### moduli spaces

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.

(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.

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.

So this is a couple of weeks backdated.  I’ve had a pretty serious cold for a while – either it was bad in its own right, or this was just a case of the difference in native viruses between two different continents that my immune system wasn’t prepared for.  Then, too, last week was Republic Day – the 100th anniversary of the middle of three revolutions (the Liberal, the Republican, and the Carnation revolution that ousted the dictatorship regime in 1974 – and let me say that it’s refreshing for a North American to be reminded that Republicanism is a refinement of Liberalism, though how the flowers fit into it is less straightforward).  So my family and I went to attend some of the celebrations downtown, which were impressive.

Anyway, with the TQFT club seminars starting up very shortly, I wanted to finish this post on the first talks I got to see here at IST, which were on pretty widely different topics.  The first was by Ivan Smith, entitled “Quadrics, 3-Manifolds and Floer Cohomology”.  The second was a recorded video talk arranged by the string theory group.  This was a recording of a talk given by Kostas Skenderis a couple of years ago, entitled “The Fuzzball Proposal for Black Holes”.

## Ivan Smith – Quadrics, 3-Manfolds and Floer Cohomology

Ivan Smith’s talk began with some motivating questions from topology, symplectic geometry, and from the study of moduli spaces.  The topological question talks about 3-manifolds $Y$ and the space of representations $Hom(\pi_1(Y),G)$ of its fundamental group into a compact Lie group $G$, which was generally $SO(3)$ or $SU(2)$.  Specifically, the question is how this space is affected by operations on $Y$ such as surgery, taking covering spaces, etc.  The symplectic geometry question asks, for a symplectic manifold $(X,\omega)$, what the “mapping class group” of symplectic transformations – that is, the group $\pi_0(Symp(X))$ of connected components of symplectomorphisms from $X$ to itself – in a sense, this is asking how much of the geometry is seen by the symplectic situation.  The question about moduli spaces asks to characterize the action of the (again, mapping class group of) diffeomorphisms of a Riemann surface on the moduli space of bundles on it.  (This space, for  $\Sigma$ with genus $g \geq 2$, look like $M_g \simeq Hom(\pi_1(\Sigma),SU(2))$ modulo conjugation.  It is the complex-manifold version of the space of flat connections which I’ve been quite interested in for purposes of TQFT, though this is a coarse quotient, not a stack-like quotient.  Lots of people are interested in this space in its various hats.)

The point of the talk being to elucidate how these all fit together.  The first part of the title, “Quadrics”, referred to the fact that, when $\Sigma$ has genus 2, the moduli space we’ll be looking at can be described as an intersection of some varieties (defined by quadric equations) in the projective space $\mathbb{CP}^5$.  Knowing this, one can describe some of its properties just by looking at intersections of curves.

In general we’re talking about complex manifolds, here.  To start with, for Riemann surfaces (one-dimensional complex manifolds), he pointed out that there is an isomorphism between the mapping class groups of symplectomorphisms and diffeomorphisms: $\pi_0(Symp(\Sigma)) \simeq \pi_0(Diff(\Sigma))$.  But in general, for example, for 3-dimensional manifolds, there is structure in the symplectic maps which is forgotten by the smooth ones – there’s still a map $\pi_0(Symp(\Sigma)) \rightarrow \pi_0(Diff(\Sigma))$, but it has a kernel – there are distinct symplectic maps that all look like the identity up to smooth deformation.

Now, our original question was what the action of the diffeomorphisms of on the moduli space $M_g$ of bundles over $\Sigma$.  An element $h$ of $\pi_0(Diff(\Sigma))$ acts (by symplectic map) on it.  The discrepancy we mentioned is that the map corresponding to $h$ will always have fixed points, but be smoothly equivalent to one that doesn’t.  So the smooth mapping class group can’t detect the property of having fixed points.  What it CAN detect, however, is information about intersections.  In particular,   as mentioned above, the moduli space of bundles over a genus 2 surface is an intersection; in this situation, there is an injective map back from the smooth mapping class group into the group of classes of symplectic maps.  So looking symplectically loses nothing from the smooth case.

Now, these symplectic maps tie into the third part of the title, “Floer Homology”, as follows.  Given a symplectic map $\phi : (X,\omega) \rightarrow (X,\omega)$, one can define a complex of vector spaces $HF(\phi)$ which is the usual cohomology of a chain complex generated by fixed points of the map $\phi$, and with a differential $\partial$ which is defined by counting certain curves.  The way this is set up, if $\phi$ is the identity so that all points are fixed points, one gets the usual cohomology of the space $X$ – except that it’s defined so as to be the quantum cohomology of $X$ (for more, check out this tutorial by Givental).  This has the same complex as the usual cohomology, but with the cup product replaced by a deformed product.  It’s an older theorem (due to Donaldson) that, at least for genus 2, the quantum cohomology of the moduli space of bundles over $\Sigma$ splits into a direct sum of rings:

$QH^*(M_2) \cong \mathbb{C} \oplus QH^*(\Sigma_2) \oplus \mathbb{C}$

So one of the key facts is that this works also with Floer homology for other maps than the identity (so this becomes a special case).  So replacing $QH^*$ in the above with $HF^*(\phi)$ for any $\phi$ (acting either on the surface $\Sigma$, or the induced action on the moduli space) still gives a true statement.  Note that this actually implies the theorem that there are fixed points in the space of bundles, since the right hand side is always nontrivial.

So at this point we have some idea of how Floer cohomology is part of what ties the original three questions together.  To take a further look at these we can start to build a category combining much of the same information.  This is the (derived) Fukaya category.  The objects are Lagrangian submanifolds of a symplectic manifold $(X,\omega)$ – ones where the symplectic form vanishes.  To start building the category, consider what we can build from pairs of such objects $(L_1,L_2)$.  This is rather like the above – we define a complex of vector spaces, which is the cohomology of another complex.  Instead of being the complex freely generated by fixed points, though, it’s generated by intersection points of $L_1$ and $L_2$.  This automatically becomes a module over $QH^*(X)$, so the category we’re building is enriched over these.

Defining the structure of this category is apparently a little bit complicated – in particular, there is a composition product $HF(L_1,L_2) \otimes HF(L_2,L_3) \rightarrow HF(L_1,L_3)$ in the form of a cohomology operation.  Furthermore, which Ivan Smith didn’t have time to describe in detail, there are other “higher” products.  These are Massey type products, which is to say higher-order cohomology operations, which involve more than two inputs.  These give the whole structure (where one takes the direct sum of all those hom-modules $HF(L_i,L_j)$ to get one big module) the structure of an $A_{\infty}$-algebra (so the Fukaya category is an $A_{\infty}$-category, I suppose).  This is one way of talking about weak higher categories (the higher products give the associator for composition, and its higher analogs), so in fact this is a pretty complex structure, which the talk didn’t dwell on in detail.  But in any case, the point is that the operations in the category correspond to cohomology operations.

Then one deals with the “derived” Fukaya category $\mathcal{DF}(X)$.  I understand derived categories to be (at least among other examples) a way of taking categories of complexes “up to homotopy”, perhaps as a way of getting rid of some of this complication.  Again, the talk didn’t elaborate too much on this.  However, the fundamental theorem about this category is a generalization of the theorem above above quantum cohomology:

$\mathcal{DF}(M_2) \cong \mathcal{DF}(pt) \oplus \mathcal{DF}(\Sigma_2) \oplus \mathcal{DF}(pt)$

That is, the derived Fukaya category for the moduli space of bundles over $\Sigma_2$ is the category for the Riemann surface itself, summed with two copies of the category for a single point (which is replacing the two copies of $\mathbb{C}$).  This reduces to the previous theorem when we’re looking at the map $\phi = id$, just as before.

So the last question Ivan Smith addressed about this is the fact that these sorts of categories are often hard to calculate explicitly, but they can be described in terms of some easily-described data.  He gave the analogy of periodic functions – which may be quite complicated, but by means of Fourier decompositions, can be easily described in terms of sines and cosines, which are easy to analyze.  In the same way, although the Fukaya categories for particular spaces might be complicated, they can be described in terms of the (derived) category of modules over the $A_{\infty}$-algebras.  In particular, every category $\mathcal{DF}(X)$ embeds in a generic example $\mathcal{D}(mod-A_{\infty}-alg)$.  So by understanding categories like this, one can understand a lot about the categories that come from spaces, which generalize quantum cohomology as described above.

I like this punchline of the analogy with Fourier analysis, as imprecise as it might be, because it suggests a nice way to approach complex entities by finding out the parts that can generate them, or simple but large things you might discover them inside.

## Fuzzballs

The Skenderis talk about black holes was interesting, in that it was a recorded version of a talk given somewhere else – I haven’t seen this done before, but apparently the String Theory group does it pretty regularly.  This has some obvious advantages – they can get a wider range of talks by many different speakers.  There was some technical problem – I suppose due to the way the video was encoded – that meant the slides were sometimes unreadably blurry, but that’s still better than not getting the speaker at all.  I don’t have the background in string theory to be able to really get at the meat of the talk, though it did involve the AdS/CFT correspondence.  However, I can at least say a few concrete things about the motivation.  First, the “fuzzball” proposal is a more-or-less specific proposal to deal with the problem of black hole entropy.

The problem, basically, is that it’s known that the thermodynamic entropy associated to a black hole – which can be computed in completely macroscopic terms – is proportional to the area of its horizon.  On the other hand, in essentially every other setting, entropy has an interpretation in terms of counting microstates, so that the entropy of a “macrostate” is proportional to the logarithm of the number of microstates.  (Or, in a thermal state, which is a statistical distribution, this is weighted by the probability of the microstate).  So, for example, with a gas in a box, there are many macrostates that correspond to a relatively even distribution of position and momentum among the molecules, and relatively few in which all molecules are all in one small corner of the box.

The reason this is a problem is that, classically, the state of a black hole is characterized by very few numbers: the mass, angular momentum, and electric charge.   There doesn’t seem to be room for “microstates” in a classical black hole.  So the overall point of the proposal is to describe what microstates would be.  The specific way this is done with “fuzzballs” is somewhat mysterious to me, but the overall idea makes sense.  One interesting consequence of this approach is that event horizons would be strictly a property of thermal states, in whatever underlying theory one takes to be the quantum theory behind classical gravity (here assumed to be some specific form of string theory – the example he was using is something called the B1-B5 black hole, which I know nothing about).  That’s because a pure state would have a single microstate, hence have zero entropy, hence no horizon.

Now, what little I do understand about the particular model relies on the fact that near a (classical) event horizon, the background metric has a component that looks like anti-deSitter space – a vacuum solution to the Einstein equations with a negative cosmological constant.  (This part isn’t so hard to see – AdS space has that “saddle-shaped” appearance of a hyperbolic surface, and so does the area around a horizon, even when you draw it like this.)  But then, there is the AdS/CFT correspondence that says states for a gravitational field in (asymptotically) anti-deSitter space correspond to states for a conformal field theory (CFT) at the boundary.  So the way to get microstates, in the “fuzzball” proposal, is to look at this CFT, and find geometries that correspond to them.  Some would be well-approximated by the classical, horizon-ridden geometry, but others would be different.  The fact that this CFT is defined at the boundary explains why entropy would be proportional to area, not volume, of the black hole – this being a manifestation of the so-called “holographic principle”.  The “fuzziness” that one throws away by reducing a thermal state that combines these many geometries to the classical “no-hair” black hole determined by just three numbers is exactly the information described by the entropy.

I couldn’t follow some parts of it, not having much string-theory background – I don’t feel qualified to judge whether string theory makes sense as physics, but it isn’t an approach I’ve studied much.  Still, this talk did reinforce my feeling that the AdS/CFT correspondence, at the very least, is something well-worth learning about and important in its own right.

Coming soon: descriptions of the TQFT club seminars which are starting up at IST.

As I mentioned in my previous post, I’ve recently started out a new postdoc at IST – the Instituto Superior Tecnico in Lisbon, Portugal.  Making the move from North America to Europe with my family was a lot of work – both before and after the move – involving lots of paperwork and shifting of heavy objects.  But Lisbon is a good city, with lots of interesting things to do, and the maths department at IST is very large, with about a hundred faculty.  Among those are quite a few people doing things that interest me.

The group that I am actually part of is coordinated by Roger Picken, and has a focus on things related to Topological Quantum Field Theory.  There are a couple of postdocs and some graduate students here associated in some degree with the group, and elsewhere than IST Aleksandar Mikovic and Joao Faria Martins.   In the coming months there should be some activity going on in this group which I will get to talk about here, including a workshop which is still in development, so I’ll hold off on that until there’s an official announcement.

## Quantales

I’ve also had a chance to talk a bit with Pedro Resende, mostly on the subject of quantales.  This is something that I got interested in while at UWO, where there is a large contingent of people interested in category theory (mostly from the point of view of homotopy theory) as well as a good group in noncommutative geometry.  Quantales were originally introduced by Chris Mulvey – I’ve been looking recently at a few papers in which he gives a nice account of the subject – here, here, and here.
The idea emerged, in part, as a way of combining two different approaches to generalising the idea of a space.  One is the approach from topos theory, and more specifically, the generalisation of topological spaces to locales.  This direction also has connections to logic – a topos is a good setting for intuitionistic, but nevertheless classical, logic, whereas quantales give an approach to quantum logics in a similar spirit.

The other direction in which they generalize space is the $C^{\star}$-algebra approach used in noncommutative geometry.  One motivation of quantales is to say that they simultaneously incorporate the generalizations made in both of these directions – so that both locales and $C^{\star}$-algebras will give examples.  In particular, a quantale is a kind of lattice, intended to have the same sort of relation to a noncommutative space as a locale has to an ordinary topological space.  So to begin, I’ll look at locales.

A locale is a lattice which formally resembles the lattice of open sets for such a space.  A lattice is a partial order with operations $\bigwedge$ (“meet”) and $\bigvee$ (“join”).  These operations take the role of the intersection and union of open sets.  So to say it formally resembles a lattice of open sets means that the lattice is closed under arbitrary joins, and finite meets, and satisfies the distributive law:

$U \bigwedge (\bigvee_i V_i) =\bigvee_i (U \bigwedge V_i)$

Lattices like this can be called either “Frames” or “Locales” – the only difference between these two categories is the direction of the arrows.  A map of lattices is a function that preserves all the structure – order, meet, and join.   This is a frame morphism, but it’s also a morphism of locales in the opposite direction.  That is, $\mathbf{Frm} = \mathbf{Loc}^{op}$.

Another name for this sort of object is a “Heyting algebra”.  One of the great things about topos theory (of which this is a tiny starting point) is that it unifies topology and logic.  So, the “internal logic” of a topos has a Heyting algebra (i.e. a locale) of truth values, where the meet and join take the place of logical operators “and” and “or”.  The usual two-valued logic is the initial object in $\mathbf{Loc}$, so while it is special, it isn’t unique.  One vital fact here is that any topological space (via the lattice of open sets) produces a locale, and the locale is enough to identify the space – so $\mathbf{Top} \rightarrow \mathbf{Loc}$ is an embedding.  (For convenience, I’m eliding over the fact that the spaces have to be “sober” – for example, Hausdorff.)  In terms of logic, we could imagine that the space is a “state space”, and the truth values in the logic identify for which states a given proposition is true.  There’s nothing particularly exotic about this: “it is raining” is a statement whose truth is local, in that it depends on where and when you happen to look.

To see locales as a generalisation of spaces, it helps to note that the embedding above is full – if $A$ and $B$ are locales that come from topological spaces, there are no extra morphisms in $\mathbf{Loc}(A,B)$ that don’t come from continuous maps in $\mathbf{Top}(A,B)$.  So the category of locales makes the category of topological spaces bigger only by adding more objects – not inventing new morphisms.  The analogous noncommutative statement turns out not to be true for quantales, which is a little red-flag warning which Pedro Resende pointed out to me.

What would this statement be?  Well, the noncommutative analogue of the idea of a topological space comes from another embedding of categories.  To start with, there is an equivalence $\mathbf{LCptHaus}^{op} \simeq \mathbf{CommC}^{\star}\mathbf{Alg}$: the category of locally compact, Hausdorff, topological spaces is (up to equivalence) the opposite of the category of commutative $C^{\star}$-algebras.  So one simply takes the larger category of all $C^{\star}$-algebras (or rather, its opposite) as the category of “noncommutative spaces”, which includes the commutative ones – the original locally compact Hausdorff spaces.  The correspondence between an algebra and a space is given by taking the algebra of functions on the space.

So what is a quantale?  It’s a lattice which is formally similar to the lattice of subspaces in some $C^{\star}$-algebra.  Special elements – “right”, “left,” or “two-sided” elements – then resemble those subspaces that happen to be ideals.  Some intuition comes from thinking about where the two generalizations coincide – a (locally compact) topological space.  There is a lattice of open sets, of course.  In the algebra of continuous functions, each open set $O$ determines an ideal – namely, the subspace of functions which vanish on $O$.  When such an ideal is norm-closed, it will correspond to an open set (it’s easy to see that continuous functions which can be approximated by those vanishing on an open set will also do so – if the set is not open, this isn’t the case).

So the definition of a quantale looks much like that for a locale, except that the meet operation $\bigwedge$ is replaced by an associative product, usually called $\&$.  Note that unlike the meet, this isn’t assumed to be commutative – this is the point where the generalization happens.  So in particular, any locate gives a quantale with $\& = \bigwedge$.  So does any $C^{\star}$-algebra, in the form of its lattice of ideals.  But there are others which don’t show up in either of these two ways, so one might hope to say this is a nice all-encompassing generalisation of the idea of space.

Now, as I said, there was a bit of a warning that comes attached to this hope.  This is that, although there is an embedding of the category of $C^{\star}$-algebras into the category of quantales, it isn’t full.  That is, not only does one get new objects, one gets new morphisms between old objects.  So, given algebras $A$ and $B$, which we think of as noncommutative spaces, and a map of algebras between them, we get a morphism between the associated quantales – lattice maps that preserve the operations.  However, unlike what happened with locales, there are quantale morphisms that don’t correspond to algebra maps.  Even worse, this is still true even in the case where the algebras are commutative, and just come from locally compact Hausdorff spaces: the associated quantales still may have extra morphisms that don’t come from continuous functions.

There seem to be three possible attitudes to this situation.  First, maybe this is just the wrong approach to generalising spaces altogether, and the hints in its favour are simply misleading.  Second, maybe quantales are absolutely the right generalisation of space, and these new morphisms are telling us something profound and interesting.  The third attitude, which Pedro mentioned when pointing out this problem to me, seems most likely, and goes as follows.  There is something special that happens with $C^{\star}$-algebras, where the analytic structure of the norm makes the algebras more rigid than one might expect.  In algebraic geometry, one can take a space (algebraic variety or scheme) and consider its algebra of global functions.  To make sure that an algebra map corresponds to a map of schemes, though, one really needs to make sure that it actually respects the whole structure sheaf for the space – which describe local functions.  When passing from a topological space to a $C^{\star}$-algebra, there is a norm structure that comes into play, which is rigid enough that all algebra morphisms will automatically do this – as I said above, the structure of ideals of the algebra tells you all about the open sets.  So the third option is to say that a quantale in itself doesn’t quite have enough information, and one needs some extra data something like the structure sheaf for a scheme.  This would then pick out which are the “good” morphisms between two quantales – namely, the ones that preserve this extra data.  What, precisely, this data ought to be isn’t so clear, though, at least to me.

So there are some complications to treating a quantale as a space.  One further point, which may or may not go anywhere, is that this type of lattice doesn’t quite get along with quantum logic in quite the same way that locales get along with (intuitionistic) classical logic (though it does have connections to linear logic).

In particular, a quantale is a distributive lattice (though taking the product, rather than $\bigwedge$, as the thing which distributes over $\bigvee$), whereas the “propositional lattice” in quantum logic need not be distributive.  One can understand the failure of distributivity in terms of the uncertainty principle.  Take a statement such as “particle $X$ has momentum $p$ and is either on the left or right of this barrier”.  Since position and momentum are conjugate variables, and momentum has been determined completely, the position is completely uncertain, so we can’t truthfully say either “particle $X$ has momentum $p$ and is on the left or “particle $X$ has momentum $p$ and is on the right”.  Thus, the combined statement that either one or the other isn’t true, even though that’s exactly what the distributive law says: “P and (Q or S) = (P and Q) or (P and S)”.

The lack of distributivity shows up in a standard example of a quantum logic.  This is one where the (truth values of) propositions denote subspaces of a vector space $V$.  “And” (the meet operation $\bigwedge$) denotes the intersection of subspaces, while “or” (the join operation $\bigvee$) is the direct sum $\oplus$.  Consider two distinct lines through the origin of $V$ – any other line in the plane they span has trivial intersection with either one, but lies entirely in the direct sum.  So the lattice of subspaces is non-distributive.  What the lattice for a quantum logic should be is orthocomplemented, which happens when $V$ has an inner product – so for any subspace $W$, there is an orthogonal complement $W^{\bot}$.

Quantum logics are not very good from a logician’s point of view, though – lacking distributivity, they also lack a sensible notion of implication, and hence there’s no good idea of a proof system.  Non-distributive lattices are fine (I just gave an example), and very much in keeping with the quantum-theoretic strategy of replacing configuration spaces with Hilbert spaces, and subsets with subspaces… but viewing them as logics is troublesome, so maybe that’s the source of the problem.

Now, in a quantale, there may be a “meet” operation, separate from the product, which is non-distributive, but if the product is taken to be the analog of “and”, then the corresponding logic is something different.  In fact, the natural form of logic related to quantales is linear logic. This is also considered relevant to quantum mechanics and quantum computation, and as a logic is much more tractable.  The internal semantics of certain monoidal categories – namely, star-autonomous ones (which have a nice notion of dual) – can be described in terms of linear logic (a fairly extensive explanation is found in this paper by Paul-André Melliès).

Part of the point in the connection seems to be resource-limitedness: in linear logic, one can only use a “resource” (which, in standard logic, might be a truth value, but in computation could be the state of some memory register) a limited number of times – often just once.  This seems to be related to the noncommutativity of $\&$ in a quantale.  The way Pedro Resende described this to me is in terms of observations of a system.  In the ordinary (commutative) logic of a locale, you can form statements such as “A is true, AND B is true, AND C is true” – whose truth value is locally defined.  In a quantale, the product operation allows you to say something like “I observed A, AND THEN observed B, AND THEN observed C”.  Even leaving aside quantum physics, it’s not hard to imagine that in a system which you observe by interacting with it, statements like this will be order-dependent.  I still don’t quite see exactly how these two frameworks are related, though.

On the other hand, the kind of orthocomplemented lattice that is formed by the subspaces of a Hilbert space CAN be recovered in (at least some) quantale settings.  Pedro gave me a nice example: take a Hilbert space $H$, and the collection of all projection operators on it, $P(H)$.  This is one of those orthocomplemented lattices again, since projections and subspaces are closely related.  There’s a quantale that can be formed out of its endomorphisms, $End(P(H))$, where the product is composition.  In any quantale, one can talk about the “right” elements (and the “left” elements, and “two sided” elements), by analogy with right/left/two-sided ideals – these are elements which, if you take the product with the maximal element, $1$, the result is less than or equal to what you started with: $a \& 1 \leq a$ means $a$ is a right element.  The right elements of the quantale I just mentioned happen to form a lattice which is just isomorphic to $P(H)$.

So in this case, the quantale, with its connections to linear logic, also has a sublattice which can be described in terms of quantum logic.  This is a more complicated situation than the relation between locales and intuitionistic logic, but maybe this is the best sort of connection one can expect here.

In short, both in terms of logic and spaces, hoping quantales will be “just” a noncommutative variation on locales seems to set one up to be disappointed as things turn out to be more complex.  On the other hand, this complexity may be revealing something interesting.

Coming soon: summaries of some talks I’ve attended here recently, including Ivan Smith on 3-manifolds, symplectic geometry, and Floer cohomology.

I recently went to California to visit Derek Wise at UC Davis – we were talking about expanding the talk he gave at Perimeter Institute into a more developed paper about ETQFT from Lie groups. Between that, the end of the Winter semester, and the beginning of the “Summer” session (in which I’m teaching linear algebra), it’s taken me a while to write up Emre Coskun’s two-part talk in our Stacks And Groupoids seminar.

Emre was explaining the theory of gerbes in terms of stacks. One way that I have often heard gerbes explained is in terms of a categorification of vector bundles – thus, the theory of “bundle gerbes”, as described by Murray in this paper here. The essential point of that point of view is that bundles can be put together by taking trivial bundles on little neighbourhoods of a space, and “gluing” them together on two-fold overlaps of those neighbourhoods – the gluing functions then have to satisfy a cocycle condition so that they agree on triple overlaps. A gerbe, on the other hand, defines line bundles (not functions) on double overlaps, and the gluing functions now live on triple overlaps. The idea is that this begins a heirarchy of concepts, each of which categorifes the previous (after “gerbe”, one just starts using terms like “2-gerbe”, “3-gerbe”, and so on). The levels of this hierarchy are supposed to be related to the various (nonabelian) cohomology groups $H^n(X,G)$ of a space $X$. I’ve mostly seen this point of view related to work by Jean-Luc Brylinski. It is a very differential-geometric sort of construction.

Emre, on the other hand, was describing another side to the theory of gerbes, which comes out of algebraic geometry, and is closely related to stacks. There’s a nice survey by Moerdijk which gives an account of gerbes from a similar point of view, though for later material, Emre said he drew on this book by Laumon and Moret-Bailly (which I can only find in the original French). As one might expect, a stack-theoretic view of gerbes thinks of them as generalizations of sheaves, rather than bundles. (The fact that there is a sheaf of sections of a bundle also generalizes to gerbes, so bundle-gerbes are a special case of this point of view).

Gerbes

So the setup is that we have some space $X$ – Emre was talking about the context of algebraic geometry, so the relevant idea of space here is scheme (which, if you’re interested, is assumed to have the etale topology – i.e. the one where covers use etale maps, the analog of local isomorphisms).  In the second talk, he generalized this to $S$-spaces: for some chosen scheme $S$.  That is, the category of “spaces” is based on the slice category $Sch/S$ of schemes equipped with maps into $S$, with the obvious morphisms.  This is a site, since there’s a notion of a cover over $S$ and so forth; an $S$-space is a sheaf (of sets) on this site.  So in particular, a scheme $X$ over $S$ determines an $S$-space, where $X : Sch/S \rightarrow Sets$ by $X(U) = Hom(U,X)$.  (That is, the usual way a space determines a representable sheaf).  There are also differential-geometric versions of gerbes.

So, whatever the right notion of space, a stack $\mathbb{F}$ over a space $X$ (in the sense of a sheaf of groupoids over $X$, which we’re assuming has the etale topology) is a gerbe if a couple of nice conditions apply:

1. There’s a cover $\{ U_i \rightarrow X \}$, such that none of the $\mathbb{F}(U_i)$ is empty.
2. Over any open $U$, all the objects $\mathbb{F}(U)$ are isomorphic (i.e. $\mathbb{F}(U)$ is connected as a category)

Notice that there doesn’t have to be a global object – that is, $\mathbb{F}(X)$ needn’t be empty – only some cover such that local objects exist – but where they do, they’re all “the same”.  These conditions can also be summarized in terms of the fibred category $\mathcal{F} \rightarrow X$.  There are two maps from $\pi, \Delta: \mathcal{F}\rightarrow \mathcal{F} \times_X \mathcal{F}$ – the projection and the diagonal.  The conditions respectively say these two maps are, locally, epi (i.e. surjective).

Emre’s first talk began by giving some examples of gerbes to motivate the rest. The first one is the “gerbe of splittings” of an Azumaya algebra. “An” Azumaya algebra $\mathcal{A}$ is actually a sheaf of algebras over some scheme $X$. The defining property is that locally it looks like the algebra of endomorphisms of a vector bundle. That is, on any neighborhood $U_i \subset X$, we have:

$\mathcal{A}(U_i) \cong End(\mathcal{E}_i)$

for some (algebraic) vector bundle $\mathcal{E}_i \rightarrow U_i$. A special case is when $X = Spec(\mathbb{R})$ is just a point, in which case an Azumaya algebra $\mathcal{A}$ is the same thing as a matrix algebra $M_n(\mathbb{R})$. So Azumaya algebras are not too complicated to describe.

The gerbe of splittings, $\mathbb{F}_{\mathcal{A}}$ for an Azumaya algebra is also not too complicated: a splitting is a way to represent an algebra as endomorphisms of a vector bundle – which in this case may only be possible locally. Over an given $U$, its objects are pairs $(E, \alpha)$, where $E$ is a vector bundle over $U$, and $\alpha : End(E) \rightarrow \mathbb{F}_{\mathcal{A}}(U)$ is an isomorphism. The morphisms are bundle isomorphisms that commute with the $\alpha$. So, roughly: if $\mathcal{A}$ is locally isomorphic to endomorphisms of vector bundles, the gerbe of splittings is the stack of all the vector bundles and isomorphisms which make this work. It’s easy to see this is a gerbe, since by definition, such bundles must exist locally, and necessarily they’re all isomorphic.

(This example – a gerbe consisting, locally, of a category of all vector bundles of a certain form – starts to suggest why one might want to think of gerbes as categorifying bundles.)

Another easily constructed gerbe in a similar spirit is found from a complex line bundle $\mathcal{L}$ over $X$ (and $n \in \mathbb{N}$). Then $\mathcal{X} \rightarrow X$ is a gerbe over $X$, where the groupoid $\mathcal{X}(U)$ over a neighborhood $U$ has, as objects, pairs $(\mathcal{M},\alpha)$ where $\alpha : \mathcal{M}^n \rightarrow \mathcal{L}$ is an isomorphism of line bundles. That is, the objects locally look like $n^{th}$ roots of $\mathcal{L}$. The gerbe is trivial (has a global object) if $\mathcal{L}$ has a root.

Cohomology

One says that a gerbe is banded by a sheaf of groups $\mathbb{G}$ on $X$ (or $\mathbb{G}$ is the band of the gerbe, or $\mathbb{F}$ is a $\mathbb{G}$-gerbe), if there are isomorphisms between the group $\mathbb{G}(U)$ and the automorphism group $Aut(u)$ for each object $u$ over $U$ (the property of a gerbe means these are all isomorphic). (These isomorphisms should also commute with the group homomorphisms induced by maps $\psi : V \rightarrow U$ of open sets.) So the band is, so to speak, the “local symmetry group over $U$” of the gerbe in a natural way.

In the case of the gerbe of splittings of $\mathcal{A}$ above, the band is $\mathbb{G}_m$, where over any given neighborhood, $\mathbb{G}_m(U) = Hom(U, G_m)$, where $G_m$ is the group of units in the base field: that is, the group $\mathbb{G}_m(U)$ consists of all the invertible sections in the structure sheaf of $X$. These get turned into bundle-automorphisms by taking a function $f$ to the automorphism that acts through multiplication by $f$. The gerbe $\mathcal{X}$ associated to a line bundle is banded by the group of $n^{th}$-roots of unity in sections in the structure sheaf.

From here, we can see how gerbes relate to cohomology. In particular, a $\mathbb{G}$-gerbe $\mathbb{F}$, we can associate a cohomology class $[F] \in H^2(X,\mathbb{G})$. This class can be thought of as “the obstruction to the existence of a global object”. So, in the case of an Azumaya algebra, it’s the obstruction to $\mathcal{A}$ being split (i.e. globally).

The way this works is, given a covering with an object $x_i$ in $\mathbb{F}(U_i)$, we take pull back this object along the morphisms corresponding to inclusions of sub-neighbourhoods, down to a triple-overlap $U_{ijk} = U_i \cup U_j \cup U_k$. Then there are isomorphisms comparing the different pullbacks: $u_{ij}^k : {x^i}_j^k \rightarrow x_i^{jk}$, and so on. (The lowered indices denote which of the $U$ we’re pulling back from).

Then we get a 2-cocycle in $\mathbb{G}(U_{ijk}$ (an isomorphism corresponding to what, for sheaves of sets, would be an identity). This is $c_{ijk} = u^i_{jk} ({u_i}^k_k)^{-1} u_{ij}^k$. The existence of this cocycle means that we’re getting an element in $H^2(X,\mathbb{G}$, which we denote $[\mathbb{F}]$. If a global object exists, then all our local objects are restrictions of a global one, the cocycle will always turn out to be the identity, so this class is trivial. A non-trivial class implies an obstruction to gluing the local objects into global ones.

Moduli Spaces

In the second talk, Emre gave some more examples of gerbes which it makes sense to think of as moduli spaces, including one which any gerbe resembles locally.

The first is the moduli space of all vector bundles $E$ over some (smooth, projective) curve $C$.  (Actually, one looks at those of some particular degree $d$ and rank $r$, and requires a condition called stability).

Actually, as discussed earlier in the seminar back in Aji’s talk, the right way to see this is that there is a “fine” moduli space – really a stack and not necessarily a space (in whichever context) – called $\mathcal{M}_C(r,d)$, and also a “coarse” moduli space called $M_C(r,d)$.  Roughly, the actual space $M_C(r,d)$ has points which are the isomorphism classes of vector bundles, while the stack remembers the whole groupoid of bundles and bundle-isomorphisms.  So there’s a map, which squashes a bundle to its isomorphism class: $\mathcal{M}_C(r,d) \rightarrow M_C(r,d)$ making the fine moduli space into a category fibred in groupoids – more than that, it’s a stack – and more than that, it’s a gerbe.  That is, there’s always a cover of $C$ such that there are some bundles locally, and (stable) bundles of a given rank and degree are always isomorphic.  In fact, this is a $\mathbb{G}_m$-gerbe, as above.

The next example is the gerbe of $G$-torsors, for a group $G$ (that is, $G$-sets which are isomorphic as $G$-sets to $G$ – the intuition is that they’re just like the group, but without a specified identity). The category $[\star / G ] = BG$ consists of $G$-torsors and their isomorphisms.  This is a gerbe over the point $\star$.  More interesting, when we’re in the context of $S$-spaces (and $S$ has a trivial action of $G$ on it), it becomes a $G$-gerbe over $S$.  Part of the point here is that any trivial gerbe (i.e. one with a section) is just such a classifying space for some group.  In particular, for the group of isomorphisms from a particular object to itself, crossed with $X$.

Since any gerbe has sections locally (that is, objects in $\mathbb{F}(U)$ for some $U$), every gerbe locally looks like one of these classifying-space gerbes.  This is the analog to the fact that any bundle locally looks like a product.

I just posted the slides for “Groupoidification and 2-Linearization”, the colloquium talk I gave at Dalhousie when I was up in Halifax last week. I also gave a seminar talk in which I described the quantum harmonic oscillator and extended TQFT as examples of these processes, which covered similar stuff to the examples in a talk I gave at Ottawa, as well as some more categorical details.

Now, in the previous post, I was talking about different notions of the “state” of a system – all of which are in some sense “dual to observables”, although exactly what sense depends on which notion you’re looking at. Each concept has its own particular “type” of thing which represents a state: an element-of-a-set, a function-on-a-set, a vector-in-(projective)-Hilbert-space, and a functional-on-operators. In light of the above slides, I wanted to continue with this little bestiary of ontologies for “states” and mention the versions suggested by groupoidification.

State as Generalized Stuff Type

This is what groupoidification introduces: the idea of a state in $Span(Gpd)$. As I said in the previous post, the key concepts behind this program are state, symmetry, and history. “State” is in some sense a logical primitive here – given a bunch of “pure” states for a system (in the harmonic oscillator, you use the nonnegative integers, representing n-photon energy states of the oscillator), and their local symmetries (the $n$-particle state is acted on by the permutation group on $n$ elements), one defines a groupoid.

So at a first approximation, this is like the “element of a set” picture of state, except that I’m now taking a groupoid instead of a set. In a more general language, we might prefer to say we’re talking about a stack, which we can think of as a groupoid up to some kind of equivalence, specifically Morita equivalence. But in any case, the image is still that a state is an object in the groupoid, or point in the stack which is just generalizing an element of a set or point in configuration space.

However, what is an “element” of a set $S$? It’s a map into $S$ from the terminal element in $\mathbf{Sets}$, which is “the” one-element set – or, likewise, in $\mathbf{Gpd}$, from the terminal groupoid, which has only one object and its identity morphism. However, this is a category where the arrows are set maps. When we introduce the idea of a “history “, we’re moving into a category where the arrows are spans, $A \stackrel{s}{\leftarrow} X \stackrel{t}{\rightarrow} B$ (which by abuse of notation sometimes gets called $X$ but more formally $(X,s,t)$). A span represents a set/groupoid/stack of histories, with source and target maps into the sets/groupoids/stacks of states of the system at the beginning and end of the process represented by $X$.

Then we don’t have a terminal object anymore, but the same object $1$ is still around – only the morphisms in and out are different. Its new special property is that it’s a monoidal unit. So now a map from the monoidal unit is a span $1 \stackrel{!}{\rightarrow} X \stackrel{\Phi}{\rightarrow} B$. Since the map on the left is unique, by definition of “terminal”, this really just given by the functor $\Phi$, the target map. This is a fibration over $B$, called here $\Phi$ for “phi”-bration, but this is appropriate, since it corresponds to what’s usually thought of as a wavefunction $\phi$.

This correspondence is what groupoidification is all about – it has to do with taking the groupoid cardinality of fibres, where a “phi”bre of $\Phi$ is the essential preimage of an object $b \in B$ – everything whose image is isomorphic to $b$. This gives an equivariant function on $B$ – really a function of isomorphism classes. (If we were being crude about the symmetries, it would be a function on the quotient space – which is often what you see in real mechanics, when configuration spaces are given by quotients by the action of some symmetry group).

In the case where $B$ is the groupoid of finite sets and bijections (sometimes called $\mathbf{FinSet_0}$), these fibrations are the “stuff types” of Baez and Dolan. This is a groupoid with something of a notion of “underlying set” – although a forgetful functor $U: C \rightarrow \mathbf{FinSet_0}$ (giving “underlying sets” for objects in a category $C$) is really supposed to be faithful (so that $C$-morphisms are determined by their underlying set map). In a fibration, we don’t necessarily have this. The special case corresponds to “structure types” (or combinatorial species), where $X$ is a groupoid of “structured sets”, with an underlying set functor (actually, species are usually described in terms of the reverse, fibre-selecting functor $\mathbf{FinSet_0} \rightarrow \mathbf{Sets}$, where the image of a finite set consists of the set of all “$\Phi$-structured” sets (such as: “graphs on set $S$“, or “trees on $S$“, etc.) The fibres of a stuff type are sets equipped with “stuff”, which may have its own nontrivial morphisms (for example, we could have the groupoid of pairs of sets, and the “underlying” functor $\Phi$ selects the first one).

Over a general groupoid, we have a similar picture, but instead of having an underlying finite set, we just have an “underlying $B$-object”. These generalized stuff types are “states” for a system with a configuration groupoid, in $Span(\mathbf{Gpd})$. Notice that the notion of “state” here really depends on what the arrows in the category of states are – histories (i.e. spans), or just plain maps.

Intuitively, such a state is some kind of “ensemble”, in statistical or quantum jargon. It says the state of affairs is some jumble of many configurations (which we apparently should see as histories starting from the vacuous unit $1$), each of which has some “underlying” pure state (such as energy level, or what-have-you). The cardinality operation turns this into a linear combination of pure states by defining weights for each configuration in the ensemble collected in $X$.

2-State as Representation

A linear combination of pure states is, as I said, an equivariant function on the objects of $B$. It’s one way to “categorify” the view of a state as a vector in a Hilbert space, or map from $\mathbb{C}$ (i.e. a point in the projective Hilbert space of lines in the Hilbert space $H = \mathbb{C}[\underline{B}]$), which is really what’s defined by one of these ensembles.

The idea of 2-linearization is to categorify, not a specific state $\phi \in H$, but the concept of state. So it should be a 2-vector in a 2-Hilbert space associated to $B$. The Hilbert space $H$ was some space of functions into $mathbb{C}$, which we categorify by taking instead of a base field, a base category, namely $\mathbf{Vect}_{\mathbb{C}}$. A 2-Hilbert space will be a category of functors into $\mathbf{Vect}_{\mathbb{C}}$ – that is, the representation category of the groupoid $B$.

(This is all fine for finite groupoids. In the inifinte case, there are some issues: it seems we really should be thinking of the 2-Hilbert space as category of representations of an algebra. In the finite case, the groupoid algebra is a finite dimensional C*-algebra – that is, just a direct sum (over iso. classes of objects) of matrix algebras, which are the group algebras for the automorphism groups at each object. In the infinite dimensional world, you probable should be looking at the representations of the von Neumann algebra completion of the C*-algebra you get from the groupoid. There are all sorts of analysis issues about measurability that lurk in this area, but they don’t really affect how you interpret “state” in this picture, so I’ll skip it.)

A “2-state”, or 2-vector in this Hilbert space, is a representation of the groupoid(-algebra) associated to the system. The “pure” states are irreducible representations – these generate all the others under the operations of the 2-Hilbert space (“sum”, “scalar product”, etc. in their 2-vector space forms). Now, an irreducible representation of a von Neumann algebra is called a “superselection sector” for a quantum system. It’s playing the role of a pure state here.

There’s an interesting connection here to the concept of state as a functional on a von Neumann algebra. As I described in the last post, the GNS representation associates a representation of the algebra to a state. In fact, the GNS representation is irreducible just when the state is a pure state. But this notion of a superselection sector makes it seem that the concept of 2-state has a place in its own right, not just by this correspondence.

So: if a quantum system is represented by an algebra $\mathcal{A}$ of operators on a Hilbert space $H$, that representation is a direct sum (or direct integral, as the case may be) of irreducible ones, which are “sectors” of the theory, in that any operator in $\mathcal{A}$ can’t take a vector out of one of these “sectors”. Physicists often associate them with conserved quantities – though “superselection” sectors are a bit more thorough: a mere “selection sector” is a subspace where the projection onto it commutes with some subalgebra of observables which represent conserved quantities. A superselection sector can equivalently be defined as a subspace whose corresponding projection operator commutes with EVERYTHING in $\mathcal{A}$. In this case, it’s because we shouldn’t have thought of the representation as a single Hilbert space: it’s a 2-vector in $\mathbb{Rep}(\mathcal{A})$ – but as a direct integral of some Hilbert bundle that lives on the space of irreps. Those projections are just part of the definition of such a bundle. The fact that $\mathcal{A}$ acts on this bundle fibre-wise is just a consequence of the fact that the total $H$ is a space of sections of the “2-state”. These correspond to “states” in usual sense in the physical interpretation.

Now, there are 2-linear maps that intermix these superselection sectors: the ETQFT picture gives nice examples. Such a map, for example, comes up when you think of two particles colliding (drawn in that world as the collision of two circles to form one circle). The superselection sectors for the particles are labelled by (in one special case) mass and spin – anyway, some conserved quantities. But these are, so to say, “rest mass” – so there are many possible outcomes of a collision, depending on the relative motion of the particles. So these 2-maps describe changes in the system (such as two particles becoming one) – but in a particular 2-Hilbert space, say $\mathbb{Rep}(X)$ for some groupoid $X$ describing the current system (or its algebra), a 2-state $\Phi$ is a representation of the of the resulting system). A 2-state-vector is a particular representation. The algebra $\mathcal{A}$ can naturally be seen as a subalgebra of the automorphisms of $\Phi$.

So anyway, without trying to package up the whole picture – here are two categorified takes on the notion of state, from two different points of view.

I haven’t, here, got to the business about Tomita flows coming from states in the von Neumann algebra sense: maybe that’s to come.

As promised in the previous post, here is a little writeup of the second conference I was at recently…

Connections in Geometry and Physics

The conference at PI was an interestingly varied cross-section of talks, with a good many of them about geometry which, to be honest, is a little over my head.  Ostensibly about “connections”, the talks actually ranged quite widely, which was interesting, and reminded me I have a lot af geometry to catch up on.  A lot of talks had to do with structures at various places along the heirarchy: (1) symplectic manifolds, (2) Kähler manifolds, and (3) Calabi-Yau manifolds.  These last are interesting to string theorists and others, in part because they satisfy a form of Einstein’s equations, while also carrying a bunch of extra structure.

Now, at least I know what all the above things are: Symplectic manifolds $(M,\omega)$ have the “symplectic form” $\omega$, a non-degenerate exact 2-form (a canonical example being $\sum dp^i \wedge dq^i$ in the cotangent space to $\mathbb{R}^n$, which happens to be the configuration space for a particle moving in $\mathbb{R}^n$ – symplectic forms often show up on configuration spaces).  A Kähler manifold is symplectic, but also has a complex structure (i.e. a way to multiply tangent vectors by $i$), which preserves the symplectic form, and a metric, which gets along with both of the above.  If the metric satisfies Einstein’s equations and is flat (this really amounts to the connection to “connections”, since this is the same as there being some flat connections, namely the Levi-Civita connection), then $M$ is a Calabi-Yau manifold.

Anyway, this sets up the kind of geometry a lot of people were talking about, and while I didn’t exactly have the background to follow everything, I got a sense of what kinds of questions people are interested in, which was good.  A lot of questions have to do with Lagrangian submanifolds of any of the above (from symplectic through Calabi-Yau).  These are submanifolds where the symplectic form gives zero when applied to any tangent, and which have the highest possible dimension consistent with this property (namely $n$, if the original thing is $2n$-dimensional).  Another theme which came up several times – for example, in the talk by Denis Auroux – has to do with “mirror symmetry” for Kähler manifolds (and Calabi-Yaus), which has to do with finding a “mirror” for the manifold $M$, called $\check{M}$ where the complex geometry on the mirror corresponds to the symplectic geometry on $M$, and vice versa.

There were some talks in the direction of physics.  One of the most obviously physical was Niky Kamran’s, talking about a project he’s worked on with F. Finster, J. Smoller, and S-T. Yau, about long-time dynamics of particles satisfying the Dirac equation, living on a background geometry described by the Kerr metric – which describes a rotating black hole.  Since I worked with Niky on a related project for my M.Sc (my thesis was basically a summary putting together a bunch of results by these same four people), I followed this talk better than many of the others.

Working on this project, I got a strong sense of how important symmetry is in studying a lot of real-world problems.  One of the essential facts about the Kerr metric is that it’s very symmetric: it’s stable in time, and rotationally symmetric.  Actually, all the black-hole solutions to Einstein’s equations are quite symmetric – there is only a small family of solutions, parametrized by mass and angular momentum (and electrical charge).  The symmetry makes differential equations written in terms of this metric much nicer – you can split things into the radial and angular parts, for example – and in particular, the wave equations Niky was talking about are integrable just because of this symmetry, so it’s possible to get exact analytic results.  (Other approaches to this kind of problem get results only numerically and approximately, but can deal with much more general backgrounds.)  The starting point (which basically is what my thesis summarizes) is to show that there are no “bound states” for the Dirac equation.  Fermions (which is what it describes) are most familiar to us in bound states: in shells orbiting the nucleus of an atom.  But if the attractive force pulling on them is gravity, rather than electical charge, this situation isn’t stable.  The work Niky was talking about deals with what happens instead: what are the long-term dynamics of a fermion near a rotating black hole?

They use spectral methods – basically, Fourier analysis – to find out.  The Dirac equation is a wave equation (for a spinor field), and you can look at the different frequencies, and get an estimate of how fast they decay.  (Since there aren’t stable orbits, the strength of the spinor field has to decay over time.)  In fact, they get a sharp estimate of the order (namely $t^{-5/6}$).  Basically, one should imagine that the wave is a superposition of “ripples” – some radiating outward from the event horizon, and some converging toward it.  Put in terms of a particle – an electron, say, or a neutrino – this says it will either fall into the black hole, or (if it has enough energy) escape off to infinity.

There were some other physics-ish talks, such as that by James Sparks, on the geometry of the “AdS/CFT” correspondence.  This correspondence has to do with two kinds of quantum field theory.  The “AdS” stands for “Anti de Sitter”, which is a sort of geometric structure for a manifold which resembles a hyperboloid – actually, all the unit vectors in $\mathbb{R}^6$ where the metric has signature (4,2): that is, the metric is something like $\Delta(1,1,1,1,-1,-1)$.  This hyperboloid is 5-dimensional, and has a metric with one timelike dimension.  Plain old “de Sitter” space is a similar thing, but using a metric with signature (5,1).  It’s possible to define some field theory on AdS space, called supersymmetric supergravity.  This theory turns out to have exactly the same algebra of observables as a different theory, “CFT” or conformal field theory, on the (conformal) boundary of Anti de Sitter space.  Sparks told us about a geometric interpretation of this.

Then there was Sergei Gukov, with a talk called “Brane Quantization”, based on this work with Ed Witten.  He was a little reticent to actually describe how this “brane quantization” actually works, preferring to refer us to that paper, but gave us a very nice, and relatively comprehensible overview of different approaches to quantizing a symplectic manifold.  (As I said, they tend to show up as configuration spaces in classical physics. A basic problem of quantization is how to turn the algebra of functions on a symplectic manifold $(M,\omega)$ into an algebra of operators on a Hilbert space $\mathcal{H}$.)  In particular, he contrasted their method with geometric quantization (which needs to make some arbitrary choices, then takes $\mathcal{H}$ to be a space of sections of some line bundle on $M$ with a connection whose curvature is $\omega$), and with deformation quantization (which needs no special choices, but only constructs an algebra of operators by algebraic deformation, and not actually $\mathcal{H}$ itself, which some people, but not Sergei Gukov, find satisfactory).  The basic idea of Brane quantization seems to be that $M$ gets complexified (somehow – it might be either impossible, or non-unique), and then studying something called an A-model of the result.  This is apparently related to, for example, Gromov-Witten theory, which I’ve written about here recently.

Finally, I’ll mention a few other talks which stood out as rather different from the rest.  Veronique Godin talked about “Relative String Topology” – string topology being a way of studying space by looking at embeddings of the circle (or of paths) into it – that is, its loop space (or path space).  Usually, invariants that come from path spaces only detect the homotopy type of the original spaces – in particular, they’re not helpful as knot invariants.  Godin talked about a clever way to detect more structure by means of an $A_{\infty}$-coalgebra structure on the cohomology groups of the path space.  The “relative” part means one’s looking at a manifold $M$ with embedded submanifold $N$ (for example, $N$ is a knot in $M=\mathbb{R}^3$), and considering only paths starting and ending on $N$.  (This is how one can get a coalgebra structure – turning one path into two paths if it crosses through $N$ again is a comultiplication – this extends to chains in the cohomology).

Chris Brav gave a talk about how braid groups act on derived categories, which I didn’t entirely follow, but subsequently he did explain to me in a pretty comprehensible way what people are trying to accomplish when they look at derived categories.  At some point I’ll have to think about this more carefully and maybe post about it.  But roughly, it’s the same sort of “nice categorical properties” I mentioned in the previous post, about smooth spaces.  Looking at derived categories of sheaves on a space, makes the objects seem more complicated, but it also makes them behave better with respect to taking things like limits and colimits.

Benjamin Young prefaced his talk, “Combinatorics Inspired by Donaldson-Thomas Theory” by pointing out that he’s a combinatorialist, not a geometer.  But Donaldson-Thomas invariants are apparently a kind of “signed count” of some geometric structures (as are a lot of invariants – the same kind of “weighted count” invariants appear in Gromov-Witten and Dijkgraaf-Witten theory, just for instance).  So he described some geometry relating to “brane tilings” – basically, embedding certain kinds of graphis in a torus – and how they give rise to structures that correspond to certain kinds of Young diagrams (“not the same Young”, he added, perhaps unnecessarily, but it got a chuckle anyway).  So the counts can be turned into a combinatorial problem of counting those Young diagrams with the appropriate sign, which can be done using a generating function.

So in any case, this conference had a whole range of talks, from several different fields.  While I found myself lost in a number of talks, I was also quite fascinating with how wide a range of topics were embraced under its umbrella – “connections” indeed!  So in the end this was one of those conferences which opened my eyes to a wider view of the field, which was certainly a good reason to go!

I’ve been to two conferences in the past two weeks, and seen a lot of interesting talks. A couple of weekends ago, I was in Ottawa at the Fields Institute workshop on “Smooth Structures in Logic, Category Theory, and Physics“. There were quite a few interesting talks, on a fairly wide range of points of view, and I had some interesting conversations as well. A good workshop overall. Another report on it by Alex Hoffnung is here on the n-Category Cafe. Then this past weekend, I attended the conference “Connections in Geometry and Physics“, at the Perimeter Institute in Waterloo (also jointly sponsored by the U of Waterloo math department, and the Fields Institute), where I gave a version of my Extended TQFT talk (if you’ve seen a previous version, this is similar, but references a few more recent variations, but is mainly distinguished as the first time I caved in and decided to use Beamer – frankly I find the graphical wingdings distracting and unhelpful, but it made sense given the facilities in the venue).

One personally interesting thing was that one of the talks in Ottawa was given by John Baez, who was my advisor for my Ph.D at UCR, and one of the talks at PI was given by Niky Kamran, who was my advisor for my M.Sc. at McGill, so I got to touch base with both of them in the space of a week. It also reminded me that I’ve worked on a fairly eclectic sampling of things, since John was talking about cartesian closed categories of smooth spaces, and Niky was talking about the long-time dynamics of the Dirac equation in the neighborhood of black holes.

In the near future I’ll make a post on the Connections conference but the following was getting long enough already…

Smooth Structures

So, as the title suggests, the Fields workshop addressed the topic of “smoothness” from several points of view, with the three mentioned being only the most obvious. To begin with, “smooth” carries the connotation of “infinitely differentiable”. So for example the space $C^{\infty}(\mathbb{R})$ of smooth functions on the real line has the property that, if $f$ is any function in it, you can take the derivative $f'$, and you get another smooth function, hence can take the derivative again, and so on. So one way to characterize the space $C^{\infty}(\mathbb{R})$ is that it has a differential operator $D$ (which satisfies some algebraic properties like the product rule etc.), and is closed under $D$.

One theme explored at the workshop has to do with finding a nice general notion of “smooth space”. The smooth structure on $\mathbb{R}$ that makes this possible is the model for smooth structures on other spaces. The most familiar way goes: (1) first extend the concept to $\mathbb{R}^n$ via partial derivatives, so we know what a smooth map $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is (or between open subsets of these), and (2) define a “smooth manifold” as a (topological) space $M$ equipped with “charts” $\phi : U \rightarrow M$ for $U$ an open subset of some $\mathbb{R}^m$, satisfying a bunch of conditions. Then we can tell smooth functions on $M$ by pulling them back to $\mathbb{R}^m$ and using the familiar concept there. We can tell smooth maps $f : N \rightarrow M$ by composing with charts and their inverses and seeing that the resulting map $\phi^{-1} \circ f \circ \psi$ is smooth. This concept is great, and underlies, just to name a couple of examples, General Relativity and gauge theory, which are the basis of 20th century physics. It’s rather brittle, though, because simple operations like taking function spaces $Man(M,N) = \{ f : M \rightarrow N | f \text{ smooth } \}$, or subspaces $A \subset M$, or quotients $M/G$, give objects which are not manifolds.

John Baez talked about some of these issues in categorical language. Those three operations illustrate the general categorical constructions of exponentials, equalizers and coequalizers (or, generally, limits and colimits). The category of differentiable manifolds has important objects, but since it lacks these constructions in general, it’s not a nice category. The idea is to find a nice category – a cartesian closed one – which contains it. John described roughly how some approaches to this problem work – there were more detailed talks by Alex Hoffnung about the Diffeological spaces of Souriau, and by Andrew Stacey summarizing how the different categories are related and making a case for Frölicher spaces – but mostly focused on examples where these categories would be useful.

One interesting case deals with orbifolds (John suggested checking out this paper of Eugene Lerman, “Orbifolds as Stacks”), which were also the subject of Dorette Pronk’s talk. Dorette described some orbifolds – sometimes they arise by taking quotients of manifolds under the action of finite groups, and sometimes they only look locally as if they did. She also talked about the right way to think about maps between orbifolds, which is basically in terms of spans. That is, the right kind of “map” between orbifolds $X$ and $Y$ consists of (certain) maps into each of them from some common orbifold $Z$.

Under “logic” (and overlapping with “categories”), the first two days started off with talks by Anders Kock about Kähler differentials and synthetic differential geometry (regarding which see e.g. Mike Schulman’s intro, or the book by Anders Kock himself). SDG is a generalization of differential geometry to be internal to some topos – in particular, getting rid of the assumption that the geometry is based on some space which has an underlying set of points (which is special to the topos $\mathbf{Sets}$), and doing everything in the internal language of the topos. Kock introduced some work with Eduardo Dubuc – describing a “Fermat Theory” (an abstract characterization of a ring with a concept of partial derivatives) and showed how it fits into SDG.

Some other talks in the “logic” world included those of Rick Blute, Robin Cockett, and Thomas Erhard, about linear logic and differential categories. The basic idea is that a category can be associated to any logic, by taking formulas as objects, and (equivalence classes under rewriting rules) of proofs as morphisms. Linear logic, which is topical to quantum computation, is interesting from this point of view because it has some standard logical facts (like the deMorgan laws, which the intuitionistic logic of toposes do not entirely have) and also good categorical properties (one has a nice monoidal category). It’s a little strange if you’re used to thinking of classical logic: “propositions” are replaced by “resources” whose truth can be consumed (think of a quantum computer with a bit stored somewhere – you can move the bit, but not read and copy it); there are both “additive” and “multiplicative” versions of connectives which in classical logic go by the names “AND” and “OR” (correspondingly, there are additive and multiplicative versions of “TRUE” and “FALSE”). The relation to smoothness is that a new variation on this adds an operator to the category which behaves like differentiation. This is formally very interesting, though I haven’t really grokked what it’s good for, but apparently the derivative acts like a quantifier.  Really!?

Among other talks, Konrad Waldorf spoke about parallel transport for extended objects – basically, this is a roundabout way of studying nonabelian gerbes (a kind of categorification of bundles), not by looking at the gerbes themselves but by directly looking at parallel transport for connections on them. Kristine Bauer gave two talks about “Functor Calculus” (in particular the Goodwillie calculus), which has to do with constructing something like a “Taylor series” for a functor into topological spaces, approximating the functor by “polynomials”, and which shows up in homotopy theory.  I also have the sense that the Goodwillie calculus generalizes to topological spaces a lot of what Joyal’s species (related to “analytic” functors in a similar sense of being representable by “polynomials”), but I don’t understand this well enough at the moment to say just how.

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