and $\pi_k(O) = \mathbb{Z}$ if $k = 3,7 (mod 8)$.  The fact that this pattern repeats mod-8 is one form of the (real) Bott Periodicity theorem.  These homotopy groups reflect that, wherever there’s nontrivial homotopy in some dimension, there’s an obstruction to contracting maps into $O$ from such a sphere.

All of this plays into the question of what kinds of nontrivial structures can be put on orthogonal bundles on manifolds of various dimensions.  In the dimensions where these homotopy groups are non-trivial, there’s an obstruction to the lifting, and therefore some interesting structure one can put on $X$ which may or may not exist.  Hisham Sati spoke of “killing” various homotopy groups – meaning, as far as I can tell, imposing conditions which get past these obstructions.  In string theory, his application of interest, one talks of “anomaly cancellation” – an anomaly being the obstruction to making these structures.  The first part of the punchline is that, since these are related to nontrivial cohomology groups, we can think of them in terms of defining structures on n-bundles or gerbes.  These structures are, essentially, connections – they tell us how to parallel-transport objects of various dimensions.  It turns out that the $\pi_k$ homotopy group is related to parallel transport along $(k-1)$-dimensional surfaces in $X$, which can be thought of as the world-sheets of $(k-2)$-dimensional “particles” (or rather, “branes”).

So, for instance, the fact that $\pi_1(O)$ is nontrivial means there’s an obstruction to a lifting in the form of a class in $H^2(X,\mathbb{Z})$, which has to do with spin structure – as above.  “Cancelling” this “anomaly” means that for a theory involving such a spin structure to be well-defined, then this characteristic class for $X$ must be zero.  The fact that $\pi_3(O) = \mathbb{Z}$ is nontrivial means there’s an obstruction to a lifting in the form of a class in $H^4(X, \mathbb{Z})$.  This has to do with “string bundles”, where the string group is a higher analog of $Spin$ in exactly the sense we’ve just described.  If such a lifting exists, then there’s a “string-structure” on $X$ which is compatible with the spin structure we lifted (and with the orientation a level below that).  Similarly, $\pi_7(O) = \mathbb{Z}$ being nontrivial, by way of an obstruction in $H^8$, means there’s an interesting notion of “five-brane” structure, and a $Fivebrane$ group, and so on.  Personally, I think of these as giving a geometric interpretation for what the higher cohomology groups actually mean.

A slight refinement of the above, and actually more directly related to “cancellation” of the anomalies, is that these structures can be defined in a “twisted” way: given a cocycle in the appropriate cohomology group, we can ask that a lifting exist, not on the nose, but as a diagram commuting only up to a higher cell, which is exactly given by the cocycle.  I mentioned, in the previous section, a situation where the cocycle gives an associator, so that instead of being exactly associative, a structure has a “twisted” associativity.  This is similar, except we’re twisting the condition that makes a spin structure (or higher spin structure) well-defined.  So if $X$ has the wrong characteristic class, we can only define one of these twisted structures at that level.

This theme of higher cohomology and gerbes, and their geometric interpretation, was another one that turned up throughout the talks in the workshop…

And speaking of that: coming up soon, some descriptions of the actual workshop.

So there’s a lot of preparations going on for the workshop HGTQGR coming up next week at IST, and the program(me) is much more developed – many of the talks are now listed, though the schedule has yet to be finalized.  This week we’ll be having a “pre-school school” to introduce the local mathematicans to some of the physics viewpoints that will be discussed at the workshop – Aleksandar Mikovic will be introducing Quantum Gravity (from the point of view of the loop/spin-foam approach), and Sebastian Guttenberg will be giving a mathematician’s introduction to String theory.

These are by no means the only approaches physicists have taken to the problem of finding a theory that incorporates both General Relativity and Quantum Field Theory.  They are, however, two approaches where lots of work has been done, and which appear to be amenable to using the mathematical tools of (higher) category theory which we’re going to be talking about at the workshop.  These are “higher gauge theory”, which very roughly is the analog of gauge theory (which includes both GR and QFT) using categorical groups, and TQFT, which is a very simple type of quantum field theory that has a natural description in terms of categories, which can be generalized to higher categories.

I’ll probably take a few posts after the workshop to write up these, and the many other talks and mini-courses we’ll be having, but right now, I’d like to say a little bit about another talk we had here recently.  Actually, the talk was in Porto, but several of us at IST in Lisbon attended by a videoconference.  This was the first time I’ve seen this for a colloquium-style talk, though I did once take a course in General Relativity from Eric Poisson that was split between U of Waterloo and U of Guelph.  I thought it was a great idea then, and it worked quite well this time, too.  This is the way of the future – and unfortunately it probably will be for some time to come…

Anyway, the talk in question was by Thomasz Brzezinski, about “Synthetic Non-Commutative Geometry” (link points to the slides).  The point here is to take two different approaches to extending differential geometry (DG) and combine the two insights.  The “Synthetic” part refers to synthetic differential geometry (SDG), which is a program for doing DG in a general topos.  One aspect of this is that in a topos where the Law of the Excluded Middle doesn’t apply, it’s possible for the real-numbers object to have infinitesimals: that is, elements which are smaller than any positive element, but bigger than zero.  This lets one take things which have to be treated in a roundabout way in ordinary DG, like $dx$, and take them at face value – as an infinitesimal change in $x$.  It also means doing geometry in a completely constructive way.

However, these aspects aren’t so important here.  The important fact about it here is that it’s based on building a theory that was originally defined in terms of sets, or topological spaces – that is, in the toposes $Sets$, or $Top$  – and transplanting it to another category.  This is because Brzezinski’s goal was to do something similar for a different extension of DG, namely non-commutative geometry (NCG).  This is a generalisation of DG which is based on the equivalence $CommAlg^{op} \simeq lCptHaus$ between the categories of commutative $C^{\star}$-algebras (and algebra maps, read “backward” as morphisms in $CommAlg^{op}$), and that of locally compact Hausdorff spaces (which, for objects, equates a space $X$ with the algebra $C(X)$ of continuous functions on it, and an algebra $A$ with its spectrum $Spec(A)$, the space of maximal ideals).  The generalization of NCG is to take structures defined for $lCptHaus$ that create DG, and make similar definitions in the category $Alg^{op}$, of not-necessarily-commutative $C^{\star}$-algebras.

This category is the one which plays the role of the topos $Top$.  It isn’t a topos, though: it’s some sort of monoidal category.  And this is what “synthetic NCG” is about: taking the definitions used in NCG and reproducing them in a generic monoidal category (to be clear, a braided monoidal category).

The way he illustrated this is by explaining what a principal bundle would be in such a generic category.

To begin with, we can start by giving a slightly nonstandard definition of the concept in ordinary DG: a principal $G$-bundle $P$ is a manifold with a free action of a (compact Lie) group $G$ on it.  The point is that this always looks like a “base space” manifold $B$, with a projection $\pi : P \rightarrow B$ so that the fibre at each point of $B$ looks like $G$.  This amounts to saying that $\pi$ is an equalizer:

$P \times G \stackrel{\longrightarrow}{\rightarrow} P \stackrel{\pi}{\rightarrow} B$

where the maps from $G\times P$ to $P$ are (a) the action, and (b) the projection onto $P$.  (Being an equalizer means that $\pi$ makes this diagram commute – has the same composite with both maps – and any other map $\phi$ that does the same factors uniquely through $\pi$.)  Another equivalent way to say this is that since $P \times G$ has two maps into $P$, then it has a map into the pullback $P \times_B P$ (the pullback of two copies of $P \stackrel{\pi}{\rightarrow} B$), and the claim is that it’s actually ismorphic.

The main points here are that (a) we take this definition in terms of diagrams and abstract it out of the category $Top$, and (b) when we do so, in general the products will be tensor products.

In particular, this means we need to have a general definition of a group object $G$ in any braided monoidal category (to know what $G$ is supposed to be like).  We reproduce the usual definition of a group objects so that $G$ must come equipped with a “multiplication” map $m : G \otimes G \rightarrow G$, an “inverse” map $\iota : G \rightarrow G$ and a “unit” map $u : I \rightarrow G$, where $I$ is the monoidal unit (which takes the role of the terminal object in a topos like $Top$, the unit for $\times$).  These need to satisfy the usual properties, such as the monoid property for multiplication:

$m \circ (m \otimes id_G) = m \circ (id_G \otimes m) : G \otimes G \otimes G \rightarrow G$

(usually given as a diagram, but I’m being lazy).

The big “however” is this: in $Sets$ or $Top$, any object $X$ is always a comonoid in a canonical way, and we use this implictly in defining some of the properties we need.  In particular, there’s always the diagonal map $\Delta : X \rightarrow X \times X$ which satisfies the dual of the monoid property:

$(id_X \times \Delta) \circ \Delta = (\Delta \times id_X) \circ \Delta$

There’s also a unique counit $\epsilon \rightarrow \star$, the map into the terminal object, which makes $(X,\Delta,\epsilon)$ a counital comonoid automatically.  But in a general braided monoidal category, we have to impose as a condition that our group object also be equipped with $\Delta : G \rightarrow G \otimes G$ and $\epsilon : G \rightarrow I$ making it a counital comonoid.  We need this property to even be able to make sense of the inverse axiom (which this time I’ll do as a diagram):

This diagram uses not only $\Delta$ but also the braiding map $\sigma_{G,G} : G \otimes G \rightarrow G \otimes G$ (part of the structure of the braided monoidal category which, in $Top$ or $Sets$ is just the “switch” map).  Now, in fact, since any object in $Set$ or $Top$ is automatically a comonoid, we’ll require that this structure be given for anything we look at: the analog of spaces (like $P$ above), or our group object $G$.  For the group object, we also must, in general, require something which comes for free in the topos world and therefore generally isn’t mentioned in the definition of a group.  Namely, the comonoid and monoid aspects of $G$ must get along.  (This comes for free in a topos essentially because the comonoid structure is given canonically for all objects.)  This means:

For a group in $Sets$ or $Top$, this essentially just says that the two ways we can go from $(x,y)$ to $(xy,xy)$ (duplicate, swap, then multiply, or on the other hand multiply then duplicate) are the same.

All these considerations about how honest-to-goodness groups are secretly also comonoids does explain why corresponding structures in noncommutative geometry seem to have more elaborate definitions: they have to explicitly say things that come for free in a topos.  So, for instance, a group object in the above sense in the braided monoidal category $Vect = (Vect_{\mathbb{F}}, \otimes_{\mathbb{F}}, \mathbb{F}, flip)$ is a Hopf algebra.  This is a nice canonical choice of category.  Another is the opposite category $Vect^{op}$ – this is a standard choice in NCG, since spaces are supposed to be algebras – this would be given the comonoid structure we demanded.

So now once we know all this, we can reproduce the diagrammatic definition of a principal $G$-bundle above: just replace the product $\times$ with the monoidal operation $\otimes$, the terminal object by $I$, and so forth.  The diagrams are understood to be diagrams of comonoids in our braided monoidal category.  In particular, we have an action $\rho : P \otimes G \rightarrow P$,which is compatible with the $\Delta$ maps – so in $Vect$ we would say that a noncommutative principal $G$-bundle $P$ is a right-module coalgebra over the Hopf algebra $G$.  We can likewise take this (in a suitably abstract sense of “algebra” or “module”) to be the definition in any braided monoidal category.

To have the “freeness” of the action, there needs to be an equalizer of:

$\rho, (id_P \otimes \epsilon) : P \otimes G \stackrel{\longrightarrow}{\rightarrow} P \stackrel{\pi}{\rightarrow} B$

The “freeness” condition for the action is likewise defined using a monoidal-category version of the pullback (fibre product) $P \times_B P$.

This was as far as Brzezinski took the idea of synthetic NCG in this particular talk, but the basic idea seems quite nice.  In SDG, one can define all sorts of differential geometric structures synthetically, that is, for a general topos: for example, Gonzalo Reyes has gone and defined the Einstein field equations synthetically.  Presumably, a lot of what’s done in NCG could also be done in this synthetic framework, and transplanted to other categories than the usual choices.

Brzezinski said he was mainly interested in the “usual” choices of category, $Vect$ and $Vect^{op}$ – so for instance in $Vect^{op}$, a “principal $G$-bundle” is what’s called a Hopf-Galois extension.  Roger Picken did, however, ask an interesting question about other possible candidates for the category to work in.  Given that one wants a braided monoidal category, a natural one to look at is the category whose morphisms are braids.  This one, as a matter of fact, isn’t quite enough (there’s no braid $m : n \otimes n \rightarrow n$, because this would be a braid with $2n$ strands in and $n$ strands out – which is impossible.  But some sort of category of tangles might make an interestingly abstract setting in which to see what NCG looks like.  So far, this doesn’t seem to have been done as far as I can see.

Marco Mackaay recently pointed me at a paper by Mikhail Khovanov, which describes a categorification of the Heisenberg algebra $H$ (or anyway its integral form $H_{\mathbb{Z}}$) in terms of a diagrammatic calculus.  This is very much in the spirit of the Khovanov-Lauda program of categorifying Lie algebras, quantum groups, and the like.  (There’s also another one by Sabin Cautis and Anthony Licata, following up on it, which I fully intend to read but haven’t done so yet. I may post about it later.)

Now, as alluded to in some of the slides I’ve from recent talks, Jamie Vicary and I have been looking at a slightly different way to answer this question, so before I talk about the Khovanov paper, I’ll say a tiny bit about why I was interested.

Groupoidification

The Weyl algebra (or the Heisenberg algebra – the difference being whether the commutation relations that define it give real or imaginary values) is interesting for physics-related reasons, being the algebra of operators associated to the quantum harmonic oscillator.  The particular approach to categorifying it that I’ve worked with goes back to something that I wrote up here, and as far as I know, originally was suggested by Baez and Dolan here.  This categorification is based on “stuff types” (Jim Dolan’s term, based on “structure types”, a.k.a. Joyal’s “species”).  It’s an example of the groupoidification program, the point of which is to categorify parts of linear algebra using the category $Span(Gpd)$.  This has objects which are groupoids, and morphisms which are spans of groupoids: pairs of maps $G_1 \leftarrow X \rightarrow G_2$.  Since I’ve already discussed the backgroup here before (e.g. here and to a lesser extent here), and the papers I just mentioned give plenty more detail (as does “Groupoidification Made Easy“, by Baez, Hoffnung and Walker), I’ll just mention that this is actually more naturally a 2-category (maps between spans are maps $X \rightarrow X'$ making everything commute).  It’s got a monoidal structure, is additive in a fairly natural way, has duals for morphisms (by reversing the orientation of spans), and more.  Jamie Vicary and I are both interested in the quantum harmonic oscillator – he did this paper a while ago describing how to construct one in a general symmetric dagger-monoidal category.  We’ve been interested in how the stuff type picture fits into that framework, and also in trying to examine it in more detail using 2-linearization (which I explain here).

Anyway, stuff types provide a possible categorification of the Weyl/Heisenberg algebra in terms of spans and groupoids.  They aren’t the only way to approach the question, though – Khovanov’s paper gives a different (though, unsurprisingly, related) point of view.  There are some nice aspects to the groupoidification approach: for one thing, it gives a nice set of pictures for the morphisms in its categorified algebra (they look like groupoids whose objects are Feynman diagrams).  Two great features of this Khovanov-Lauda program: the diagrammatic calculus gives a great visual representation of the 2-morphisms; and by dealing with generators and relations directly, it describes, in some sense1, the universal answer to the question “What is a categorification of the algebra with these generators and relations”.  Here’s how it works…

Heisenberg Algebra

One way to represent the Weyl/Heisenberg algebra (the two terms refer to different presentations of isomorphic algebras) uses a polynomial algebra $P_n = \mathbb{C}[x_1,\dots,x_n]$.  In fact, there’s a version of this algebra for each natural number $n$ (the stuff-type references above only treat $n=1$, though extending it to “$n$-sorted stuff types” isn’t particularly hard).  In particular, it’s the algebra of operators on $P_n$ generated by the “raising” operators $a_k(p) = x_k \cdot p$ and the “lowering” operators $b_k(p) = \frac{\partial p}{\partial x_k}$.  The point is that this is characterized by some commutation relations.  For $j \neq k$, we have:

$[a_j,a_k] = [b_j,b_k] = [a_j,b_k] = 0$

but on the other hand

$[a_k,b_k] = 1$

So the algebra could be seen as just a free thing generated by symbols $\{a_j,b_k\}$ with these relations.  These can be understood to be the “raising and lowering” operators for an $n$-dimensional harmonic oscillator.  This isn’t the only presentation of this algebra.  There’s another one where $[p_k,q_k] = i$ (as in $i = \sqrt{-1}$) has a slightly different interpretation, where the $p$ and $q$ operators are the position and momentum operators for the same system.  Finally, a third one – which is the one that Khovanov actually categorifies – is skewed a bit, in that it replaces the $a_j$ with a different set of $\hat{a}_j$ so that the commutation relation actually looks like

$[\hat{a}_j,b_k] = b_{k-1}\hat{a}_{j-1}$

It’s not instantly obvious that this produces the same result – but the $\hat{a}_j$ can be rewritten in terms of the $a_j$, and they generate the same algebra.  (Note that for the one-dimensional version, these are in any case the same, taking $a_0 = b_0 = 1$.)

Diagrammatic Calculus

To categorify this, in Khovanov’s sense (though see note below1), means to find a category $\mathcal{H}$ whose isomorphism classes of objects correspond to (integer-) linear combinations of products of the generators of $H$.  Now, in the $Span(Gpd)$ setup, we can say that the groupoid $FinSet_0$, or equvialently $\mathcal{S} = \coprod_n \mathcal{S}_n$, represents Fock space.  Groupoidification turns this into the free vector space on the set of isomorphism classes of objects.  This has some extra structure which we don’t need right now, so it makes the most sense to describe it as $\mathbb{C}[[t]]$, the space of power series (where $t^n$ corresponds to the object $[n]$).  The algebra itself is an algebra of endomorphisms of this space.  It’s this algebra Khovanov is looking at, so the monoidal category in question could really be considered a bicategory with one object, where the monoidal product comes from composition, and the object stands in formally for the space it acts on.  But this space doesn’t enter into the description, so we’ll just think of $\mathcal{H}$ as a monoidal category.  We’ll build it in two steps: the first is to define a category $\mathcal{H}'$.

The objects of $\mathcal{H}'$ are defined by two generators, called $Q_+$ and $Q_-$, and the fact that it’s monoidal (these objects will be the categorifications of $a$ and $b$).  Thus, there are objects $Q_+ \otimes Q_- \otimes Q_+$ and so forth.  In general, if $\epsilon$ is some word on the alphabet $\{+,-\}$, there’s an object $Q_{\epsilon} = Q_{\epsilon_1} \otimes \dots \otimes Q_{\epsilon_m}$.

As in other categorifications in the Khovanov-Lauda vein, we define the morphisms of $\mathcal{H}'$ to be linear combinations of certain planar diagrams, modulo some local relations.  (This type of formalism comes out of knot theory – see e.g. this intro by Louis Kauffman).  In particular, we draw the objects as sequences of dots labelled $+$ or $-$, and connect two such sequences by a bunch of oriented strands (embeddings of the interval, or circle, in the plane).  Each $+$ dot is the endpoint of a strand oriented up, and each $-$ dot is the endpoint of a strand oriented down.  The local relations mean that we can take these diagrams up to isotopy (moving the strands around), as well as various other relations that define changes you can make to a diagram and still represent the same morphism.  These relations include things like:

which seems visually obvious (imagine tugging hard on the ends on the left hand side to straighten the strands), and the less-obvious:

and a bunch of others.  The main ingredients are cups, caps, and crossings, with various orientations.  Other diagrams can be made by pasting these together.  The point, then, is that any morphism is some $\mathbf{k}$-linear combination of these.  (I prefer to assume $\mathbf{k} = \mathbb{C}$ most of the time, since I’m interested in quantum mechanics, but this isn’t strictly necessary.)

The second diagram, by the way, are an important part of categorifying the commutation relations.  This would say that $Q_- \otimes Q_+ \cong Q_+ \otimes Q_- \oplus 1$ (the commutation relation has become a decomposition of a certain tensor product).  The point is that the left hand sides show the composition of two crossings $Q_- \otimes Q_+ \rightarrow Q_+ \otimes Q_-$ and $Q_+ \otimes Q_- \rightarrow Q_- \otimes Q_+$ in two different orders.  One can use this, plus isotopy, to show the decomposition.

That diagrams are invariant under isotopy means, among other things, that the yanking rule holds:

(and similar rules for up-oriented strands, and zig zags on the other side).  These conditions amount to saying that the functors $- \otimes Q_+$ and $- \otimes Q_-$ are two-sided adjoints.  The two cups and caps (with each possible orientation) give the units and counits for the two adjunctions.  So, for instance, in the zig-zag diagram above, there’s a cup which gives a unit map $\mathbf{k} \rightarrow Q_- \otimes Q_+$ (reading upward), all tensored on the right by $Q_-$.  This is followed by a cap giving a counit map $Q_+ \otimes Q_- \rightarrow \mathbf{k}$ (all tensored on the left by $Q_-$).  So the yanking rule essentially just gives one of the identities required for an adjunction.  There are four of them, so in fact there are two adjunctions: one where $Q_+$ is the left adjoint, and one where it’s the right adjoint.

Karoubi Envelope

Now, so far this has explained where a category $\mathcal{H}'$ comes from – the one with the objects $Q_{\epsilon}$ described above.  This isn’t quite enough to get a categorification of $H_{\mathbb{Z}}$: it would be enough to get the version with just one $a$ and one $b$ element, and their powers, but not all the $a_j$ and $b_k$.  To get all the elements of the (integral form of) the Heisenberg algebras, and in particular to get generators that satisfy the right commutation relations, we need to introduce some new objects.  There’s a convenient way to do this, though, which is to take the Karoubi envelope of $\mathcal{H}'$.

The Karoubi envelope of any category $\mathcal{C}$ is a universal way to find a category $Kar(\mathcal{C})$ that contains $\mathcal{C}$ and for which all idempotents split (i.e. have corresponding subobjects).  Think of vector spaces, for example: a map $p \in End(V)$ such that $p^2 = p$ is a projection.  That projection corresponds to a subspace $W \subset V$, and $W$ is actually another object in $Vect$, so that $p$ splits (factors) as $V \rightarrow W subset V$.  This might not happen in any general $\mathcal{C}$, but it will in $Kar(\mathcal{C})$.  This has, for objects, all the pairs $(C,p)$ where $p : C \rightarrow C$ is idempotent (so $\mathcal{C}$ is contained in $Kar(\mathcal{C})$ as the cases where $p=1$).  The morphisms $f : (C,p) \rightarrow (C',p')$ are just maps $f : C \rightarrow C'$ with the compatibility condition that $p' f = p f = f$ (essentially, maps between the new subobjects).

So which new subobjects are the relevant ones?  They’ll be subobjects of tensor powers of our $Q_{\pm}$.  First, consider $Q_{+^n} = Q_+^{\otimes n}$.  Obviously, there’s an action of the symmetric group $\mathcal{S}_n$ on this, so in fact (since we want a $\mathbf{k}$-linear category), its endomorphisms contain a copy of $\mathbf{k}[\mathcal{S}_n]$, the corresponding group algebra.  This has a number of different projections, but the relevant ones here are the symmetrizer,:

$e_n = \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} \sigma$

which wants to be a “projection onto the symmetric subspace” and the antisymmetrizer:

$e'_n = \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} sign(\sigma) \sigma$

which wants to be a “projection onto the antisymmetric subspace” (if it were in a category with the right sub-objects). The diagrammatic way to depict this is with horizontal bars: so the new object $S^n_+ = (Q_{+^n}, e)$ (the symmetrized subobject of $Q_+^{\oplus n}$) is a hollow rectangle, labelled by $n$.  The projection from $Q_+^{\otimes n}$ is drawn with $n$ arrows heading into that box:

The antisymmetrized subobject $\Lambda^n_+ = (Q_{+^n},e')$ is drawn with a black box instead.  There are also $S^n_-$ and $\Lambda^n_-$ defined in the same way (and drawn with downward-pointing arrows).

The basic fact – which can be shown by various diagram manipulations, is that $S^n_- \otimes \Lambda^m_+ \cong (\Lambda^m_+ \otimes S^n_-) \oplus (\Lambda_+^{m-1} \otimes S^{n-1}_-)$.  The key thing is that there are maps from the left hand side into each of the terms on the right, and the sum can be shown to be an isomorphism using all the previous relations.  The map into the second term involves a cap that uses up one of the strands from each term on the left.

There are other idempotents as well – for every partition $\lambda$ of $n$, there’s a notion of $\lambda$-symmetric things – but ultimately these boil down to symmetrizing the various parts of the partition.  The main point is that we now have objects in $\mathcal{H} = Kar(\mathcal{H}')$ corresponding to all the elements of $H_{\mathbb{Z}}$.  The right choice is that the $\hat{a}_j$  (the new generators in this presentation that came from the lowering operators) correspond to the $S^j_-$ (symmetrized products of “lowering” strands), and the $b_k$ correspond to the $\Lambda^k_+$ (antisymmetrized products of “raising” strands).  We also have isomorphisms (i.e. diagrams that are invertible, using the local moves we’re allowed) for all the relations.  This is a categorification of $H_{\mathbb{Z}}$.

Some Generalities

This diagrammatic calculus is universal enough to be applied to all sorts of settings where there are functors which are two-sided adjoints of one another (by labelling strands with functors, and the regions of the plane with categories they go between).  I like this a lot, since biadjointness of certain functors is essential to the 2-linearization functor $\Lambda$ (see my link above).  In particular, $\Lambda$ uses biadjointness of restriction and induction functors between representation categories of groupoids associated to a groupoid homomorphism (and uses these unit and counit maps to deal with 2-morphisms).  That example comes from the fact that a (finite-dimensional) representation of a finite group(oid) is a functor into $Vect$, and a group(oid) homomorphism is also just a functor $F : H \rightarrow G$.  Given such an $F$, there’s an easy “restriction” $F^* : Fun(G,Vect) \rightarrow Fun(H,Vect)$, that just works by composing with $F$.  Then in principle there might be two different adjoints $Fun(H,Vect) \rightarrow Fun(G,Vect)$, given by the left and right Kan extension along $F$.  But these are defined by colimits and limits, which are the same for (finite-dimensional) vector spaces.  So in fact the adjoint is two-sided.

Khovanov’s paper describes and uses exactly this example of biadjointness in a very nice way, albeit in the classical case where we’re just talking about inclusions of finite groups.  That is, given a subgroup $H < G$, we get a functors $Res_G^H : Rep(G) \rightarrow Rep(H)$, which just considers the obvious action of $H$ act on any representation space of $G$.  It has a biadjoint $Ind^G_H : Rep(H) \rightarrow Rep(G)$, which takes a representation $V$ of $H$ to $\mathbf{k}[G] \otimes_{\mathbf{k}[H]} V$, which is a special case of the formula for a Kan extension.  (This formula suggests why it’s also natural to see these as functors between module categories $\mathbf{k}[G]-mod$ and $\mathbf{k}[H]-mod$).  To talk about the Heisenberg algebra in particular, Khovanov considers these functors for all the symmetric group inclusions $\mathcal{S}_n < \mathcal{S}_{n+1}$.

Except for having to break apart the symmetric groupoid as $S = \coprod_n \mathcal{S}_n$, this is all you need to categorify the Heisenberg algebra.  In the $Span(Gpd)$ categorification, we pick out the interesting operators as those generated by the $- \sqcup \{\star\}$ map from $FinSet_0$ to itself, but “really” (i.e. up to equivalence) this is just all the inclusions $\mathcal{S}_n < \mathcal{S}_{n+1}$ taken at once.  However, Khovanov’s approach is nice, because it separates out a lot of what’s going on abstractly and uses a general diagrammatic way to depict all these 2-morphisms (this is explained in the first few pages of Aaron Lauda’s paper on ambidextrous adjoints, too).  The case of restriction and induction is just one example where this calculus applies.

There’s a fair bit more in the paper, but this is probably sufficient to say here.

1 There are two distinct but related senses of “categorification” of an algebra $A$ here, by the way.  To simplify the point, say we’re talking about a ring $R$.  The first sense of a categorification of $R$ is a (monoidal, additive) category $C$ with a “valuation” in $R$ that takes $\otimes$ to $\times$ and $\oplus$ to $+$.  This is described, with plenty of examples, in this paper by Rafael Diaz and Eddy Pariguan.  The other, typical of the Khovanov program, says it is a (monoidal, additive) category $C$ whose Grothendieck ring is $K_0(C) = R$.  Of course, the second definition implies the first, but not conversely.  The objects of the Grothendieck ring are isomorphism classes in $C$.  A valuation may identify objects which aren’t isomorphic (or, as in groupoidification, morphisms which aren’t 2-isomorphic).

So a categorification of the first sort could be factored into two steps: first take the Grothendieck ring, then take a quotient to further identify things with the same valuation.  If we’re lucky, there’s a commutative square here: we could first take the category $C$, find some surjection $C \rightarrow C'$, and then find that $K_0(C') = R$.  This seems to be the relation between Khovanov’s categorification of $H_{\mathbb{Z}}$ and the one in $Span(Gpd)$. This is the sense in which it seems to be the “universal” answer to the problem.

A more substantial post is upcoming, but I wanted to get out this announcement for a conference I’m helping to organise, along with Roger Picken, João Faria Martins, and Aleksandr Mikovic.  Its website: https://sites.google.com/site/hgtqgr/home has more details, and will have more as we finalise them, but here are some of them:

## ﻿Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity

Lisbon, 10-13 February, 2011 (Workshop), 7-13 February, 2011 (School)

Description from the website:

Higher gauge theory is a fascinating generalization of ordinary abelian and non-abelian gauge theory, involving (at the first level) connection 2-forms, curvature 3-forms and parallel transport along surfaces. This ladder can be continued to connection forms of higher degree and transport along extended objects of the corresponding dimension. On the mathematical side, higher gauge theory is closely tied to higher algebraic structures, such as 2-categories, 2-groups etc., and higher geometrical structures, known as gerbes or n-gerbes with connection. Thus higher gauge theory is an example of the categorification phenomenon which has been very influential in mathematics recently.

There have been a number of suggestions that higher gauge theory could be related to (4D) quantum gravity, e.g. by Baez-Huerta (in the QG^2 Corfu school lectures), and Baez-Baratin-Freidel-Wise in the context of state-sums. A pivotal role is played by TQFTs in these approaches, in particular BF theories and variants thereof, as well as extended TQFTs, constructed from suitable geometric or algebraic data. Another route between higher gauge theory and quantum gravity is via string theory, where higher gauge theory provides a setting for n-form fields, worldsheets for strings and branes, and higher spin structures (i.e. string structures and generalizations, as studied e.g. by Sati-Schreiber-Stasheff). Moving away from point particles to higher-dimensional extended objects is a feature both of loop quantum gravity and string theory, so higher gauge theory should play an important role in both approaches, and may allow us to probe a deeper level of symmetry, going beyond normal gauge symmetry.

Thus the moment seems ripe to bring together a group of researchers who could shed some light on these issues. Apart from the courses and lectures given by the invited speakers, we plan to incorporate discussion sessions in the afternoon throughout the week, for students to ask questions and to stimulate dialogue between participants from different backgrounds.

Provisional list of speakers:

• Paolo Aschieri (Alessandria)
• Benjamin Bahr (Cambridge)
• Aristide Baratin (Paris-Orsay)
• John Barrett (Nottingham)
• Rafael Diaz (Bogotá)
• Bianca Dittrich (Potsdam)
• Laurent Freidel (Perimeter)
• John Huerta (California)
• Branislav Jurco (Prague)
• Thomas Krajewski (Marseille)
• Tim Porter (Bangor)
• Hisham Sati (Maryland)
• Christopher Schommer-Pries (MIT)
• Urs Schreiber (Utrecht)
• Jamie Vicary (Oxford)
• Derek Wise (Erlangen)
• Christoph Wockel (Hamburg)

The workshop portion will have talks by the speakers above (those who can make it), and any contributed talks.  The “school” portion is, roughly, aimed at graduate students in a field related to the topics, but not necessarily directly in them.  You don’t need to be a student to attend the school, of course, but they are the target audience.  The only course that has been officially announced so far will be given by Christopher Schommer-Pries, on TQFT.  We hope/expect to also have minicourses on Higher Gauge Theory, and Quantum Gravity as well, but details aren’t settled yet.

If you’re interested, the deadline to register is Jan 8 (hence the rush to announce).  Some funding is available for those who need it.

In the most recent TQFT Club seminar, we had a couple of talks – one was the second in a series of three by Marco Mackaay, which as promised previously I’ll write up together after the third one.

The other was by Björn Gohla, a student of João Faria Martins, giving an overview on the subject of “Tricategories and Trifunctors”, a mostly expository talk explaining some definitions.  Actually, this was a bit more specific than a general introduction – the point of it was to describe a certain kind of mapping space.  I’ve talked here before about representing the “configuration space” of a gauge theory as a groupoid: the objects are (optionally, flat) connections on a manifold $M$, and the morphisms are gauge transformations taking one connection to another.  The reason for the things Björn was talking about is analogous, except that in this case, the goal is to describe the configuration space of a higher gauge theory.

There are at least two ways I know of to talk about higher gauge theory.  One is in terms of categorical (or n-categorical) groups – which makes it a “categorification” of gauge theory in the sense of reproducing in $\mathbf{Cat}$ (or $\mathbf{nCat}$) an analog of a sturcture, gauge theory, originally formulated in $\mathbf{Set}$.  Among other outlines, you might look at this one by John Baez and John Huerta for an introduction.  Another uses the lingo of crossed modules or crossed complexes.  In either case, the essential point is the same: there is some collection of groups (or groupoids, but let’s say groups to keep everything clear) which play the role of the single gauge group in ordinary gauge theory.

In the first language, we can speak of a “2-group”, or “categorical group” – a group internal to $\mathbf{Cat}$, or what is equivalent, a category internal to $\mathbf{Grp}$, which would have a group of objects and a group of morphisms (and, in higher settings still, groups of 2-morphisms, 3-morphisms, and so on).  The structure maps of the category (source, target, composition, etc.) have to live in the category of groups.

A crossed complex of groups (again, we could generalize to groupoids, but I won’t) is a nonabelian variation on a chain complex: a sequence of groups with maps from one to the next.  There are also a bunch more structures, which ultimately serve to reproduce all the kind of composition, source, and target maps in the $n$-categorical groups: some groups act on others, there are “bracket” operations on one group valued in another, and so forth.  This paper by Brown and Higgins explains how the two concepts are related when most of the groups are abelian, and there’s a lot more about crossed complexes and related stuff in Tim Porter’s “Crossed Menagerie“.

The point of all this right now is that these things play the role of the gauge group in higher gauge theory.  The idea is that in gauge theory, you have a connection.  Typically this is described in terms of a form valued in the Lie algebra of the gauge group.  Then a (thin) homotopy classes of curves, gets a holonomy valued in the group by integrating that form.  Alternatively, you can just think of the path groupoid of a manifold $\mathcal{P}_1(M)$, where those classes of curves form the morphisms between the objects, which are just points of $M$.  Then a connection defines a functor $\Gamma : \mathcal{P}_1(M) \rightarrow G$, where $G$ is the gauge group thought of as a category (groupoid in fact) with one object.  Or, you can just define a connection that way in the first place.  In higher gauge theory, a similar principle exists: begin with the $n$-path groupoid $\mathcal{P}_n(M)$ where the morphisms are (thin homotopy classes of) paths, the 2-morphisms are surfaces (really homotopy classes of homotopies of paths), and so on, so the $k$-morphisms are $k$-dimensional bits of $M$.  Then you could define an $n$-connection as a $n$-functor into an $n$-group as defined above.  OR, you could define it in terms of a tower of differential $k$-forms valued in the crossed complex of Lie algebras associated to the crossed complex of Lie groups that replaces the gauge group.  You can then use an integral to get an element of the group at level $k$ of the complex for any given $k$-morphism in $\mathcal{P}_n(M)$, which (via the equivalence I mentioned) amounts to the same thing as the other definition of connection.

João Martins has done some work on this sort of thing when $n$ is dimension 2 (with Tim Porter) and 3 (with Roger Picken), which I guess is how Björn came to work on this question.  The question is, roughly, how to describe the moduli space of these connections.  The gist of the answer is that it’s a functor $n$-category $[\mathcal{P}_n(M),\mathcal{G}]$, where $\mathcal{G}$ is the $n$-group.  A little more generally, the question is how to describe mapping spaces for higher categories.  In particular, he was talking about the case $n=3$, which is where certain tricky issues start to show up.  In particular every bicategory (the weakest form of 2-category) is (bi)equivalent to a strict 2-category, so there’s no real need to worry about weakening things like associativity so that they only work up to isomorphism – these are all equalities.  With 3-categories, this fails: the weakest kind of 3-category is a tricategory (introduced by Gordon, Power and Street, though also see the references beyond that link).  These are always tri-equivalent to something stricter than general, but not completely strict: Gray-categories.  The only equation from 2-categories which has to be weakened to an isomorphism here is the interchange law: given a square of four morphisms, we can either compose vertically first, and then horizontally, or vice versa.  In a Gray-category, there’s an “interchanger” isomorphism

$I_{\alpha,\alpha ',\beta,\beta'} : (\alpha \circ \beta) \cdot (\alpha ' \circ \beta ') \Rightarrow (\alpha \cdot \alpha ') \circ (\beta \cdot \beta ')$

where $\cdot$ is vertical composition of 2-cells, and $\circ$ is horizontal (i.e. the same direction as 1-cells).  This is supposed to satisfy a compatibility condition.  It’s essentially the only one you can come up with starting with $(\alpha \cdot \alpha ') \circ \beta$ (and composing it in different orders by throwing in identities in various places).

There’s another way to look at things, as Björn explained, in terms of enriched category theory.  If you have a monoidal category $(\mathcal{V},\otimes)$, then a $(\mathcal{V},\otimes)$-enriched category $\mathbb{G}$ is one in which, for any two objects $x,y$, there is an object $\mathbb{G}(x,y) \in \mathcal{V}$ of morphisms, and composition gives morphisms $\circ_{x,y,z} : \mathbb{G}(y,z) \otimes \mathbb{G}(x,y) \rightarrow \mathbb{G}(x,z)$.  A strict 3-category is enriched in $\mathbf{2Cat}$, with its usual tensor product, dual to its internal hom $[-,-]$ (which gives the mapping 2-category of functors, natural transformations, and modifications, between any two 2-categories).  A Gray category is similar, except that it is enriched in $\mathbf{Gray}$, a version of $\mathbf{2Cat}$ with a different tensor product, dual to the hom functor $[-,-]'$ which gives the mapping 2-category with pseudonatural transformations (the weak version of the concept, where the naturality square only has to commute up to a specified 2-cell) as morphisms.  These are not the same, which is where the unavoidability of weakening 3-categories “really” comes from.   The upshot of this is as above: it matters which order we compose things in.

Having defined Gray-categories, let’s say $A$ and $B$ (which, in the applications I mentioned above, tend to actually be Gray-groupoids, though this doesn’t change the theory substantially), the point is to talk about “mapping spaces” – that is, Gray-categories of Gray-functors (etc.) from $A$ to $B$.

Since they’ve been defined terms of enriched category theory, one wants to use the general theory of enriched functors, transformations, and so forth – which is a lot easier than trying to work out the correct definitions from scratch using a low-level description.  So then a Gray-functor $F : A \rightarrow B$ has an object map $F_0 : A_0 \rightarrow B_0$, mapping objects of $A$ to objects of $B$, and then for each $x,y \in A_0$, a morphism in $\mathbf{Gray}$ (which is our $\mathcal{V}$), namely $F_{x,y} : A(x,y) \rightarrow B(F(x),F(y))$.  There are a bunch of compatibility conditions, which can be expressed for any monoidal category $\mathcal{V}$ (since they involve diagrams with the map $\circ_{x,y,z}$ for any triple, and the like).  Similar comments apply to defining $\mathcal{V}$-natural transformations.

There is a slight problem here, which is that in this case, $\mathcal{V} = \mathbf{Gray}$ is a 2-category, so we really need to use a form of weakly enriched categories…  All the compatibility diagrams should have 2-cells in them, and so forth.  This, too, gets complicated.  So Björn explained is a shortcut from drawing $n$-dimensional diagrams for these mapping $n$-categories in terms of the arrow category $\vec{B}$. This is the category whose objects are the morphisms of $B$, and whose morphisms are commuting squares, or when $B$ is a 2-category, squares with a 2-cell, so a morphism in $\vec{B}$ from $f: x \rightarrow y$ to $f' : x' \rightarrow y'$ is a triple $g = (g_x,g_y,g_f)$ like so:

Morphism in arrow category

The 2-morphisms in $\vec{B}$ are commuting “pillows”, where the front and back face are morphisms like the above. So $\beta : g \Rightarrow g'$ is $\beta = (\beta_x,\beta_y)$, where $\beta_x : g_x \Rightarrow g_{x'}$ is a 2-cell, and the whole “pillow” commutes.  When $B$ is a tricategory, then we need to go further – these 2-morphsims should be triples including a 3-cell $\beta_f$ filling the “pillow”, and then 3-morphisms are commuting structures between these. These diagrams get hard to draw pretty quickly. This is the point of having an ordinary 2D diagram with at most 1-dimensional cells: pushing all the nasty diagrams into these arrow categories, we can replace a 2-cell representing a natural transformation with a diagram involving the arrow category.

This uses that there are source and target maps (which are Gray-functors, of course) which we’ll call $d_0, d_1: \vec{B} \rightarrow B$. So then here (in one diagram) we have two ways of depicting a natural transformation $\alpha : F \rightarrow G$ between functors $F,G : A \Rightarrow B$:

One is the 2-cell, and the other is the functor into $\vec{B}$, such that $d_0 \circ \alpha = F$ and $d_1 \circ \alpha = G$.
To depict a modification between natural transformations (a 3-cell between 2-cells) just involves building the arrow category of $\vec{B}$, say $\vec{\vec{B}}$, and drawing an arrow from $A$ into it. And so on: in principle, there is a tower above $B$ built by iterating the arrow category construction, and all the different levels of “functor”, “natural transformation”, “modification”, and all the higher equivalents are just functors into different levels of this tower.  (The generic term for the $k^{th}$ level of maps-between-maps-etc between $n$-categories is “$(n,k)$-transfor“, a handy term coined here.)
The advantage here is that at least the general idea can be extended pretty readily to higher values of $n$ than 3.  Naturally, no matter which way one decides to do it, things will get complicated – either there’s a combinatorial explosion of things to consider, or one has to draw higher-dimensional diagrams, or whatever.  This exploding complexity of $n$-categories (in this case, globular ones) is one of the reasons why simplicial appreaches – quasicategories or $\infty$-categories – are good.  They allow you to avoid talking about those problems, or at least fold them into fairly well-understood aspects of simplicial sets.  A lot of things – limits, colimits, mapping spaces, etc. are pretty well understood in that case (see, for instance, the first chapter of Joshua Nicholls-Barrer’s thesis for the basics, or Jacob Lurie’s humongous book for something more comprehensive).  But sometimes, as in this case, they just don’t happen to be the things you want for your application.  So here we have some tools for talking about mapping spaces in the world of globular $n$-categories – and as the work by Martins/Porter/Picken show, it’s motivated by some fairly specific work about invariants of manifolds, differential geometry, and so on.

In the first week of November, I was in Montreal for the biannual meeting of the Philosophy of Science Association, at the invitation of Hans Halvorson and Steve Awodey.  This was for a special session called “Category Theoretical Reflections on the Foundations of Physics”, which also had talks by Bob Coecke (from Oxford), Klaas Landsman (from Radboud University in Nijmegen), and Gonzalo Reyes (from the University of Montreal).  Slides from the talks in this session have been collected here by Steve Awodey.  The meeting was pretty big, and there were a lot of talks on a lot of different topics, some more technical, and some less.  There were enough sessions relating to physics that I had a full schedule just attending those, although for example there were sessions on biology and cognition which I might otherwise have been interested in sitting in on, with titles like “Biology: Evolution, Genomes and Biochemistry”, “Exploring the Complementarity between Economics and Recent Evolutionary Theory”, “Cognitive Sciences and Neuroscience”, and “Methodological Issues in Cognitive Neuroscience”.  And, of course, more fundamental philosophy of science topics like “Fictions and Scientific Realism” and “Kinds: Chemical, Biological and Social”, as well as socially-oriented ones such as “Philosophy of Commercialized Science” and “Improving Peer Review in the Sciences”.  However, interesting as these are, one can’t do everything.

In some ways, this was a really great confluence of interests for me – physics and category theory, as seen through a philosophical lens.  I don’t know exactly how this session came about, but Hans Halvorson is a philosopher of science who started out in physics (and has now, for example, learned enough category theory to teach the course in it offered at Princeton), and Steve Awodey is a philosopher of mathematics who is interested in category theory in its own right.  They managed to get this session brought in to present some of the various ideas about the overlap between category theory and physics to an audience mostly consisting of philosophers, which seems like a good idea.  It was also interesting for me to get a view into how philosophers approach these subjects – what kind of questions they ask, how they argue, and so on.  As with any well-developed subject, there’s a certain amount of jargon and received ideas that people can refer to – for example, I learned the word and current usage (though not the basic concept) of supervenience, which came up, oh, maybe 5-10 times each day.

There are now a reasonable number of people bringing categorical tools to bear on physics – especially quantum physics.  What people who think about the philosophy of science can bring to this research is the usual: careful, clear thinking about the fundamental concepts involved in a way that tries not to get distracted by the technicalities and keep the focus on what is important to the question at hand in a deep way.  In this case, the question at hand is physics.  Philosophy doesn’t always accomplish this, of course, and sometimes get sidetracked by what some might call “pseudoquestions” – the kind of questions that tend to arise when you use some folk-theory or simple intuitive understanding of some subtler concept that is much better expressed in mathematics.  This is why anyone who’s really interested in the philosophy of science needs to learn a lot about science in its own terms.  On the whole, this is what they actually do.

And, of course, both mathematicians and physicists try to do this kind of thinking themselves, but in those fields it’s easy – and important! – to spend a lot of time thinking about some technical question, or doing extensive computations, or working out the fiddly details of a proof, and so forth.  This is the real substance of the work in those fields – but sometimes the bigger “why” questions, that address what it means or how to interpret the results, get glossed over, or answered on the basis of some superficial analogy.  Mind you – one often can’t really assess how a line of research is working out until you’ve been doing the technical stuff for a while.  Then the problem is that people who do such thinking professionally – philosophers – are at a loss to understand the material because it’s recent and technical.  This is maybe why technical proficiency in science has tended to run ahead of real understanding – people still debate what quantum mechanics “means”, even though we can use it competently enough to build computers, nuclear reactors, interferometers, and so forth.

Anyway – as for the substance of the talks…  In our session, since every speaker was a mathematician in some form, they tended to be more technical.  You can check out the slides linked to above for more details, but basically, four views of how to draw on category theory to talk about physics were represented.  I’ve actually discussed each of them in previous posts, but in summary:

• Bob Coecke, on “Quantum Picturalism”, was addressing the monoidal dagger-category point of view, which looks at describing quantum mechanical operations (generally understood to be happening in a category of Hilbert spaces) purely in terms of the structure of that category, which one can see as a language for handling a particular kind of logic.  Monoidal categories, as Peter Selinger as painstakingly documented, can be described using various graphical calculi (essentially, certain categories whose morphisms are variously-decorated “strands”, considered invariant under various kinds of topological moves, are the free monoidal categories with various structures – so anything you can prove using these diagrams is automatically true for any example of such categories).  Selinger has also shown that, for the physically interesting case of dagger-compact closed monoidal categories, a theorem is true in general if and only if it’s true for (finite dimensional) Hilbert spaces, which may account for why Hilbert spaces play such a big role in quantum mechanics.  This program is based on describing as much of quantum mechanics as possible in terms of this kind of diagrammatic language.  This stuff has, in some ways, been explored more through the lens of computer science than physics per se – certainly Selinger is coming from that background.  There’s also more on this connection in the “Rosetta Stone” paper by John Baez and Mike Stay,
• My talk (actually third, but I put it here for logical flow) fits this framework, more or less.  I was in some sense there representing a viewpoint whose current form is due to Baez and Dolan, namely “groupoidification”.  The point is to treat the category $Span(Gpd)$ as a “categorification” of (finite dimensional) Hilbert spaces in the sense that there is a representation map $D : Span(Gpd) \rightarrow Hilb$ so that phenomena living in $Hilb$ can be explained as the image of phenomena in $Span(Gpd)$.  Having done that, there is also a representation of $Span(Gpd)$ into 2-Hilbert spaces, which shows up more detail (much more, at the object level, since Tannaka-Krein reconstruction means that the monoidal 2-Hilbert space of representations of a groupoid is, at least in nice cases, enough to completely reconstruct it).  This gives structures in $2Hilb$ which “conceptually” categorify the structures in $Hilb$, and are also directly connected to specific Hilbert spaces and maps, even though taking equivalence classes in $2Hilb$ definitely doesn’t produce these.  A “state” in a 2-Hilbert space is an irreducible representation, though – so there’s a conceptual difference between what “state” means in categorified and standard settings.  (There’s a bit more discussion in my notes for the talk than in the slides above.)
• Klaas Landsman was talking about what he calls “Bohrification“, which, on the technical side, makes use of Topos theory.  The philosophical point comes from Niels Bohr’s “doctrine of classical concepts” – that one should understand quantum systems using concepts from the classical world.  In practice, this means taking a (noncommutative) von Neumann algebra $A$ which describes the observables a quantum system and looking at it via its commutative subalgebras.  These are organized into a lattice – in fact, a site.  The idea is that the spectrum of $A$ lives in the topos associated to this site: it’s a presheaf that, over each commutative subalgebra $C \subset A$, just gives the spectrum of $C$.  This is philosophically nice in that the “Bohrified” propositions actually behave in a logically sensible way.  The topos approach comes from Chris Isham, developed further with Andreas Doring. (Note the series of four papers by both from 2007.  Their approach is in some sense dual to that of Lansman, Heunen and Spitters, in the sense that they look at the same site, but look at dual toposes – one of sheaves, the other of cosheaves.  The key bit of jargon in Isham and Doring’s approach is “daseinization”, which is a reference to Heidegger’s “Being and Time”.  For some reason this makes me imagine Bohr and Heidegger in a room, one standing on the ceiling, one on the floor, disputing which is which.)
• Gonzalo Reyes talked about synthetic differential geometry (SDG) as a setting for building general relativity.  SDG is a way of doing differential geometry in a category where infinitesimals are actually available, that is, there is a nontrivial set $D = \{ x \in \mathbb{R} | x^2 = 0 \}$.  This simplifies discussions of vector fields (tangent vectors will just be infinitesimal vectors in spacetime).  A vector field is really a first order DE (and an integral curve tangent to it is a solution), so it’s useful to have, in SDG, the fact that any differentiable curve is, literally, infinitesimally a line.  Then the point is that while the gravitational “field” is a second-order DE, so not a field in this sense, the arguments for GR can be reproduced nicely in SDG by talking about infinitesimally-close families of curves following geodesics.  Gonzalo’s slides are brief by necessity, but happily, more details of this are in his paper on the subject.

The other sessions I went to were mostly given by philosophers, rather than physicists or mathematicians, though with exceptions.  I’ll briefly present my own biased and personal highlights of what I attended.  They included sessions titled:

Quantum Physics“: Edward Slowik talked about the “prehistory of quantum gravity”, basically revisiting the debate between Newton and Leibniz on absolute versus relational space, suggesting that Leibniz’ view of space as a classification of the relation of his “monads” is more in line with relational theories such as spin foams etc.  M. Silberstein and W. Stuckey – gave a talk about their “relational blockworld” (described here) which talks about QFT as an approximation to a certain discrete theory, built on a graph, where the nodes of the graph are spacetime events, and using an action functional on the graph.

Meinard Kuhlmann gave an interesting talk about “trope bundles” and AQFTTrope ontology is an approach to “entities” that doesn’t assume there’s a split between “substrates” (which have no properties themselves), and “properties” which they carry around.  (A view of ontology that goes back at least to Aristotle’s “substance” and “accident” distinction, and maybe further for all I know).  Instead, this is a “one-category” ontology – the basic things in this ontology are “tropes”, which he defined as “individual property instances” (i.e. as opposed to abstract properties that happen to have instances).  “Things” then, are just collections of tropes.  To talk about the “identity” of a thing means to pick out certain of the tropes as the core ones that define that thing, and others as peripheral.  This struck me initially as a sort of misleading distinction we impose (say, “a sphere” has a core trope of its radial symmetry, and incidental tropes like its colour – but surely the way of picking the object out of the world is human-imposed), until he gave the example from AQFT.  To make a long story short, in this setup, the key entites are something like elementary particles, and the core tropes are those properties that define an irreducible representation of a $C^{\star}$-algebra (things like mass, spin, charge, etc.), whereas the non-core tropes are those that identify a state vector within such a representation: the attributes of the particle that change over time.

I’m not totally convinced by the “trope” part of this (surely there are lots of choices of the properties which determine a representation, but I don’t see the need to give those properties the burden of being the only ontologically primaries), but I also happen to like the conclusions because in the 2Hilbert picture, irreducible representations are states in a 2-Hilbert space, which are best thought of as morphisms, and the state vectors in their components are best thought of in terms of 2-morphisms.  An interpretation of that setup says that the 1-morphism states define which system one’s talking about, and the 2-morphism states describe what it’s doing.

New Directions Concerning Quantum Indistinguishability“: I only caught a couple of the talks in this session, notably missing Nick Huggett’s “Expanding the Horizons of Quantum Statistical Mechanics”.  There were talks by John Earman (“The Concept of Indistinguishable Particles in Quantum
Mechanics”), and by Adam Caulton (based on work with Jeremy Butterfield) on “On the Physical Content of the Indistinguishability Postulate”.  These are all about the idea of indistinguishable particles, and the statistics thereof.  Conventionally, in QM you only talk about bosons and fermions – one way to say what this means is that the permutation group $S_n$ naturally acts on a system of $n$ particles, and it acts either trivially (not altering the state vector at all), or by sign (each swap of two particles multiplies the state vector by a minus sign).  This amounts to saying that only one-dimensional representations of $S_n$ occur.  It is usually justified by the “spin-statistics theorem“, relating it to the fact that particles have either integer or half-integer spins (classifying representations of the rotation group).  But there are other representations of $S_n$, labelled by Young diagrams, though they are more than one-dimensional.  This gives rise to “paraparticle” statistics.  On the other hand, permuting particles in two dimensions is not homotopically trivial, so one ought to use the braid group $B_n$, rather than $S_n$, and this gives rise again to different statistics, called “anyonic” statistics.

One recurring idea is that, to deal with paraparticle statistics, one needs to change the formalism of QM a bit, and expand the idea of a “state vector” (or rather, ray) to a “generalized ray” which has more dimensions – corresponding to the dimension of the representation of $S_n$ one wants the particles to have.  Anyons can be dealt with a little more conventionally, since a 2D system may already have them.  Adam Caulton’s talk described how this can be seen as a topological phenomenon or a dynamical one – making an analogy with the Bohm-Aharonov effect, where the holonomy of an EM field around a solenoid can be described either dynamically with an interacting Lagrangian on flat space, or topologically with a free Lagrangian in space where the solenoid has been removed.

Quantum Mechanics“: A talk by Elias Okon and Craig Callender about QM and the Equivalence Principle, based on this.  There has been some discussion recently as to whether quantum mechanics is compatible with the principle that relates gravitational and inertial mass.  They point out that there are several versions of this principle, and that although QM is incompatible with some versions, these aren’t the versions that actually produce general relativity.  (For example, objects with large and small masses fall differently in quantum physics, because though the mean travel time is the same, the variance is different.  But this is not a problem for GR, which only demands that all matter responds dynamically to the same metric.)  Also, talks by Peter Lewis on problems with the so-called “transactional interpretation” of QM, and Bryan Roberts on time-reversal.

Why I Care About What I Don’t Yet Know“:  A funny name for a session about time-asymmetry, which is the essentially philosophical problem of why, if the laws of physics are time-symmetric (which they approximately are for most purposes), what we actually experience isn’t.  Personally, the best philosophical account of this I’ve read is Huw Price’s “Time’s Arrow“, though Reichenbach’s “The Direction of Time” has good stuff in it also, and there’s also Zeh’s more technical “The Physical Basis of the Direction of Time“. In the session, Chris Suhler and Craig Callender gave an account of how, given causal asymmetry, our subjective asymmetry of values for the future and the past can arise (the intuitively obvious point being that if we can influence the future and not the past, we tend to value it more).  Mathias Frisch talked about radiation asymmetry (the fact that it’s equally possible in EM to have waves converging on a source than spreading out from it, yet we don’t see this).  Owen Maroney argued that “There’s No Route from Thermodynamics to the Information Asymmetry” by describing in principle how to construct a time-reversed (probabilisitic) computer.  David Wallace spoke on “The Logic of the Past Hypothesis”, the idea inspired by Boltzmann that we see time-asymmetry because there is a point in what we call the “past” where entropy was very low, and so we perceive the direction away from that state as “forward” it time because the world tends to move toward equilibrium (though he pointed out that for dynamical reasons, the world can easily stay far away from equilibrium for a long time).  He went on to discuss the logic of this argument, and the idea of a “simple” (i.e. easy-to-describe) distribution, and the conjecture that the evolution of these will generally be describable in terms of an evolution that uses “coarse graining” (i.e. that repeatedly throws away microscopic information).

The Emergence of Spacetime in Quantum Theories of Gravity“:  This session addressed the idea that spacetime (or in some cases, just space) might not be fundamental, but could emerge from a more basic theory.  Christian Wüthrich spoke about “A-Priori versus A-Posteriori” versions of this idea, mostly focusing on ideas such as LQG and causal sets, which start with discrete structures, and get manifolds as approximations to them.  Nick Huggett gave an overview of noncommutative geometry for the philosophically minded audience, explaining how an algebra of observables can be treated like space by means of all the concepts from geometry which can be imported into the theory of $C^{\star}$-algebras, where space would be an approximate description of the algebra by letting the noncommutativity drop out of sight in some limit (which would be described as a “large scale” limit).  Sean Carroll discussed the possibility that “Space is Not Fundamental – But Time Might Be”, pointing out that even in classical mechanics, space is not a fundamental notion (since it’s possible to reformulate even Hamiltonian classical mechanics without making essential distinctions between position and momentum coordinates), and suggesting that space arises from the dynamics of an actual physical system – a Hamiltonian, in this example – by the principle “Position Is The Thing In Which Interactions Are Local”.  Finally, Sean Maudlin gave an argument for the fundamentality of time by showing how to reconstruct topology in space from a “linear structure” on points saying what a (directed!) path among the points is.