### double categories

The main thing happening in my end of the world is that it’s relocated from Europe back to North America. I’m taking up a teaching postdoc position in the Mathematics and Computer Science department at Mount Allison University starting this month. However, amidst all the preparations and moving, I was also recently in Edinburgh, Scotland for a workshop on Higher Gauge Theory and Higher Quantization, where I gave a talk called 2-Group Symmetries on Moduli Spaces in Higher Gauge Theory. That’s what I’d like to write about this time.

Edinburgh is a beautiful city, though since the workshop was held at Heriot-Watt University, whose campus is outside the city itself, I only got to see it on the Saturday after the workshop ended. However, John Huerta and I spent a while walking around, and as it turned out, climbing a lot: first the Scott Monument, from which I took this photo down Princes Street:

And then up a rather large hill called Arthur’s Seat, in Holyrood Park next to the Scottish Parliament.

The workshop itself had an interesting mix of participants. Urs Schreiber gave the most mathematically sophisticated talk, and mine was also quite category-theory-minded. But there were also some fairly physics-minded talks that are interesting to me as well because they show the source of these ideas. In this first post, I’ll begin with my own, and continue with David Roberts’ talk on constructing an explicit string bundle. …

### 2-Group Symmetries of Moduli Spaces

My own talk, based on work with Roger Picken, boils down to a couple of observations about the notion of symmetry, and applies them to a discrete model in higher gauge theory. It’s the kind of model you might use if you wanted to do lattice gauge theory for a BF theory, or some other higher gauge theory. But the discretization is just a convenience to avoid having to deal with infinite dimensional spaces and other issues that don’t really bear on the central point.

Part of that point was described in a previous post: it has to do with finding a higher analog for the relationship between two views of symmetry: one is “global” (I found the physics-inclined part of the audience preferred “rigid”), to do with a group action on the entire space; the other is “local”, having to do with treating the points of the space as objects of a groupoid who show how points are related to each other. (Think of trying to describe the orbit structure of just the part of a group action that relates points in a little neighborhood on a manifold, say.)

In particular, we’re interested in the symmetries of the moduli space of connections (or, depending on the context, flat connections) on a space, so the symmetries are gauge transformations. Now, here already some of the physically-inclined audience objected that these symmetries should just be eliminated by taking the quotient space of the group action. This is based on the slogan that “only gauge-invariant quantities matter”. But this slogan has some caveats: in only applies to closed manifolds, for one. When there are boundaries, it isn’t true, and to describe the boundary we need something which acts as a representation of the symmetries. Urs Schreiber pointed out a well-known example: the Chern-Simons action, a functional on a certain space of connections, is not gauge-invariant. Indeed, the boundary terms that show up due to this not-invariance explain why there is a Wess-Zumino-Witt theory associated with the boundaries when the bulk is described by Chern-Simons.

Now, I’ve described a lot of the idea of this talk in the previous post linked above, but what’s new has to do with how this applies to moduli spaces that appear in higher gauge theory based on a 2-group $\mathcal{G}$. The points in these space are connections on a manifold $M$. In particular, since a 2-group is a group object in categories, the transformation groupoid (which captures global symmetries of the moduli space) will be a double category. It turns out there is another way of seeing this double category by local descriptions of the gauge transformations.

In particular, general gauge transformations in HGT are combinations of two special types, described geometrically by $G$-valued functions, or $Lie(H)$-valued 1-forms, where $G$ is the group of objects of $\mathcal{G}$, and $H$ is the group of morphisms based at $1_G$. If we think of connections as functors from the fundamental 2-groupoid $\Pi_2(M)$ into $\mathcal{G}$, these correspond to pseudonatural transformations between these functors. The main point is that there are also two special types of these, called “strict”, and “costrict”. The strict ones are just natural transformations, where the naturality square commutes strictly. The costrict ones, also called ICONs (for “identity component oplax natural transformations” – see the paper by Steve Lack linked from the nlab page above for an explanation of “costrictness”). They assign the identity morphism to each object, but the naturality square commutes only up to a specified 2-cell. Any pseudonatural transformation factors into a strict and costrict part.

The point is that taking these two types of transformation to be the horizontal and vertical morphisms of a double category, we get something that very naturally arises by the action of a big 2-group of symmetries on a category. We also find something which doesn’t happen in ordinary gauge theory: that only the strict gauge transformations arise from this global symmetry. The costrict ones must already be the morphisms in the category being acted on. This category plays the role of the moduli space in the normal 1-group situation. So moving to 2-groups reveals that in general we should distinguish between global/rigid symmetries of the moduli space, which are strict gauge transformations, and costrict ones, which do not arise from the global 2-group action and should be thought of as intrinsic to the moduli space.

### String Bundles

David Roberts gave a rather interesting talk called “Constructing Explicit String Bundles”. There are some notes for this talk here. The point is simply to give an explicit construction of a particular 2-group bundle. There is a lot of general abstract theory about 2-bundles around, and a fair amount of work that manipulates physically-motivated descriptions of things that can presumably be modelled with 2-bundles. There has been less work on giving a mathematically rigorous description of specific, concrete 2-bundles.

This one is of interest because it’s based on the String 2-group. Details are behind that link, but roughly the classifying space of $String(G)$ (a homotopy 2-type) is fibred over the classifying space for $G$ (a 1-type). The exact map is determined by taking a pullback along a certain characteristic class (which is a map out of $BG$). Saying “the” string 2-group is a bit of a misnomer, by the way, since such a 2-group exists for every simply connected compact Lie group $G$. The group that’s involved here is a $String(n)$, the string 2-group associated to $Spin(n)$, the universal cover of the rotation group $SO(n)$. This is the one that determines whether a given manifold can support a “string structure”. A string structure on $M$, therefore, is a lift of a spin structure, which determines whether one can have a spin bundle over $M$, hence consistently talk about a spin connection which gives parallel transport for spinor fields on $M$. The string structure determines if one can consistently talk about a string-bundle over $M$, and hence a 2-group connection giving parallel transport for strings.

In this particular example, the idea was to find, explicitly, a string bundle over Minkowski space – or its conformal compactification. In point of fact, this particular one is for $latek String(5)$, and is over 6-dimensional Minkowski space, whose compactification is $M = S^5 \times S^1$. This particular $M$ is convenient because it’s possible to show abstractly that it has exactly one nontrivial class of string bundles, so exhibiting one gives a complete classification. The details of the construction are in the notes linked above. The technical details rely on the fact that we can coordinatize $M$ nicely using the projective quaternionic plane, but conceptually it relies on the fact that $S^5 \cong SU(3)/SU(2)$, and because of how the lifting works, this is also $String(SU(3))/String(SU(2))$. This quotient means there’s a string bundle $String(SU(3)) \rightarrow S^5$ whose fibre is $String(SU(2))$.

While this is only one string bundle, and not a particularly general situation, it’s nice to see that there’s a nice elegant presentation which gives such a bundle explicitly (by constructing cocycles valued in the crossed module associated to the string 2-group, which give its transition functions).

(Here endeth Part I of this discussion of the workshop in Edinburgh. Part II will talk about Urs Schreiber’s very nice introduction to Higher Geometric Quantization)

(This ends the first part of this update – the next will describe the physics-oriented talks, and the third will describe Urs Schreiber’s series on higher geometric quantization)

So it’s been a while since I last posted – the end of 2013 ended up being busy with a couple of visits to Jamie Vicary in Oxford, and Roger Picken in Lisbon. In the aftermath of the two trips, I did manage to get a major revision of this paper submitted to a journal, and put this one out in public. A couple of others will be coming down the pipeline this year as well.

I’m hoping to get back to a post about motives which I planned earlier, but for the moment, I’d like to write a little about the second paper, with Roger Picken.

### Global and Local Symmetry

The upshot is that it’s about categorifying the concept of symmetry. More specifically, it’s about finding the analog in the world of categories for the interplay between global and local symmetry which occurs in the world of set-based structures (sets, topological spaces, vector spaces, etc.) This distinction is discussed in a nice way by Alan Weinstein in this article from the Notices of the AMS from

The global symmetry of an object $X$ in some category $\mathbf{C}$ can be described in terms of its group of automorphisms: all the ways the object can be transformed which leave it “the same”. This fits our understanding of “symmetry” when the morphisms can really be interpreted as transformations of some sort. So let’s suppose the object is a set with some structure, and the morphisms are set-maps that preserve the structure: for example, the objects could be sets of vertices and edges of a graph, so that morphisms are maps of the underlying data that preserve incidence relations. So a symmetry of an object is a way of transforming it into itself – and an invertible one at that – and these automorphisms naturally form a group $Aut(X)$. More generally, we can talk about an action of a group $G$ on an object $X$, which is a map $\phi : G \rightarrow Aut(X)$.

“Local symmetry” is different, and it makes most sense in a context where the object $X$ is a set – or at least, where it makes sense to talk about elements of $X$, so that $X$ has an underlying set of some sort.

Actually, being a set-with-structure, in a lingo I associate with Jim Dolan, means that the forgetful functor $U : \mathbf{C} \rightarrow \mathbf{Sets}$ is faithful: you can tell morphisms in $\mathbf{C}$ (in particular, automorphisms of $X$) apart by looking at what they do to the underlying set. The intuition is that the morphisms of $\mathbf{C}$ are exactly set maps which preserve the structure which $U$ forgets about – or, conversely, that the structure on objects of $\mathbf{C}$ is exactly that which is forgotten by $U$. Certainly, knowing only this information determines $\mathbf{C}$ up to equivalence. In any case, suppose we have an object like this: then knowing about the symmetries of $X$ amounts to knowing about a certain group action, namely the action of $Aut(X)$, on the underlying set $U(X)$.

From this point of view, symmetry is about group actions on sets. The way we represent local symmetry (following Weinstein’s discussion, above) is to encode it as a groupoid – a category whose morphisms are all invertible. There is a level-slip happening here, since $X$ is now no longer seen as an object inside a category: it is the collection of all the objects of a groupoid. What makes this a representation of “local” symmetry is that each morphism now represents, not just a transformation of the whole object $X$, but a relationship under some specific symmetry between one element of $X$ and another. If there is an isomorphism between $x \in X$ and $y \in X$, then $x$ and $y$ are “symmetric” points under some transformation. As Weinstein’s article illustrates nicely, though, there is no assumption that the given transformation actually extends to the entire object $X$: it may be that only part of $X$ has, for example, a reflection symmetry, but the symmetry doesn’t extend globally.

### Transformation Groupoid

The “interplay” I alluded to above, between the global and local pictures of symmetry, is to build a “transformation groupoid” (or “action groupoid“) associated to a group $G$ acting on a set $X$. The result is called $X // G$ for short. Its morphisms consist of pairs such that  $(g,x) : x \rightarrow (g \rhd x)$ is a morphism taking $x$ to its image under the action of $g \in G$. The “local” symmetry view of $X // G$ treats each of these symmetry relations between points as a distinct bit of data, but coming from a global symmetry – that is, a group action – means that the set of morphisms comes from the product $G \times X$.

Indeed, the “target” map in $X // G$ from morphisms to objects is exactly a map $G \times X \rightarrow X$. It is not hard to show that this map is an action in another standard sense. Namely, if we have a real action $\phi : G \rightarrow Hom(X,X)$, then this map is just $\hat{\phi} : G \times X \rightarrow X$, which moves one of the arguments to the left side. If $\phi$ was a functor, then $\hat{\phi}$ satisfies the “action” condition, namely that the following square commutes:

(Here, $m$ is the multiplication in $G$, and this is the familiar associativity-type axiom for a group action: acting by a product of two elements in $G$ is the same as acting by each one successively.

So the starting point for the paper with Roger Picken was to categorify this. It’s useful, before doing that, to stop and think for a moment about what makes this possible.

First, as stated, this assumed that $X$ either is a set, or has an underlying set by way of some faithful forgetful functor: that is, every morphism in $Aut(X)$ corresponds to a unique set map from the elements of $X$ to itself. We needed this to describe the groupoid $X // G$, whose objects are exactly the elements of $X$. The diagram above suggests a different way to think about this. The action diagram lives in the category $\mathbf{Set}$: we are thinking of $G$ as a set together with some structure maps. $X$ and the morphism $\hat{\phi}$ must be in the same category, $\mathbf{Set}$, for this characterization to make sense.

So in fact, what matters is that the category $X$ lived in was closed: that is, it is enriched in itself, so that for any objects $X,Y$, there is an object $Hom(X,Y)$, the internal hom. In this case, it’s $G = Hom(X,X)$ which appears in the diagram. Such an internal hom is supposed to be a dual to $\mathbf{Set}$‘s monoidal product (which happens to be the Cartesian product $\times$): this is exactly what lets us talk about $\hat{\phi}$.

So really, this construction of a transformation groupoid will work for any closed monoidal category $\mathbf{C}$, producing a groupoid in $\mathbf{C}$. It may be easier to understand in cases like $\mathbf{C}=\mathbf{Top}$, the category of topological spaces, where there is indeed a faithful underlying set functor. But although talking explicitly about elements of $X$ was useful for intuitively seeing how $X//G$ relates global and local symmetries, it played no particular role in the construction.

### Categorify Everything

In the circles I run in, a popular hobby is to “categorify everything“: there are different versions, but what we mean here is to turn ideas expressed in the world of sets into ideas in the world of categories. (Technical aside: all the categories here are assumed to be small). In principle, this is harder than just reproducing all of the above in any old closed monoidal category: the “world” of categories is $\mathbf{Cat}$, which is a closed monoidal 2-category, which is a more complicated notion. This means that doing all the above “strictly” is a special case: all the equalities (like the commutativity of the action square) might in principle be replaced by (natural) isomorphisms, and a good categorification involves picking these to have good properties.

(In our paper, we left this to an appendix, because the strict special case is already interesting, and in any case there are “strictification” results, such as the fact that weak 2-groups are all equivalent to strict 2-groups, which mean that the weak case isn’t as much more general as it looks. For higher $n$-categories, this will fail – which is why we include the appendix to suggest how the pattern might continue).

Why is this interesting to us? Bumping up the “categorical level” appeals for different reasons, but the ones matter most to me have to do with taking low-dimensional (or -codimensional) structures, and finding analogous ones at higher (co)dimension. In our case, the starting point had to do with looking at the symmetries of “higher gauge theories” – which can be used to describe the transport of higher-dimensional surfaces in a background geometry, the way gauge theories can describe the transport of point particles. But I won’t ask you to understand that example right now, as long as you can accept that “what are the global/local symmetries of a category like?” is a possibly interesting question.

So let’s categorify the discussion about symmetry above… To begin with, we can just take our (closed monoidal) category to be $\mathbf{Cat}$, and follow the same construction above. So our first ingredient is a 2-group $\mathcal{G}$. As with groups, we can think of a 2-group either as a 2-category with just one object $\star$, or as a 1-category with some structure – a group object in $\mathbf{Cat}$, which we’ll call $C(\mathcal{G})$ if it comes from a given 2-group. (In our paper, we keep these distinct by using the term “categorical group” for the second. The group axioms amount to saying that we have a monoidal category $(\mathcal{G}, \otimes, I)$. Its objects are the morphisms of the 2-group, and the composition becomes the monoidal product $\otimes$.)

(In fact, we often use a third equivalent definition, that of crossed modules of groups, but to avoid getting into that machinery here, I’ll be changing our notation a little.)

### 2-Group Actions

So, again, there are two ways to talk about an action of a 2-group on some category $\mathbf{C}$. One is to define an action as a 2-functor $\Phi : \mathcal{G} \rightarrow \mathbf{Cat}$. The object being acted on, $\mathbf{C} \in \mathbf{Cat}$, is the unique object $\Phi(\star)$ – so that the 2-functor amounts to a monoidal functor from the categorical group $C(\mathcal{G})$ into $Aut(\mathbf{C})$. Notice that here we’re taking advantage of the fact that $\mathbf{Cat}$ is closed, so that the hom-“sets” are actually categories, and the automorphisms of $\mathbf{C}$ – invertible functors from $\mathbf{C}$ to itself – form the objects of a monoidal category, and in fact a categorical group. What’s new, though, is that there are also 2-morphisms – natural transformations between these functors.

To begin with, then, we show that there is a map $\hat{\Phi} : \mathcal{G} \times \mathbf{C} \rightarrow \mathbf{C}$, which corresponds to the 2-functor $\Phi$, and satisfies an action axiom like the square above, with $\otimes$ playing the role of group multiplication. (Again, remember that we’re only talking about the version where this square commutes strictly here – in an appendix of the paper, we talk about the weak version of all this.) This is an intuitive generalization of the situation for groups, but it is slightly more complicated.

The action $\Phi$ directly gives three maps. First, functors $\Phi(\gamma) : \mathbf{C} \rightarrow \mathbf{C}$ for each 2-group morphism $\gamma$ – each of which consists of a function between objects of $\mathbf{C}$, together with a function between morphisms of $\mathbf{C}$. Second, natural transformations $\Phi(\eta) : \Phi(\gamma) \rightarrow \Phi(\gamma ')$ for 2-morphisms $\eta : \gamma \rightarrow \gamma'$ in the 2-group – each of which consists of a function from objects to morphisms of $\mathbf{C}$.

On the other hand, $\hat{\Phi} : \mathcal{G} \times \mathbf{C} \rightarrow \mathbf{C}$ is just a functor: it gives two maps, one taking pairs of objects to objects, the other doing the same for morphisms. Clearly, the map $(\gamma,x) \mapsto x'$ is just given by $x' = \Phi(\gamma)(x)$. The map taking pairs of morphisms $(\eta,f) : (\gamma,x) \rightarrow (\gamma ', y)$ to morphisms of $\mathbf{C}$ is less intuitively obvious. Since I already claimed $\Phi$ and $\hat{\Phi}$ are equivalent, it should be no surprise that we ought to be able to reconstruct the other two parts of $\Phi$ from it as special cases. These are morphism-maps for the functors, (which give $\Phi(\gamma)(f)$ or $\Phi(\gamma ')(f)$), and the natural transformation maps (which give $\Phi(\eta)(x)$ or $\Phi(\eta)(y)$). In fact, there are only two sensible ways to combine these four bits of information, and the fact that $\Phi(\eta)$ is natural means precisely that they’re the same, so:

$\hat{\Phi}(\eta,f) = \Phi(\eta)(y) \circ \Phi(\gamma)(f) = \Phi(\gamma ')(f) \circ \Phi(\eta)(x)$

Given the above, though, it’s not so hard to see that a 2-group action really involves two group actions: of the objects of $\mathcal{G}$ on the objects of $\mathbf{C}$, and of the morphisms of $\mathcal{G}$ on objects of $\mathbf{C}$. They fit together nicely because objects can be identified with their identity morphisms: furthermore, $\Phi$ being a functor gives an action of $\mathcal{G}$-objects on $\mathbf{C}$-morphisms which fits in between them nicely.

But what of the transformation groupoid? What is the analog of the transformation groupoid, if we repeat its construction in $\mathbf{Cat}$?

### The Transformation Double Category of a 2-Group Action

The answer is that a category (such as a groupoid) internal to $\mathbf{Cat}$ is a double category. The compact way to describe it is as a “category in $\mathbf{Cat}$“, with a category of objects and a category of morphisms, each of which of course has objects and morphisms of its own. For the transformation double category, following the same construction as for sets, the object-category is just $\mathbf{C}$, and the morphism-category is $\mathcal{G} \times \mathbf{C}$, and the target functor is just the action map $\hat{\Phi}$. (The other structure maps that make this into a category in $\mathbf{Cat}$ can similarly be worked out by following your nose).

This is fine, but the internal description tends to obscure an underlying symmetry in the idea of double categories, in which morphisms in the object-category and objects in the morphism-category can switch roles, and get a different description of “the same” double category, denoted the “transpose”.

A different approach considers these as two different types of morphism, “horizontal” and “vertical”: they are the morphisms of horizontal and vertical categories, built on the same set of objects (the objects of the object-category). The morphisms of the morphism-category are then called “squares”. This makes a convenient way to draw diagrams in the double category. Here’s a version of a diagram from our paper with the notation I’ve used here, showing what a square corresponding to a morphism $(\chi,f) \in \mathcal{G} \times \mathbf{C}$ looks like:

The square (with the boxed label) has the dashed arrows at the top and bottom for its source and target horizontal morphisms (its images under the source and target functors: the argument above about naturality means they’re well-defined). The vertical arrows connecting them are the source and target vertical morphisms (its images under the source and target maps in the morphism-category).

### Horizontal and Vertical Slices of $\mathbf{C} // \mathcal{G}$

So by construction, the horizontal category of these squares is just the object-category $\mathbf{C}$.  For the same reason, the squares and vertical morphisms, make up the category $\mathcal{G} \times \mathbf{C}$.

On the other hand, the vertical category has the same objects as $\mathbf{C}$, but different morphisms: it’s not hard to see that the vertical category is just the transformation groupoid for the action of the group of $\mathbf{G}$-objects on the set of $\mathbf{C}$-objects, $Ob(\mathbf{C}) // Ob(\mathcal{G})$. Meanwhile, the horizontal morphisms and squares make up the transformation groupoid $Mor(\mathbf{C}) // Mor(\mathcal{G})$. These are the object-category and morphism-category of the transpose of the double-category we started with.

We can take this further: if squares aren’t hip enough for you – or if you’re someone who’s happy with 2-categories but finds double categories unfamiliar – the horizontal and vertical categories can be extended to make horizontal and vertical bicategories. They have the same objects and morphisms, but we add new 2-cells which correspond to squares where the boundaries have identity morphisms in the direction we’re not interested in. These two turn out to feel quite different in style.

First, the horizontal bicategory extends $\mathbf{C}$ by adding 2-morphisms to it, corresponding to morphisms of $\mathcal{G}$: roughly, it makes the morphisms of $\mathbf{C}$ into the objects of a new transformation groupoid, based on the action of the group of automorphisms of the identity in $\mathcal{G}$ (which ensures the square has identity edges on the sides.) This last point is the only constraint, and it’s not a very strong one since $Aut(1_G)$ and $G$ essentially determine the entire 2-group: the constraint only relates to the structure of $\mathcal{G}$.

The constraint for the vertical bicategory is different in flavour because it depends more on the action $\Phi$. Here we are extending a transformation groupoid, $Ob(\mathbf{C}) // Ob(\mathcal{G})$. But, for some actions, many morphisms in $\mathcal{G}$ might just not show up at all. For 1-morphisms $(\gamma, x)$, the only 2-morphisms which can appear are those taking $\gamma$ to some $\gamma '$ which has the same effect on $x$ as $\gamma$. So, for example, this will look very different if $\Phi$ is free (so only automorphisms show up), or a trivial action (so that all morphisms appear).

In the paper, we look at these in the special case of an adjoint action of a 2-group, so you can look there if you’d like a more concrete example of this difference.

### Speculative Remarks

The starting point for this was a project (which I talked about a year ago) to do with higher gauge theory – see the last part of the linked post for more detail. The point is that, in gauge theory, one deals with connections on bundles, and morphisms between them called gauge transformations. If one builds a groupoid out of these in a natural way, it turns out to result from the action of a big symmetry group of all gauge transformations on the moduli space of connections.

In higher gauge theory, one deals with connections on gerbes (or higher gerbes – a bundle is essentially a “0-gerbe”). There are now also (2-)morphisms between gauge transformations (and, in higher cases, this continues further), which Roger Picken and I have been calling “gauge modifications”. If we try to repeat the situation for gauge theory, we can construct a 2-groupoid out of these, which expresses this local symmetry. The thing which is different for gerbes (and will continue to get even more different if we move to $n$-gerbes and the corresponding $(n+1)$-groupoids) is that this is not the same type of object as a transformation double category.

Now, in our next paper (which this one was written to make possible) we show that the 2-groupoid is actually very intimately related to the transformation double category: that is, the local picture of symmetry for a higher gauge theory is, just as in the lower-dimensional situation, intimately related to a global symmetry of an entire moduli 2-space, i.e. a category. The reason this wasn’t obvious at first is that the moduli space which includes only connections is just the space of objects of this category: the point is that there are really two special kinds of gauge transformations. One should be thought of as the morphisms in the moduli 2-space, and the other as part of the symmetries of that 2-space. The intuition that comes from ordinary gauge theory overlooks this, because the phenomenon doesn’t occur there.

Physically-motivated theories are starting to use these higher-categorical concepts more and more, and symmetry is a crucial idea in physics. What I’ve sketched here is presumably only the start of a pattern in which “symmetry” extends to higher-categorical entities. When we get to 3-groups, our simplifying assumptions that use “strictification” results won’t even be available any more, so we would expect still further new phenomena to show up – but it seems plausible that the tight relation between global and local symmetry will still exist, but in a way that is more subtle, and refines the standard understanding we have of symmetry today.

## Hamburg

Since I moved to Hamburg,   Alessandro Valentino and I have been organizing one series of seminar talks whose goal is to bring people (mostly graduate students, and some postdocs and others) up to speed on the tools used in Jacob Lurie’s big paper on the classification of TQFT and proof of the Cobordism Hypothesis.  This is part of the Forschungsseminar (“research seminar”) for the working groups of Christoph Schweigert, Ingo Runkel, and Christoph Wockel.  First, I gave one introducing myself and what I’ve done on Extended TQFT. In our main series We’ve had a series of four so far – two in which Alessandro outlined a sketch of what Lurie’s result is, and another two by Sebastian Novak and Marc Palm that started catching our audience up on the simplicial methods used in the theory of $(\infty,n)$-categories which it uses.  Coming up in the New Year, Nathan Bowler and I will be talking about first $(\infty,1)$-categories, and then $(\infty,n)$-categories.   I’ll do a few posts summarizing the talks around then.

Some people in the group have done some work on quantum field theories with defects, in relation to which, there’s this workshop coming up here in February!  The idea here is that one could have two regions of space where different field theories apply, which are connected along a boundary. We might imagine these are theories which are different approximations to what’s going on physically, with a different approximation useful in each region.  Whatever the intuition, the regions will be labelled by some category, and boundaries between regions are labelled by functors between categories.  Where different boundary walls meet, one can have natural transformations.  There’s a whole theory of how a 3D TQFT can be associated to modular tensor categories, in sort of the same sense that a 2D TQFT is associated to a Frobenius algebra. This whole program is intimately connected with the idea of “extending” a given TQFT, in the sense that it deals with theories that have inputs which are spaces (or, in the case of defects, sub-spaces of given ones) of many different dimensions.  Lurie’s paper describing the n-dimensional cobordism category, is very much related to the input to a theory like this.

## Brno Visit

This time, I’d like to mention something which I began working on with Roger Picken in Lisbon, and talked about for the first time in Brno, Czech Republic, where I was invited to visit at Masaryk University.  I was in Brno for a week or so, and on Thursday, December 13, I gave this talk, called “Higher Gauge Theory and 2-Group Actions”.  But first, some pictures!

This fellow was near the hotel I stayed in:

Since this sculpture is both faceless and hard at work on nonspecific manual labour, I assume he’s a Communist-era artwork, but I don’t really know for sure.

The Christmas market was on in Náměstí Svobody (Freedom Square) in the centre of town.  This four-headed dragon caught my eye:

On the way back from Brno to Hamburg, I met up with my wife to spend a couple of days in Prague.  Here’s the Christmas market in the Old Town Square of Prague:

Anyway, it was a good visit to the Czech Republic.  Now, about the talk!

### Moduli Spaces in Higher Gauge Theory

The motivation which I tried to emphasize is to define a specific, concrete situation in which to explore the concept of “2-Symmetry”.  The situation is supposed to be, if not a realistic physical theory, then at least one which has enough physics-like features to give a good proof of concept argument that such higher symmetries should be meaningful in nature.  The idea is that Higher Gauge theory is a field theory which can be understood as one in which the possible (classical) fields on a space/spacetime manifold consist of maps from that space into some target space $X$.  For the topological theory, they are actually just homotopy classes of maps.  This is somewhat related to Sigma models used in theoretical physics, and mathematically to Homotopy Quantum Field Theory, which considers these maps as geometric structure on a manifold.  An HQFT is a functor taking such structured manifolds and cobordisms into Hilbert spaces and linear maps.  In the paper Roger and I are working on, we don’t talk about this stage of the process: we’re just considering how higher-symmetry appears in the moduli spaces for fields of this kind, which we think of in terms of Higher Gauge Theory.

Ordinary topological gauge theory – the study of flat connections on $G$-bundles for some Lie group $G$, can be looked at this way.  The target space $X = BG$ is the “classifying space” of the Lie group – homotopy classes of maps in $Hom(M,BG)$ are the same as groupoid homomorphisms in $Hom(\Pi_1(M),G)$.  Specifically, the pair of functors $\Pi_1$ and $B$ relating groupoids and topological spaces are adjoints.  Now, this deals with the situation where $X = BG$ is a homotopy 1-type, which is to say that it has a fundamental groupoid $\Pi_1(X) = G$, and no other interesting homotopy groups.  To deal with more general target spaces $X$, one should really deal with infinity-groupoids, which can capture the whole homotopy type of $X$ – in particular, all its higher homotopy groups at once (and various relations between them).  What we’re talking about in this paper is exactly one step in that direction: we deal with 2-groupoids.

We can think of this in terms of maps into a target space $X$ which is a 2-type, with nontrivial fundamental groupoid $\Pi_1(X)$, but also interesting second homotopy group $\pi_2(X)$ (and nothing higher).  These fit together to make a 2-groupoid $\Pi_2(X)$, which is a 2-group if $X$ is connected.  The idea is that $X$ is the classifying space of some 2-group $\mathcal{G}$, which plays the role of the Lie group $G$ in gauge theory.  It is the “gauge 2-group”.  Homotopy classes of maps into $X = B \mathcal{G}$ correspond to flat connections in this 2-group.

For practical purposes, we use the fact that there are several equivalent ways of describing 2-groups.  Two very directly equivalent ways to define them are as group objects internal to $\mathbf{Cat}$, or as categories internal to $\mathbf{Grp}$ – which have a group of objects and a group of morphisms, and group homomorphisms that define source, target, composition, and so on.  This second way is fairly close to the equivalent formulation as crossed modules $(G,H,\rhd,\partial)$.  The definition is in the slides, but essentially the point is that $G$ is the group of objects, and with the action $G \rhd H$, one gets the semidirect product $G \ltimes H$ which is the group of morphisms.  The map $\partial : H \rightarrow G$ makes it possible to speak of $G$ and $H$ acting on each other, and that these actions “look like conjugation” (the precise meaning of which is in the defining properties of the crossed module).

The reason for looking at the crossed-module formulation is that it then becomes fairly easy to understand the geometric nature of the fields we’re talking about.  In ordinary gauge theory, a connection can be described locally as a 1-form with values in $Lie(G)$, the Lie algebra of $G$.  Integrating such forms along curves gives another way to describe the connection, in terms of a rule assigning to every curve a holonomy valued in $G$ which describes how to transport something (generally, a fibre of a bundle) along the curve.  It’s somewhat nontrivial to say how this relates to the classic definition of a connection on a bundle, which can be described locally on “patches” of the manifold via 1-forms together with gluing functions where patches overlap.  The resulting categories are equivalent, though.

In higher gauge theory, we take a similar view. There is a local view of “connections on gerbes“, described by forms and gluing functions (the main difference in higher gauge theory is that the gluing functions related to higher cohomology).  But we will take the equivalent point of view where the connection is described by $G$-valued holonomies along paths, and $H$-valued holonomies over surfaces, for a crossed module $(G,H,\rhd,\partial)$, which satisfy some flatness conditions.  These amount to 2-functors of 2-categories $\Pi_2(M) \rightarrow \mathcal{G}$.

The moduli space of all such 2-connections is only part of the story.  2-functors are related by natural transformations, which are in turn related by “modifications”.  In gauge theory, the natural transformations are called “gauge transformations”, and though the term doesn’t seem to be in common use, the obvious term for the next layer would be “gauge modifications”. It is possible to assemble a 2-groupoid $Hom(\Pi_2(M),\mathcal{G}$, whose space of objects is exactly the moduli space of 2-connections, and whose 1- and 2-morphisms are exactly these gauge transformations and modifications.  So the question is, what is the meaning of the extra information contained in the 2-groupoid which doesn’t appear in the moduli space itself?

Our claim is that this information expresses how the moduli space carries “higher symmetry”.

### 2-Group Actions and the Transformation Double Category

What would it mean to say that something exhibits “higher” symmetry? A rudimentary way to formalize the intuition of “symmetry” is to say that there is a group (of “symmetries”) which acts on some object. One could get more subtle, but this should be enough to begin with. We already noted that “higher” gauge theory uses 2-groups (and beyond into $n$-groups) in the place of ordinary groups.  So in this context, the natural way to interpret it is by saying that there is an action of a 2-group on something.

Just as there are several equivalent ways to define a 2-group, there are different ways to say what it means for it to have an action on something.  One definition of a 2-group is to say that it’s a 2-category with one object and all morphisms and 2-morphisms invertible.  This definition makes it clear that a 2-group has to act on an object of some 2-category $\mathcal{C}$. For our purposes, just as we normally think of group actions on sets, we will focus on 2-group actions on categories, so that $\mathcal{C} = \mathbf{Cat}$ is the 2-category of interest. Then an action is just a map:

$\Phi : \mathcal{G} \rightarrow \mathbf{Cat}$

The unique object of $\mathcal{G}$ – let’s call it $\star$, gets taken to some object $\mathbf{C} = \Phi(\star) \in \mathbf{Cat}$.  This object $\mathbf{C}$ is the thing being “acted on” by $\mathcal{G}$.  The existence of the action implies that there are automorphisms $\Phi(g) : \mathbf{C} \rightarrow \mathbf{C}$ for every morphism in $\mathbf{G}$ (which correspond to the elements of the group $G$ of the crossed module).  This would be enough to describe ordinary symmetry, but the higher symmetry is also expressed in the images of 2-morphisms $\Phi( \eta : g \rightarrow g') = \Phi(\eta) : \Phi(g) \rightarrow \Phi(g')$, which we might call 2-symmetries relating 1-symmetries.

What we want to do in our paper, which the talk summarizes, is to show how this sort of 2-group action gives rise to a 2-groupoid (actually, just a 2-category when the $\mathbf{C}$ being acted on is a general category).  Then we claim that the 2-groupoid of connections can be seen as one that shows up in exactly this way.  (In the following, I have to give some credit to Dany Majard for talking this out and helping to find a better formalism.)

To make sense of this, we use the fact that there is a diagrammatic way to describe the transformation groupoid associated to the action of a group $G$ on a set $S$.  The set of morphisms is built as a pullback of the action map, $\rhd : (g,s) \mapsto g(s)$.

This means that morphisms are pairs $(g,s)$, thought of as going from $s$ to $g(s)$.  The rule for composing these is another pullback.  The diagram which shows how it’s done appears in the slides.  The whole construction ends up giving a cubical diagram in $\mathbf{Sets}$, whose top and bottom faces are mere commuting diagrams, and whose four other faces are all pullback squares.

To construct a 2-category from a 2-group action is similar. For now we assume that the 2-group action is strict (rather than being given by $\Phi$ a weak 2-functor).  In this case, it’s enough to think of our 2-group $\mathcal{G}$ not as a 2-category, but as a group-object in $\mathbf{Cat}$ – the same way that a 1-group, as well as being a category, can be seen as a group object in $\mathbf{Set}$.  The set of objects of this category is the group $G$ of morphisms of the 2-category, and the morphisms make up the group $G \ltimes H$ of 2-morphisms.  Being a group object is the same as having all the extra structure making up a 2-group.

To describe a strict action of such a $\mathcal{G}$ on $\mathbf{C}$, we just reproduce in $\mathbf{Cat}$ the diagram that defines an action in $\mathbf{Sets}$:

The fact that $\rhd$ is an action just means this commutes. In principle, we could define a weak action, which would mean that this commutes up to isomorphism, but we won’t be looking at that here.

Constructing the same diagram which describes the structure of a transformation groupoid (p29 in the slides for the talk), we get a structure with a “category of objects” and a “category of morphisms”.  The construction in $\mathbf{Set}$ gives us directly a set of morphisms, while $S$ itself is the set of objects. Similarly, in $\mathbf{Cat}$, the category of objects is just $\mathbf{C}$, while the construction gives a category of morphisms.

The two together make a category internal to $\mathbf{Cat}$, which is to say a double category.  By analogy with $S / \!\! / G$, we call this double category $\mathbf{C} / \!\! / \mathcal{G}$.

We take $\mathbf{C}$ as the category of objects, as the “horizontal category”, whose morphisms are the horizontal arrows of the double category. The category of morphisms of $\mathbf{C} /\!\!/ \mathcal{G}$ shows up by letting its objects be the vertical arrows of the double category, and its morphisms be the squares.  These look like this:

The vertical arrows are given by pairs of objects $(\gamma, x)$, and just like the transformation 1-groupoid, each corresponds to the fact that the action of $\gamma$ takes $x$ to $\gamma \rhd x$. Each square (morphism in the category of morphisms) is given by a pair $( (\gamma, \eta), f)$ of morphisms, one from $\mathcal{G}$ (given by an element in $G \rtimes H$), and one from $\mathbf{C}$.

The horizontal arrow on the bottom of this square is:

$(\partial \eta) \gamma \rhd f \circ \Phi(\gamma,\eta)_x = \Phi(\gamma,\eta)_y \circ \gamma \rhd f$

The fact that these are equal is exactly the fact that $\Phi(\gamma,\eta)$ is a natural transformation.

The double category $\mathbf{C} /\!\!/ \mathcal{G}$ turns out to have a very natural example which occurs in higher gauge theory.

### Higher Symmetry of the Moduli Space

The point of the talk is to show how the 2-groupoid of connections, previously described as $Hom(\Pi_2(M),\mathcal{G})$, can be seen as coming from a 2-group action on a category – the objects of this category being exactly the connections. In the slides above, for various reasons, we did this in a discretized setting – a manifold with a decomposition into cells. This is useful for writing things down explicitly, but not essential to the idea behind the 2-symmetry of the moduli space.

The point is that there is a category we call $\mathbf{Conn}$, whose objects are the connections: these assign $G$-holonomies to edges of our discretization (in general, to paths), and $H$-holonomies to 2D faces. (Without discretization, one would describe these in terms of $Lie(G)$-valued 1-forms and $Lie(H)$-valued 2-forms.)

The morphisms of $\mathbf{Conn}$ are one type of “gauge transformation”: namely, those which assign $H$-holonomies to edges. (Or: $Lie(H)$-valued 1-forms). They affect the edge holonomies of a connection just like a 2-morphism in $\mathcal{G}$.  Face holonomies are affected by the $H$-value that comes from the boundary of the face.

What’s physically significant here is that both objects and morphisms of $\mathbf{Conn}$ describe nonlocal geometric information.  They describe holonomies over edges and surfaces: not what happens at a point.  The “2-group of gauge transformations”, which we call $\mathbf{Gauge}$, on the other hand, is purely about local transformations.  If $V$ is the vertex set of the discretized manifold, then $\mathbf{Gauge} = \mathcal{G}^V$: one copy of the gauge 2-group at each vertex.  (Keeping this finite dimensional and avoiding technical details was one main reason we chose to use a discretization.  In principle, one could also talk about the 2-group of $\mathcal{G}$-valued functions, whose objects and morphisms, thinking of it as a group object in $\mathbf{Cat}$, are functions valued in morphisms of $\mathcal{G}$.)

Now, the way $\mathbf{Gauge}$ acts on $\mathbf{Conn}$ is essentially by conjugation: edge holonomies are affected by pre- and post-multiplication by the values at the two vertices on the edge – whether objects or morphisms of $\mathbf{Gauge}$.  (Face holonomies are unaffected).  There are details about this in the slides, but the important thing is that this is a 2-group of purely local changes.  The objects of $\mathbf{Gauge}$ are gauge transformations of this other type.  In a continuous setting, they would be described by $G$-valued functions.  The morphisms are gauge modifications, and could be described by $H$-valued functions.

The main conceptual point here is that we have really distinguished between two kinds of gauge transformation, which are the horizontal and vertical arrows of the double category $\mathbf{Conn} /\!\!/ \mathbf{Gauge}$.  This expresses the 2-symmetry by moving some gauge transformations into the category of connections, and others into the 2-group which acts on it.  But physically, we would like to say that both are “gauge transformations”.  So one way to do this is to “collapse” the double category to a bicategory: just formally allow horizontal and vertical arrows to compose, so that there is only one kind of arrow.  Squares become 2-cells.

So then if we collapse the double category expressing our 2-symmetry relation this way, the result is exactly equivalent to the functor category way of describing connections.  (The morphisms will all be invertible because $\mathbf{Conn}$ is a groupoid and $\mathbf{Gauge}$ is a 2-group).

I’m interested in this kind of geometrical example partly because it gives a good way to visualize something new happening here.  There appears to be some natural 2-symmetry on this space of fields, which is fairly easy to see geometrically, and distinguishes in a fundamental way between two types of gauge transformation.  This sort of phenomenon doesn’t occur in the world of $\mathbf{Sets}$ – a set $S$ has no morphisms, after all, so the transformation groupoid for a group action on it is much simpler.

In broad terms, this means that 2-symmetry has qualitatively new features that familiar old 1-symmetry doesn’t have.  Higher categorical versions – $n$-groups acting on $n$-groupoids, as might show up in more complicated HQFT – will certainly be even more complicated.  The 2-categorical version is just the first non-trivial situation where this happens, so it gives a nice starting point to understand what’s new in higher symmetry that we didn’t already know.

Since the last post, I’ve been busily attending some conferences, as well as moving to my new job at the University of Hamburg, in the Graduiertenkolleg 1670, “Mathematics Inspired by String Theory and Quantum Field Theory”.  The week before I started, I was already here in Hamburg, at the conference they were organizing “New Perspectives in Topological Quantum Field Theory“.  But since I last posted, I was also at the 20th Oporto Meeting on Geometry, Topology, and Physics, as well as the third Higher Structures in China workshop, at Jilin University in Changchun.  Right now, I’d like to say a few things about some of the highlights of that workshop.

Higher Structures in China III

So last year I had a bunch of discussions I had with Chenchang Zhu and Weiwei Pan, who at the time were both in Göttingen, about my work with Jamie Vicary, which I wrote about last time when the paper was posted to the arXiv.  In that, we showed how the Baez-Dolan groupoidification of the Heisenberg algebra can be seen as a representation of Khovanov’s categorification.  Chenchang and Weiwei and I had been talking about how these ideas might extend to other examples, in particular to give nice groupoidifications of categorified Lie algebras and quantum groups.

That is still under development, but I was invited to give a couple of talks on the subject at the workshop.  It was a long trip: from Lisbon, the farthest-west of the main cities of (continental) Eurasia all the way to one of the furthest-East.   (Not quite the furthest, but Changchun is in the northeast of China, just a few hours north of Korea, and it took just about exactly 24 hours including stopovers to get there).  It was a long way to go for a three day workshop, but as there were also three days of a big excursion to Changbai Mountain, just on the border with North Korea, for hiking and general touring around.  So that was a sort of holiday, with 11 other mathematicians.  Here is me with Dany Majard, in a national park along the way to the mountains:

Here’s me with Alex Hoffnung, on Changbai Mountain (in the background is China):

And finally, here’s me a little to the left of the previous picture, where you can see into the volcanic crater.  The lake at the bottom is cut out of the picture, but you can see the crater rim, of which this particular part is in North Korea, as seen from China:

Well, that was fun!

Anyway, the format of the workshop involved some talks from foreigners and some from locals, with a fairly big local audience including a good many graduate students from Jilin University.  So they got a chance to see some new work being done elsewhere – mostly in categorification of one kind or another.  We got a chance to see a little of what’s being done in China, although not as much as we might have. I gather that not much is being done yet that fit the theme of the workshop, which was part of the reason to organize the workshop, and especially for having a session aimed specially at the graduate students.

### Categorified Algebra

This is a sort of broad term, but certainly would include my own talk.  The essential point is to show how the groupoidification of the Heisenberg algebra is a representation of Khovanov’s categorification of the same algebra, in a particular 2-category.  The emphasis here is on the fact that it’s a representation in a 2-category whose objects are groupoids, but whose morphisms aren’t just functors, but spans of functors – that is, composites of functors and co-functors.  This is a pretty conservative weakening of “representations on categories” – but it lets one build really simple combinatorial examples.  I’ve discussed this general subject in recent posts, so I won’t elaborate too much.  The lecture notes are here, if you like, though – they have more detail than my previous post, but are less technical than the paper with Jamie Vicary.

Aaron Lauda gave a nice introduction to the program of categorifying quantum groups, mainly through the example of the special case $U_q(sl_2)$, somewhat along the same lines as in his introductory paper on the subject.  The story which gives the motivation is nice: one has knot invariants such as the Jones polynomial, based on representations of groups and quantum groups.  The Jones polynomial can be categorified to give Khovanov homology (which assigns a complex to a knot, whose graded Euler characteristic is the Jones polynomial) – but also assigns maps of complexes to cobordisms of knots.  One then wants to categorify the representation theory behind it – to describe actions of, for instance, quantum $sl_2$ on categories.  This starting point is nice, because it can work by just mimicking the construction of $sl_2$ and $U_q(sl_2)$ representations in terms of weight spaces: one gets categories $V_{-N}, \dots, V_N$ which correspond to the “weight spaces” (usually just vector spaces), and the $E$ and $F$ operators give functors between them, and so forth.

Finding examples of categories and functors with this structure, and satisfying the right relations, gives “categorified representations” of the algebra – the monoidal categories of diagrams which are the “categorifications of the algebra” then are seen as the abstraction of exactly which relations these are supposed to satisfy.  One such example involves flag varieties.  A flag, as one might eventually guess from the name, is a nested collection of subspaces in some $n$-dimensional space.  A simple example is the Grassmannian $Gr(1,V)$, which is the space of all 1-dimensional subspaces of $V$ (i.e. the projective space $P(V)$), which is of course an algebraic variety.  Likewise, $Gr(k,V)$, the space of all $k$-dimensional subspaces of $V$ is a variety.  The flag variety $Fl(k,k+1,V)$ consists of all pairs $W_k \subset W_{k+1}$, of a $k$-dimensional subspace of $V$, inside a $(k+1)$-dimensional subspace (the case $k=2$ calls to mind the reason for the name: a plane intersecting a given line resembles a flag stuck to a flagpole).  This collection is again a variety.  One can go all the way up to the variety of “complete flags”, $Fl(1,2,\dots,n,V)$ (where $V$ is $n$-dimenisonal), any point of which picks out a subspace of each dimension, each inside the next.

The way this relates to representations is by way of geometric representation theory. One can see those flag varieties of the form $Fl(k,k+1,V)$ as relating the Grassmanians: there are projections $Fl(k,k+1,V) \rightarrow Gr(k,V)$ and $Fl(k,k+1,V) \rightarrow Gr(k+1,V)$, which act by just ignoring one or the other of the two subspaces of a flag.  This pair of maps, by way of pulling-back and pushing-forward functions, gives maps between the cohomology rings of these spaces.  So one gets a sequence $H_0, H_1, \dots, H_n$, and maps between the adjacent ones.  This becomes a representation of the Lie algebra.  Categorifying this, one replaces the cohomology rings with derived categories of sheaves on the flag varieties – then the same sort of “pull-push” operation through (derived categories of sheaves on) the flag varieties defines functors between those categories.  So one gets a categorified representation.

Heather Russell‘s talk, based on this paper with Aaron Lauda, built on the idea that categorified algebras were motivated by Khovanov homology.  The point is that there are really two different kinds of Khovanov homology – the usual kind, and an Odd Khovanov Homology, which is mainly different in that the role played in Khovanov homology by a symmetric algebra is instead played by an exterior (antisymmetric) algebra.  The two look the same over a field of characteristic 2, but otherwise different.  The idea is then that there should be “odd” versions of various structures that show up in the categorifications of $U_q(sl_2)$ (and other algebras) mentioned above.

One example is the fact that, in the “even” form of those categorifications, there is a natural action of the Nil Hecke algebra on composites of the generators.  This is an algebra which can be seen to act on the space of polynomials in $n$ commuting variables, $\mathbb{C}[x_1,\dots,x_n]$, generated by the multiplication operators $x_i$, and the “divided difference operators” based on the swapping of two adjacent variables.  The Hecke algebra is defined in terms of “swap” generators, which satisfy some $q$-deformed variation of the relations that define the symmetric group (and hence its group algebra).   The Nil Hecke algebra is so called since the “swap” (i.e. the divided difference) is nilpotent: the square of the swap is zero.  The way this acts on the objects of the diagrammatic category is reflected by morphisms drawn as crossings of strands, which are then formally forced to satisfy the relations of the Nil Hecke algebra.

The ODD Nil Hecke algebra, on the other hand, is an analogue of this, but the $x_i$ are anti-commuting, and one has different relations satisfied by the generators (they differ by a sign, because of the anti-commutation).  This sort of “oddification” is then supposed to happen all over.  The main point of the talk was to to describe the “odd” version of the categorified representation defined using flag varieties.  Then the odd Nil Hecke algebra acts on that, analogously to the even case above.

Marco Mackaay gave a couple of talks about the $sl_3$ web algebra, describing the results of this paper with Weiwei Pan and Daniel Tubbenhauer.  This is the analog of the above, for $U_q(sl_3)$, describing a diagram calculus which accounts for representations of the quantum group.  The “web algebra” was introduced by Greg Kuperberg – it’s an algebra built from diagrams which can now include some trivalent vertices, along with rules imposing relations on these.  When categorifying, one gets a calculus of “foams” between such diagrams.  Since this is obviously fairly diagram-heavy, I won’t try here to reproduce what’s in the paper – but an important part of is the correspondence between webs and Young Tableaux, since these are labels in the representation theory of the quantum group – so there is some interesting combinatorics here as well.

### Algebraic Structures

Some of the talks were about structures in algebra in a more conventional sense.

Jiang-Hua Lu: On a class of iterated Poisson polynomial algebras.  The starting point of this talk was to look at Poisson brackets on certain spaces and see that they can be found in terms of “semiclassical limits” of some associative product.  That is, the associative product of two elements gives a power series in some parameter $h$ (which one should think of as something like Planck’s constant in a quantum setting).  The “classical” limit is the constant term of the power series, and the “semiclassical” limit is the first-order term.  This gives a Poisson bracket (or rather, the commutator of the associative product does).  In the examples, the spaces where these things are defined are all spaces of polynomials (which makes a lot of explicit computer-driven calculations more convenient). The talk gives a way of constructing a big class of Poisson brackets (having some nice properties: they are “iterated Poisson brackets”) coming from quantum groups as semiclassical limits.  The construction uses words in the generating reflections for the Weyl group of a Lie group $G$.

Li Guo: Successors and Duplicators of Operads – first described a whole range of different algebra-like structures which have come up in various settings, from physics and dynamical systems, through quantum field theory, to Hopf algebras, combinatorics, and so on.  Each of them is some sort of set (or vector space, etc.) with some number of operations satisfying some conditions – in some cases, lots of operations, and even more conditions.  In the slides you can find several examples – pre-Lie and post-Lie algebras, dendriform algebras, quadri- and octo-algebras, etc. etc.  Taken as a big pile of definitions of complicated structures, this seems like a terrible mess.  The point of the talk is to point out that it’s less messy than it appears: first, each definition of an algebra-like structure comes from an operad, which is a formal way of summing up a collection of operations with various “arities” (number of inputs), and relations that have to hold.  The second point is that there are some operations, “successor” and “duplicator”, which take one operad and give another, and that many of these complicated structures can be generated from simple structures by just these two operations.  The “successor” operation for an operad introduces a new product related to old ones – for example, the way one can get a Lie bracket from an associative product by taking the commutator.  The “duplicator” operation takes existing products and introduces two new products, whose sum is the previous one, and which satisfy various nice relations.  Combining these two operations in various ways to various starting points yields up a plethora of apparently complicated structures.

Dany Majard gave a talk about algebraic structures which are related to double groupoids, namely double categories where all the morphisms are invertible.  The first part just defined double categories: graphically, one has horizontal and vertical 1-morphisms, and square 2-morphsims, which compose in both directions.  Then there are several special degenerate cases, in the same way that categories have as degenerate cases (a) sets, seen as categories with only identity morphisms, and (b) monoids, seen as one-object categories.  Double categories have ordinary categories (and hence monoids and sets) as degenerate cases.  Other degenerate cases are 2-categories (horizontal and vertical morphisms are the same thing), and therefore their own special cases, monoidal categories and symmetric monoids.  There is also the special degenerate case of a double monoid (and the extra-special case of a double group).  (The slides have nice pictures showing how they’re all degenerate cases).  Dany then talked about some structure of double group(oids) – and gave a list of properties for double groupoids, (such as being “slim” – having at most one 2-cell per boundary configuration – as well as two others) which ensure that they’re equivalent to the semidirect product of an abelian group with the “bicrossed product”  $H \bowtie K$ of two groups $H$ and $K$ (each of which has to act on the other for this to make sense).  He gave the example of the Poincare double group, which breaks down as a triple bicrossed product by the Iwasawa decomposition:

$Poinc = (SO(3) \bowtie (SO(1; 1) \bowtie N)) \ltimes \mathbb{R}_4$

($N$ is certain group of matrices).  So there’s a unique double group which corresponds to it – it has squares labelled by $\mathbb{R}_4$, and the horizontial and vertical morphisms by elements of $SO(3)$ and $N$ respectively.  Dany finished by explaining that there are higher-dimensional analogs of all this – $n$-tuple categories can be defined recursively by internalization (“internal categories in $(n-1)$-tuple-Cat”).  There are somewhat more sophisticated versions of the same kind of structure, and finally leading up to a special class of $n$-tuple groups.  The analogous theorem says that a special class of them is just the same as the semidirect product of an abelian group with an $n$-fold iterated bicrossed product of groups.

Also in this category, Alex Hoffnung talked about deformation of formal group laws (based on this paper with various collaborators).  FGL’s are are structures with an algebraic operation which satisfies axioms similar to a group, but which can be expressed in terms of power series.  (So, in particular they have an underlying ring, for this to make sense).  In particular, the talk was about formal group algebras – essentially, parametrized deformations of group algebras – and in particular for Hecke Algebras.  Unfortunately, my notes on this talk are mangled, so I’ll just refer to the paper.

### Physics

I’m using the subject-header “physics” to refer to those talks which are most directly inspired by physical ideas, though in fact the talks themselves were mathematical in nature.

Fei Han gave a series of overview talks intorducing “Equivariant Cohomology via Gauged Supersymmetric Field Theory”, explaining the Stolz-Teichner program.  There is more, using tools from differential geometry and cohomology to dig into these theories, but for now a summary will do.  Essentially, the point is that one can look at “fields” as sections of various bundles on manifolds, and these fields are related to cohomology theories.  For instance, the usual cohomology of a space $X$ is a quotient of the space of closed forms (so the $k^{th}$ cohomology, $H^{k}(X) = \Omega^{k}$, is a quotient of the space of closed $k$-forms – the quotient being that forms differing by a coboundary are considered the same).  There’s a similar construction for the $K$-theory $K(X)$, which can be modelled as a quotient of the space of vector bundles over $X$.  Fei Han mentioned topological modular forms, modelled by a quotient of the space of “Fredholm bundles” – bundles of Banach spaces with a Fredholm operator around.

The first two of these examples are known to be related to certain supersymmetric topological quantum field theories.  Now, a TFT is a functor into some kind of vector spaces from a category of $(n-1)$-dimensional manifolds and $n$-dimensional cobordisms

$Z : d-Bord \rightarrow Vect$

Intuitively, it gives a vector space of possible fields on the given space and a linear map on a given spacetime.  A supersymmetric field theory is likewise a functor, but one changes the category of “spacetimes” to have both bosonic and fermionic dimension.  A normal smooth manifold is a ringed space $(M,\mathcal{O})$, since it comes equipped with a sheaf of rings (each open set has an associated ring of smooth functions, and these glue together nicely).  Supersymmetric theories work with manifolds which change this sheaf – so a $d|\delta$-dimensional space has the sheaf of rings where one introduces some new antisymmetric coordinate functions $\theta_i$, the “fermionic dimensions”:

$\mathcal{O}(U) = C^{\infty}(U) \otimes \bigwedge^{\ast}[\theta_1,\dots,\theta_{\delta}]$

Then a supersymmetric TFT is a functor:

$E : (d|\delta)-Bord \rightarrow STV$

(where $STV$ is the category of supersymmetric topological vector spaces – defined similarly).  The connection to cohomology theories is that the classes of such field theories, up to a notion of equivalence called “concordance”, are classified by various cohomology theories.  Ordinary cohomology corresponds then to $0|1$-dimensional extended TFT (that is, with 0 bosonic and 1 fermionic dimension), and $K$-theory to a $1|1$-dimensional extended TFT.  The Stoltz-Teichner Conjecture is that the third example (topological modular forms) is related in the same way to a $2_1$-dimensional extended TFT – so these are the start of a series of cohomology theories related to various-dimension TFT’s.

Last but not least, Chris Rogers spoke about his ideas on “Higher Geometric Quantization”, on which he’s written a number of papers.  This is intended as a sort of categorification of the usual ways of quantizing symplectic manifolds.  I am still trying to catch up on some of the geometry This is rooted in some ideas that have been discussed by Brylinski, for example.  Roughly, the message here is that “categorification” of a space can be thought of as a way of acting on the loop space of a space.  The point is that, if points in a space are objects and paths are morphisms, then a loop space $L(X)$ shifts things by one categorical level: its points are loops in $X$, and its paths are therefore certain 2-morphisms of $X$.  In particular, there is a parallel to the fact that a bundle with connection on a loop space can be thought of as a gerbe on the base space.  Intuitively, one can “parallel transport” things along a path in the loop space, which is a surface given by a path of loops in the original space.  The local description of this situation says that a 1-form (which can give transport along a curve, by integration) on the loop space is associated with a 2-form (giving transport along a surface) on the original space.

Then the idea is that geometric quantization of loop spaces is a sort of higher version of quantization of the original space. This “higher” version is associated with a form of higher degree than the symplectic (2-)form used in geometric quantization of $X$.   The general notion of n-plectic geometry, where the usual symplectic geometry is the case $n=1$, involves a $(n+1)$-form analogous to the usual symplectic form.  Now, there’s a lot more to say here than I properly understand, much less can summarize in a couple of paragraphs.  But the main theorem of the talk gives a relation between n-plectic manifolds (i.e. ones endowed with the right kind of form) and Lie n-algebras built from the complex of forms on the manifold.  An important example (a theorem of Chris’ and John Baez) is that one has a natural example of a 2-plectic manifold in any compact simple Lie group $G$ together with a 3-form naturally constructed from its Maurer-Cartan form.

At any rate, this workshop had a great proportion of interesting talks, and overall, including the chance to see a little more of China, was a great experience!

So this paper of mine was recently accepted by the Journal of Homotopy and Related Structures (the version that was accepted should be reflected on the arXiv by tomorrow – i.e. July 10 – I’m not sure about the journal ). It’s been a while since I sent out the earliest version, and most of the changes have involved figuring out who the audience is, and consequently what could be left out. I guess that’s a side-effect of taking an excerpt from my thesis, which was much longer. In any case, it now seems to have reached a final point. Some of what was in it – the section about cobordisms – is now in a paper (in progress) about TQFT. I don’t see anywhere else to include the other missing bit, however, which has to do with Lawvere theories, and since I just wrote a bunch about MakkaiFest, I thought I might include some of that here.

The paper came about because I was trying to write my thesis, which describes an extended TQFT as a 2-functor (and considers how it could produce a version of 3D quantum gravity). The 2-functor

$Z_G : nCob_2 \rightarrow 2Vect$

(or into $2Hilb$) is an ETQFT. The construction of the 2-functor uses the fact that you can get spans of groupoids out of cospans of manifolds – and in particular, out of cobordisms. One problem is how to describe $nCob_2$ so that this works. It’s actually most naturally a cubical 2-category of some kind. The strict version of this concept is a double category – which has (in principle separate) categories of horizontal and vertical of morphisms, as well as square 2-cells. Ideally, one would like a “weak” version, where composition of squares and morphisms can be only weakly associative (and have weak unit laws). A “pseudocategory” implements this where the only higher-dimensional morphisms are the squares, but it turns out to be strict in one direction, and weak in the other. As it happens, it’s a big pain to use only squares for the 2-morphisms.

Initially it seemed I would have to define a whole new structure to get weak composition in both directions, because in both directions, composition represents gluing bits of manifolds together along boundaries – using a diffeomorphism (or a smooth homeomorphism, depending on which kind of manifolds we’re dealing with). I called it a “double bicategory” and started trying to define it along the same lines as a double category. It then turned out that Dominic Verity had already defined a “double bicategory” – you can read the paper where I talk about how the notions are related. Here I want to talk about a few aspects which I cut out of the paper along the way.

The idea is that there are two ways of “categorifying”: internalization, and enrichment. A bicategory is a category enriched in $Cat$, the category of categories – for any two elements, there’s a whole hom-category of morphisms (and 2-morphisms). A double category is a category internal to $Cat$. This means you can think of it as a category of objects and a category of morphisms, equipped with functors satisfying all the usual properties for the maps in the definition of a category: composition functors, unit functors, and so forth. This definition turns out to be equivalent to the usual one. So I thought: why not do the same with bicategories?

Thus, the way I defined double bicategory was: “A bicategory internal to $Bicat$“. In the paper as it stands, that’s all I say. What I cut out was a sort of dangling loose end pointing toward Lawvere theories – or rather, a variant thereof – finite limit theories (for something more detailed, see this recent paper by Lack and Rosicky). As I mentioned in the previous post, a Lawvere theory is an approach to universal algebra – it formally defines a kind of object (e.g. group, ring, abelian group, etc.) as a functor from a category $T$ which is the “theory” of such objects, while the functor is a “model” of the theory.

What makes it “universal” algebra is that it can involve definitions with many sorts of objects, many operations, given as arrows, of different arities (number of inputs and outputs). This last makes sense in the monoidal context, and in particular Cartesian. Making decisions like this – what class of categories and functors we’re dealing with – specifies which doctrine the theory lives in. In the case of bicategories, this is the doctrine of categories with finite limits. In a Lawvere theory in the original sense, the doctrine is categories with finite products – so if there’s an object $G$, there are also objects $G^n$ for all $n$. Then there are things like multiplication maps $m : G^2 \rightarrow G$ and so on. For a category or bicategory, multiplication might be partial – so we need finite limits. A model of a theory in this doctrine is a limit-preserving functor.

So what does the theory of bicategories look like? It’s easy enough to see if you think that a (small) bicategory is a “bicategory in $Sets$“, and reproduce the usual definition, omitting reference to sets. It has objects $Ob$, $Mor$, and $2Mor$. (This fact already means this is a “multi-sorted” theory, which goes beyond what can be done with another approach to universal algebra based on monads). Funthermore, there are maps between these objects, interpreted as source, target, and identity maps of various sorts. These form diagrams, and since we’re in a finite limit theory, there must be various objects like $Pairs = Mor \times_{Ob} Mor$ which for sets would have the interpretation “pairs of composable morphisms”. Then there’s a composition map $\circ : Pairs \rightarrow Mor$… and so on. In short, in describing the axioms for a bicategory in a “nice” way (i.e. in terms of arrows, commuting diagrams, etc.), we’re giving a presentation of a certain category, $Th(Bicat)$, in generators and relations. Then a model of the theory is a functor $Th(Bicat) \rightarrow \mathcal{C}$ – picking out a “bicategory in $\mathcal{C}$“.

Now, a bicategory in $Sets$ is a bicategory. But a bicategory in $Bicat$ is another matter. First of all, I should say there’s something kind of odd here, since $Bicat$ is most naturally regarded as a tricategory. However, we can regard it as a category by disregarding higher morphisms and taking 2-functors only up to equivalence to make $Bicat$ into an honest category with associative composition. Thus, if we have a functor $F : Th(Bicat) \rightarrow Bicat$, we have:

• Bicategories $F(Ob)$, latex $F(Mor)$, and $F(2Mor)$
• 2-Functors $F(s)$, $F(\circ)$ and so on
• satisfying conditions implied by the bicategory axioms

But each of those bicategories (in $Sets$!) has sets of objects, morphisms, and 2-morphisms, and one can break all the functors apart into three collections of maps acting on each of these three levels. They’ll satisfy all the conditions from the axioms – in fact, they make three new bicategories. So, for example, the object-sets of the bicategories $F(Ob)$, $F(Mor)$ and $F(2Mor)$ form a bicategory using the object maps of the 2-functors $F(s)$ and so on.

So if we say the original bicategories $F(Ob)$ and so on are “horizontal”, and these new ones are “vertical”, we have something resembling a double category, but weak (since bicategories are weak) in both directions. The result is most naturally a four-dimensional structure (the 2-morphisms in $2Mor$ are most conveniently drawn as 4d, which is shown in Table 2 of the paper).

Now, the paper as it is describes all this structure without explicitly mentioning the theory $Th(Bicat)$ except in passing – one can define “internal bicategory” without it. This is why this is a “loose end” of this paper: a major benefit of using Lawvere-style theories is the availability of morphisms of theories, which don’t come up here.

In any case, with this 4D structure in hand, what I do in the paper is (a) get some conditions that allow one to decategorify it down to Verity’s version of “double bicategory” (and even down to a bicategory); and (b) show that couble cospans are an example (double spans would do equally well, but the application is to cobordisms, which are cospans). My own reason for wanting to get down to a 2D structure is the application to extended TQFT, which means we want a 2-category of cobordisms, thought of in terms of (co)spans.

Maybe in a subsequent post I’ll talk about the example itself, but one point about internalization does occur to me. Double cospans give an example of a double bicategory in the sense above – a strict model of $Th(Bicat)$ in $Bicat$. In fact, they consist of “(co)spans of (co)spans” in a way that Marco Grandis formalized in terms of powers $\Lambda^n$, where $\Lambda$ is the diagram (i.e. category) $\bullet \leftarrow \bullet \rightarrow \bullet$. One can actually think of this in terms of internalization: these are spans in a category whose objects are spans in $\mathcal{C}$, and whose morphisms are triples of maps in $C$ linking two spans (likewise for the span-map 2-morphisms). Yet it’s manifestly edge-symmetric: both the horizontal and vertical bicategories are the same.

As I mentioned in the previous post, there are lots of nice examples of double categories which are not edge-symmetric – sets, functions, and relations; or rings, homomorphisms, and bimodules, say. In fact, the second is only a pseudocategory – weak in one direction (composition of bimodules by tensor product is really only defined up to isomorphism). This is a significant thing about non-edge-symmetric examples. There’s much less motive for assuming both directions are equally strict. It’s also more natural in some ways: a pseudocategory is a weak model of $Th(Cat)$ in $Cat$ – equations in the theory are represented by (coherent) isomorphisms. This is the most general situation, and a strict model is a special case.

In the bicategory world, as I said, $Bicat$ is a tricategory, so weaker models than the one I’ve given are possible – though they’re not symmetric, and so while one direction has composition and units as weak as a bicategory, the other direction will be weaker still. Robert Paré, in a conversation at MakkaiFest, suggested that a nice definition for a cubical n-category might have each direction being one step weaker than the previous one – a natural generalization of pseudocategories. Maybe there’s a way to make this seem natural in terms of internalization? One can iterate internalizing: having defined double bicategories, collect them together and find models of $Th(Bicat)$ in $DblBicat$, and so forth. Maybe doing this as weakly as possible would give this tower of increasing weakness.

Now, I don’t have a great punchline to sum all this up, except that internalization seems to be an interesting lens with which to look at cubical n-categories.