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 . 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 -particle state is acted on by the permutation group on 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 ? It’s a map into from the terminal element in , which is “the” one-element set – or, likewise, in , 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, (which by abuse of notation sometimes gets called but more formally ). 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 .

Then we don’t have a terminal object anymore, but the same object 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 . Since the map on the left is unique, by definition of “terminal”, this really just given by the functor , the target map. This is a fibration over , called here for “phi”-bration, but this is appropriate, since it corresponds to what’s usually thought of as a wavefunction .

This correspondence is what groupoidification is all about – it has to do with taking the groupoid cardinality of fibres, where a “phi”bre of is the essential preimage of an object – everything whose image is isomorphic to . This gives an equivariant function on – 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 is the groupoid of finite sets and bijections (sometimes called ), 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 (giving “underlying sets” for objects in a category ) is really supposed to be faithful (so that -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 is a groupoid of “structured sets”, with an underlying set functor (actually, species are usually described in terms of the reverse, fibre-selecting functor , where the image of a finite set consists of the set of all “$\Phi$-structured” sets (such as: “graphs on set “, or “trees on “, 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 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 -object”. These generalized stuff types are “states” for a system with a configuration groupoid, in . 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 ), 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 .

2-State as Representation

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

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

(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 of operators on a Hilbert space , 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 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 . 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 – 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 acts on this bundle fibre-wise is just a consequence of the fact that the total 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 for some groupoid describing the current system (or its algebra), a 2-state is a representation of the of the resulting system). A 2-state-vector is a particular representation. The algebra can naturally be seen as a subalgebra of the automorphisms of .

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.

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

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

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

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

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

Thus, the way I defined double bicategory was: “A bicategory internal to “. In the paper as it stands, that’s all I say. What I cut out was a sort of dangling loose end pointing toward Lawvere theories – or rather, a variant thereof – finite limit theories (for something more detailed, see this recent paper by Lack and Rosicky). As I mentioned in the previous post, a Lawvere theory is an approach to universal algebra – it formally defines a kind of object (e.g. group, ring, abelian group, etc.) as a functor from a category which is the “theory” of such objects, while the functor is a “model” of the theory.

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

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

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

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

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

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

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

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

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

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

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

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

I spent most of last week attending four of the five days of the workshop “Categories, Quanta, Concepts”, at the Perimeter Institute.  In the next few days I plan to write up many of the talks, but it was quite a lot.  For the moment, I’d like to do a little writeup on the talk I gave.  I wasn’t originally expecting to speak, but the organizers wanted the grad students and postdocs who weren’t talking in the scheduled sessions to give little talks.  So I gave a short version of this one which I gave in Ottawa but as a blackboard talk, so I have no slides for it.

Now, the workshop had about ten people from Oxford’s Comlab visiting, including Samson Abramsky and Bob Coecke, Marni Sheppard, Jamie Vicary, and about half a dozen others.  Many folks in this group work in the context of dagger compact categories, which is a nice abstract setting that captures a lot of the features of the category which are relevant to quantum mechanics.  Jamie Vicary had, earlier that day, given a talk about n-dimensional TQFT’s and n-categories – specifically, n-Hilbert spaces.  I’ll write up their talks in a later,  but it was a nice context in which to give the talk.

The point of this talk is to describe, briefly, – as a category and as a 2-category; to explain why it’s a good conceptual setting for quantum theory; and to show how it bridges the gap between Hilbert spaces and 2-Hilbert spaces.

History and Symmetry

In the course of an afternoon discussion session, we were talking about the various approaches people are taking in fundamentals of quantum theory, and in trying to find a “quantum theory of gravity” (whatever that ends up meaning).  I raised a question about robust ideas: basically, it seems to me that if an idea shows up across many different domains, that’s probably a sign it belongs in a good theory.  I was hoping people knew of a number of these notions, because there are really only two I’ve seen in this light, and really there probably should be more.

The two physical  notions that motivate everything here are (1) symmetry, and (2) emphasis on histories.  Both ideas are applied to states: states have symmetries; histories link starting states to ending states.  Combining them suggests histories should have symmetries of their own, which ought to get along with the symmetries of the states they begin and end with.

Both concepts are rather fundamental. Hermann Weyl wrote a whole book, “Symmetry”, about the first, and wrote: As far as I can see, all a-priori statements in physics are based on symmetry. From diffeomorphism invariance in general relativity, to gauge symmetry in quantum field theory, to symmetric tensor products involved in Fock space, through classical examples like Noether’s theorem. Noether’s theorem is also about histories: it applies when a symmetry holds along an entire history of a system: in fact, Langrangian mechanics generally is all about histories, and how they’re selected to be “real” in a classical system (by having a critical value of the action functional). The Lagrangian point of view appears in quantum theory (and this was what Richard Feynman did in his thesis) as the famous “sum over histories”, or path integral. General relativity embraces histories as real – they’re spacetimes, which is what GR is all about. So these concepts seem to hold up rather well across different contexts.

I began by drawing this table:

The names are all those of categories. Moving left to right moves from a category describing collections of states, to one describing states-and-histories. It so happens that it also takes a cartesian category (or 2-category) to a symmetric monoidal one. Moving from top to bottom goes from a setting with no symmetry to one with symmetry. In both cases, the key concept is naturally expressed with a category, and shows up in morphisms. Now, since groupoids are already categories, both of the bottom entries properly ought to be 2-categories, but when we choose to, we can ignore that fact.

Why Spans?

I’ve written a bunch on spans here before, but to recap, a span in a category is a diagram like: . Say we’re in , so all these objects are sets: we interpret and as sets of states. Each one describes some system by collecting all its possible (“pure”) states. (To be better, we could start with a different base category – symplectic manifolds, say – and see if the rest of the analysis goes through). For now, we just realize that is a set of histories leading the system to the system (notice there’s no assumption the system is the same). The maps are source and target maps: they specify the unique state where a history starts and where it ends.

If has pullbacks (or at least any we may need), we can use them to compose spans:

The pullback – a fibred product if we’re in – picks out pairs of histories in which match at . This should be exactly the possible histories taking to .

I’ve included an arrow to the category : this is the category whose objects are sets, and whose morphisms are relations. A number of people at CQC mentioned as an example of a monoidal category which supports toy models having some but not all features of quantum mechanics. It happens to be a quotient of . A relation is an equivalence class of spans, where we only notice whether the set of histories connecting to is empty or not. is more like quantum mechanics, because its composition is just like matrix multiplication: counting the number of histories from to turns the span into a matrix – so we can think of and as being like vector spaces.

In fact, there’s a map taking an object to and a span to the matrix I just mentioned, which faithfully represents . A more conceptual way to say this is: a function can be transported across the span. It lifts to as . Getting down the other leg, we add all the contributions of each history ending at a given : .

This “sum over histories” is what matrix multiplication actually is.

Why Groupoids?

The point of groupoids is that they represent sets with a notion of (local) symmetry. A groupoid is a category with invertible morphisms. Each such isomorphism tells us that two states are in some sense “the same”. The beginning example is the “action groupoid” that comes from a group acting on a set , which we call (or the “weak quotient” of by ).

This suggests how groupoids come into the physical picture – the intuition is that is the set (or, in later variations, space) of states, and is a group of symmetries.  For example, could be a group of coordinate transformations: states which can be transformed into each other by a rotation, say, are formally but not physically different.  The Extended TQFT example comes from the case where is a set of connections, and the group of gauge transformations.  Of course, not all physically interesting cases come from a single group action: for the harmonic oscillator, the states (“pure states”) are just energy levels – nonnegative integers.  On each state , there is an action of the permutation group – a “local” symmetry.

One nice thing about groupoids is that one often really only wants to think about them up to equivalence – as a result, it becomes a matter of convention whether formally different but physically indistinguishable states are really considered different.  There’s a side effect, though: is a 2-category.  In particular, this has two consequences for : it ought to have 2-morphisms, so we stop thinking about spans up to isomorphism.  Instead, we allow spans of span maps as 2-morphisms.  Also, when composing spans (which are no longer taken up to isomorphism) we have to use a weak pullback, not an ordinary one.  I didn’t have time to say much about the 2-morphism level in the CQC talk, but the slides above do.

In any case, moving into means that the arrows in the spans are now functors – in particular, a symmetry of a history  now has to map to a symmetry of the start and end states, and .  In particular, the functors give homomorphisms of the symmetry groups of each object.

Physics in Hilb and 2Hilb

So the point of the above is really to motivate the claim that there’s a clear physical meaning to groupoids (states and symmetries), and spans of them (putting histories on an even footing with states).  There’s less obvious physical meaning to the usual setting of quantum theory, the category – but it’s a slightly nicer category than .  For one thing, there is a concept of a “dual” of a span – it’s the same span, with the roles of and interchanged.  However (as Jamie Vicary pointed out to me), it’s not an “adjoint” in in the technical sense.  In particular, is a symmetric monoidal category, like , but it’s not “dagger compact”, the kind of category all the folks from Oxford like so much.

Now, groupoidification lets us generalize the map to groupoids making as few changes as possible.  We still use Hilbert space , but now is the set of isomorphism classes of objects in the groupoid.  The “sum over histories” – in other words, the linear map associated to a span – is found in almost the same way, but histories now have “weights” found using groupoid cardinality (see any of the papers on groupoidification, or my slides above, for the details).  This reproduces a lot of known physics (see my paper on the harmonic oscillator; TQFT’s can also be defined this way).

While this is “as much like” linearization of as possible in some sense, it’s not exactly analogous.  It also is rather violent to the structure of the groupoids: at the level of objects it treats as . At the morphism level, it ignores everything about the structure of symmetries in the system except how many of them there are.   Since a groupoid is a category, the more direct analogy for – the set of functions (fancier versions use, say, functions only) from to is – the category of functors from a groupoid into .  That is, representations of .

One of the attractions here is that, because of a generalization of Tanaka-Krein duality, this category will actually be enough to reconstruct the groupoid if it’s reasonably nice.  The representation of in , unlike in is actually faithful for objects, at least for compact or finite groupoids.

Then you can “pull and push” a representation across a span to get – using , the adjoint functor to pulling back.  This is the 1-morphism level of the 2-functor I call , generalizing the functor in the world of sets.  The result is still a “direct sum over histories” – but because we’re dealing with pushing representations through homomorphisms, this adjoint is a bit more complicated than in the 0-category world of .  (See my slides or paper for the details).  But it remains true that the weights and so forth used in ordinary groupoidification show up here at the level of 2-morphisms.  So the representation in is not a faithful representation of the (intuitively meaningful) category either.  But it does capture a fair bit more than Hilbert spaces.

One point of my talk was to try to motivate the use of 2-Hilbert spaces in physics from an a-priori point of view.  One thing I think is nice, for this purpose, is to see how our physical intuitions motivate – a nice point itself – and then observe that there is this “higher level” span around:

Further Thoughts

Where can one take this?  There seem to be theories whose states and symmetries naturally want to form n-groupoids: in “higher gauge theory“, a sort of  gauge theory for categorical groups, one would have connections as states, gauge transformations as symmetries, and some kind of  “symmetry of symmetries”, rather as 2-categories have functors, natural transformations between them, and modifications of these.  Perhaps these could be organized into n-dimensional spans-of-spans-of-spans… of n-groupoids.  Then representations of an n-groupoid – namely, n-functors into – could be subjected to the kind of “pull-push” process we’ve just looked at.

Finally, part of the point here was to see how some fundamental physical notions – symmetry and histories – appear across physics, and lead to .  Presumably these two aren’t enough.  The next principle that looks appealing – because it appears across domains – is some form of an action principle.

But that would be a different talk altogether.

So for my inaugural blog post of 2009, I thought I would step back and comment about the big picture of the motivation behind what I’ve been talking about here, and other things which I haven’t. I recently gave a talk at the University of Ottawa, which tries to give some of the mathematical/physical context. It describes both “degroupoidification” and “2-linearization” as maps from spans of groupoids into (a) vector spaces, and (b) 2-vector spaces. I will soon write a post setting out the new thing in case (b) that I was hung up on for a while until I learned some more representation theory. However, in this venue I can step even further back than that.

Over the Xmas/New Year break, I was travelling about “The Corridor” (the densely populated part of Canada – London, where I live, is toward one end, and I visited Montreal, Ottawa, Toronto, Kitchener, and some of the areas in between, to see family and friends). Between catching up with friends – who, naturally, like to know what I’m up to – and the New Year impulse to summarize, and the fact that I’m applying for jobs these days, I’ve had occasion to think through the answer to the question “What do you work on?” on a few different levels. So what I thought i’d do here is give the “Cocktail Party Version” of what it is I’m working on (a less technical version of my research statement, with some philosophical asides, I guess).

In The Middle

The first thing I usually have to tell people is that what I work on lives in the middle – somewhere between mathematics and physics. Having said that, I have to clear up the fact that I’m a mathematician, rather than a physicist. I approach questions with a mathematician’s point of view – I’m interested in making concepts precise, proving facts about them rigorously, and so on. But I do find it helps to motivate this activity to suppose that the concepts in question apply to the real world – by which I mean, the physical world.

(That’s a contentious position in itself, obviously. Platonists, Cartesian dualists, and people who believe in the supernatural generally don’t accept it, for example. For most purposes it doesn’t matter, but my choice about what to work on is definitely influenced by the view that mathematical concepts don’t exist independently of human thought, but the physical world does, and the concepts we use today have been selected – unconsciously sometimes, but for the most part, I think, on purpose – for their use in describing it. This is how I account for the supposedly unreasonable effectiveness of mathematics – not really any more surprising than the remarkable effectiveness of car engines at turning gasoline into motion, or that steel girders and concrete can miraculously hold up a building. You can be surprised that anything at all might work, but it’s less amazing that the thing selected for the job does it well.)

Physics

The physical world, however, is just full of interesting things one could study, even as a mathematician. Biology is a popular subject these days, which is being brought into mathematics departments in various ways. This involves theoretical study of non-equilibrium thermodynamics, the dynamics of networks (of chemical reactions, for example), and no doubt a lot of other things I know nothing about. It also involves a lot of detailed modelling and computer simulation. There’s a lot of profound mathematical engagement with the physical world here, and I think this stuff is great, but it’s not what I work on. My taste in research questions is a lot more foundational. These days, the physical side of the questions I’m thinking about has more to do with foundations of quantum mechanics (in the guise of 2-Hilbert spaces), and questions related to quantum gravity.

Now, recently, I’ve more or less come around to the opinion that these are related: that part of the difficulty of finding a good theory accomodating quantum mechanics and general relativity comes from not having a proper understanding of the foundations of quantum mechanics itself. It’s constantly surprising that there are still controversies, even, over whether QM should be understood as an ontological theory describing what the world is like, or an epistemological theory describing the dynamics of the information about the world known to some observer. (Incidentally – I’m assuming here that the cocktail party in question is one where you can use the word “ontological” in polite company. I’m told there are other kinds.)

Furthermore, some of the most intractable problems surrounding quantum gravity involve foundational questions. Since the language of quantum mechanics deals with the interactions between a system and an observer, so applying it to the entire universe (quantum cosmology) is problematic. Then there’s the problem of time: quantum mechanics (and field theory), both old-fashioned and relativistic, assume a pre-existing notion of time (either a coordinate, or at least a fixed background geometry), when calculating how systems (including fields) evolve. But if the field in question is the gravitational field, then the right notion of time will depend on which solution you’re looking at.

Category Theory

So having said the above, I then have to account for why it is that I think category theory has anything to say to these fundamental issues. This being the cocktail party version, this has to begin with an explanation of what category theory is, which is probably the hardest part. Not so much because the concept of a category is hard, but because as a concept, it’s fairly abstract. The odd thing is, individual categories themselves are in some ways more concrete than the “decategorified” nubbins we often deal with. For example, finite sets and set maps are quite concrete: here are four sheep, and here four rocks, and here is a way of matching sheep with rocks. Contrast that with the abstract concept of the pure number “four” – an element in the set of cardinalities of finite sets, which gets addition and multiplication (abstractly defined operations) from the very concrete concepts of union and product (set of pairs) of sets. Part of the point of categorification is to restore our attention to things which are “more real” in this way, by giving them names.

One philosophical point about categories is that they treat objects and morphisms (which, for cocktail party purposes, I would describe as “relations between objects”) as equally real. Since I’ve already used the word, I’ll say this is an ontological commitment (at least in some domain – here’s an issue where computer science offers some nicely structured terminology) to the existence of relations as real. It might be surprising to hear someone say that relations between things are just as “real” as things themselves – or worse, more real, albeit less tangible.  Most of us are used to thinking of relations as some kind of derivative statement about real things. On the other hand, relations (between subject and object, system and observer) are what we have actual empirical evidence for. So maybe this shouldn’t be such a surprising stance.

Now, there are different ways category theory can enter into this discussion. Just to name one: the causal structure of a spacetime (a history) is a category – in particular, a poset (though we might want to refine that into a timelike-path category – or a double category where the morphisms are timelike and spacelike paths). Another way category theory may come in is as the setting for representation theory, which comes up in what I’ve been looking at. Here, there is some category representing a specific physical system – for example, a groupoid which represents the pure states of a system and their symmetries. Then we want to describe that system in a more universal way – for example, studying it by looking at maps (functors) from that category into one like Hilb, which isn’t tied to the specific system. The underlying point here is to represent something physical in terms of the sort of symbolic/abstract structures which we can deal with mathematically. Then there’s a category of such representations, whose morphisms (intertwiners in some suitably general sense) are ways of “changing coordinates” which get along with what’s important about the system.

The Point

So by “The Point”, I mean: how this all addresses questions in quantum mechanics and gravity, which I previously implied it did (or could). Let me summarize it by describing what happens in the 3D quantum gravity toy model developed in my thesis. There, the two levels (object and morphism) give us two concepts of “state”: a state in a 2-Hilbert space is an object in a category. Then there’s a “2-state” (which is actually more like the usual QM concept of a state): this is a vector in a Hilbert space, which happens to be a component in a 2-linear map between 2-vector spaces. In particular, a “state” specifies the geometry of space (albeit, in 3D, it does this by specifying boundary conditions only). A “2-state” describes a state of a quantum field theory which lives on that background.

Here is a Big Picture conjecture (which I can in no way back up at the moment, and reserve the right to second-guess): the division between “state and 2-state” as I just outlined it should turn out to resolve the above questions about the “problem of time”, and other philosophical puzzles of quantum gravity. This distinction is most naturally understood via categorification.

(Maybe. It appears to work that way in 3D. In the real world, gravity isn’t topological – though it has a limit that is.)

About a week ago (of November 22-23) I was in Riverside, California for Groupoidfest ‘08. (Slides for the talk I gave here, and also in pdf.) It’s taken me a while to write it up, because I’ve been, among other things, applying for jobs.

This would be the second time I’ve been to Groupoidfest, and the first time I’ve been back in Riverside since I graduated from UCR last summer. While I was there, I also had the chance to talk to John Baez and some of his other students, past and present, and also to attend Alan Weinstein’s colloquium talk on Friday. On top of that, I managed to see a couple of my other friends in town, so all in all, it was a good trip.

There were quite a few talks, several of which were fairly short, so I’ll comment on a few examples which I found particularly relevant to me. So for instance Alan Paterson’s talk on Equivariant K-Theory for Proper Groupoids: here’s a case where I’m seeing familiar issues from a different direction. K-theory studies objects by looking at categories of vector bundles. Equivariant K-theory can be taken to mean that these bundles come with isomorphisms between fibres which come from a group action, or more generally the morphisms of a groupoid. It’s a kind of categorification of equivariant cohomology. Alan Paterson’s talk was quite extensive, but there’s a whole vocabulary here I’m still learning. The culmination of the talk dealt with Hilbert bundles (a little more structured than vector bundles), and the the Hilbert bundle (where is the space of morphisms of a groupoid – so this can be treated as a bundle over the space of objects induced by, say, the map taking a morphism to its target object). This bundle has the nice “stabilization” property that taking a direct sum with any other bundle leaves it unchanged.

John Quigg also spoke about Hilbert bundles, and “Fell bundles” (he spoke about these last year, too), but since John Baez described this in more detail in his report on GFest, I’ll just remark on another aspect of this talk, where he was using a “disintegration theorem”, which was more familiar to me. This says that every representation of the convolution algebra of a groupoid comes from direct-integrating some Hilbert bundle. This is reminiscent of the decomposition of any von Neumann algebra as a direct integral of “factors” (which are each subalgebras of the algebra of operators on fibres of some Hilbert bundle). There seem to be a lot of these “disintegration theorems” involving direct integrals.  I have some ideas about this, but I’ll hold off on them until they’re a little more developed.

There were a number of other talks with interesting elements, but many were a bit too short for me to get much more than an awareness that there’s interesting work being done that I’d like to learn more about: Xiang Tang’s talk on “Group Extensions and Duality of Gerbes” seemed to be perhaps related to what I would describe as 2-vector spaces generated by -groupoids, but using (blush) a more standard language; Joris Vankerschaner talked about classical mechanics on Lie Groupoids – in particular, discrete field theories valued in groupoids.

Now, the colloqium talk by Alan Weinstein was titled “Groupoid Symmetry for Einstein’s Equation?”, including the querulous punctuation, since some of it was speculative. The basic idea behind this talk was to apply groupoids to General Relativity, thought of as an evolution equation. The Hamiltonian formulation of GR describes a spacelike hypersurface evolving in time – this was described by Arnowitt, Deser and Misner, or ADM, from whom we likewise get the “ADM mass”, which can be thought of as the energy of the worldsheet, as it’s seen by an observer at spacelike infinity. This formulation doesn’t describe all solutions of Einstein’s equations – in particular, nothing with closed timelike curves, and unless I misremember, really only makes sense for asymptotically flat spacetimes – certainly that’s true for the ADM mass. But it does fit with our usual intuitions about systems evolving in time, and makes some initial-value problems – including local ones – more or less tractable, which is good for practical purposes. (There are still further technical provisos to ensure the result actually satisfies Einstein’s equations.)

(Note: on looking the asymptotic flatness issue up in Wald’s book, it seems that even for compact space slices, although it naively appears the Hamiltonian vanishes, this can possibly be resolved by some tricky “deparametrization” Wald doesn’t entirely explain. The restriction against closed timelike curves alone probably won’t dissuade anyone who isn’t dead set on building a time machine.)

Anyway, the groupoid symmetry Weinstein was suggesting involves taking space slices (or some slight variation thereon, such as thickened slices, or slices equipped with a metric) to be objects, and considering diffeomorphisms between slices as morphisms. This would make a Lie groupoid, and the corresponding Lie algebroid would reproduce an algebroid structure which it’s natural to associate with the phase space for the system (specifically, one associated with the Poisson structure. There are some links on this correspondence over on the n-Category Cafe – basically, a Poisson algebra is like a Lie bracket structure on the tangent Lie algebroid to a manifold).

Where one can go with this idea, I’m not sure. More clear to me, since I’ve thought about it more, was the content of Weinstein’s other talk – about the volume of a differentiable stack – which he gave at the conference proper. This is a smooth/differentiable version of groupoid cardinality, and therefore has all sorts of applications to groupoidification in both the vector-space and 2-vector-space flavours. The basic point is that groupoid cardinality – for finite groupoids – involves measures in two ways. One way to find it is as a sum over all the objects; the sum is of the quantities , the reciprocals of the numbers of morphisms ending at object . There are other, equivalent, ways, but they all amount to sums of reciprocals of numbers found by counting. Both these numbers, and the sums themselves, can be seen as integrals – using the counting measure on a discrete, finite set. The first point of the talk was that counting measure should be replaced by two kinds of measure – one on the object spaces, and one for morphisms – when one passes to a differentiable stack.

(There are a variety of ways to think about stacks – one is that a stack is a groupoid, thought of up to equivalence. In which case, the real information in a stack consists of the set of isomorphism classes of object, or orbits, and also the automorphism, or isotropy, groups for objects in each orbit. One good thing about stacks is that they keep track of information which is lost when taking quotients – if a point is fixed under a group action, for instance, it still has nontrivial isotropy when taking the “stacky” quotient of a set by a group action. A nice representative groupoid for this quotient is keeps all points, but adds morphisms corresponding to motions under the action.)

There were also a bunch of talks about groupoidification of linear algebra, by John Baez and his current students – about groupoidification of linear algebra. Since I’ve written about this a lot here anyway, I’ll just remark that Christopher Walker introduced the concept, Alex Hoffnung talked about applications to Hecke algebras and incidence geometries (also discussed in their seminar starting here), while John spoke about Jim Dolan’s ideas for groupoidifying the harmonic oscillator, which have also been written up and slightly expanded by me. My own talk is also sort of about groupoidification, albeit a higher-dimensional version thereof.

At any rate, that’s about all I have time to say about GFest ‘08, although there were many other talks which reinforced my desire to keep learning more about all the wonderful stuff known to people who study groupoids, and especially Lie groupoids.

There haven’t been many colloquium talks here this term, but there was one a week ago (Thursday) by Joel Kamnitzer from University of Toronto (and contributor to the Secret Blogging Seminar), who gave a talk called “Categorical Actions and Equivalence of Categories”.

As it turns out, I have at least two things in common with Joel Kamnitzer. First, we were both President of the University of Waterloo Pure Math Club (which became the Pure Math, Applied Math, and Combinatorics and Optimization club ’round about my time, when we noticed the other two math faculties at Waterloo no longer had their own undergraduate clubs). Second, we both did math Ph.D’s in California.  And while that’s probably a coincidence, there were several themes in the talk that overlap things I’ve talked about here.

The basic idea behind the talk was roughly this: when there’s an action of the Lie algebra (i.e. trace-zero 2-by-2 matrices) on a space, that space can be decomposed into some eigenspaces, and one can get isomorphisms between certain pairs of them. So the question is whether this can be categorified: if there’s an action of a categorical on a category, can it be decomposed into subcategories which generate it, such that certain pairs can be shown to be equivalent?

So first he reminded/informed us of some of the non-categorified examples. The main thing is to show an equivalent way of describing an action. This uses that is generated by three matrices:

and and

These satisfy some commutation relations: , and . These relations specify up to isomorphism, so one can describe an action on a set by specifying what , , and do (satisfying the commutation relations, of course).  It’s a classical fact from Lie theory that representations of all look similar: they’re direct sums of eigenspaces of the generator (for integer eigenvalues ), and the generators and act as “raising” and “lowering” operators, and .  (All of which is key to describing spins of fundamental particles, due to being the cover of the Lorentz group , though that’s beside the point just at the moment.

We heard three examples, of which for me the most intuitively nice involves an action on the vector space generated by the power set of a fixed finite set of size .  Then is a (modified) counting operator – its eigenspaces are the subspaces generated by subsets of size (where ).  The operator takes a set of size and maps it to the sum over all with of size (all ways to “add one element” to );  takes to the sum of all subsets of size contained in (all ways to “remove one element” from .  (This all seems very familiar to me from the combinatorial interpretation of the Weyl algebra, which I talk about here.)  These satisfy the commutation relations .

Now, the “equivalences” in the talk will be categorified versions of some obvious isomorphisms here, namely (that is, subsets are in bijection with -subsets).  These turn out to be imposed by the fact that we have a representation of , which lifts to a representation of in .  The isomorphism is given by restricting the action of to .

There is a more algebraic-geometry version of this example which replaces the power set of a set with the union of the Grassman varieties of subspaces of .  Instead of the vector space generated by subsets of size , one builds out of the cohomology of the tangent bundle to the variety, with .

Now, the thing I find interesting about this picture is that, as with the Weyl algebra setup I mention above, it represets the raising and lowering operators in terms of transfer through a span.  Since this seems to pop up everywhere, it’s important enough to think on for a moment.  The span in question goes from to .

To say what goes in the middle, we use the fact that an element of the cotangent bundle amounts to a pair , where is a -dimensional subspace (a point on ) and is a tangent vector at .  As it turns out amounts to a map which annihilates itself.  So then we have the variety where , and and are cotangent vectors.  This has projection maps to the two cotangent bundles: .

Then the point is that the cohomology spaces are build from maps into , so we call “pull-push” them through the span by .  This defines , and is similar, going the other way.

…

So much for actions of “old-school” : what about “categorical” ?  To begin with, what does that even mean?  Well, Aaron Lauda has described a “categorified” version of (actually, of Lusztig’s presentation of the enveloping algebra – a quantum version, though that won’t enter into this).  This is a categorification of the generators , , and , and of their commutation relations (which now become isomorphisms, which may have to satisfy some coherence laws – the details here being incredibly important, but not very enlightening at first).  These , and are now functors, rather than maps.

As a side note, this is not precisely a categorification of the Lie algebra , but actually a categorification of a particular presentation of .  Though, since I’m mentioning this, I’ll remark it’s much more like the categorification of the Weyl algebra which is involved in the groupoidification of the quantum harmonic oscillator.

In any case, Joel went on to describe categorical actions of .  Actually, he distinguished “weak” and “strong” versions, which is apparently a common usage, though not the one I’m used to.  “Weak” means things are specified up to unspecified isomorphisms required to exist, and “strong” means things are defined up to specified (presumably coherent) isomorphisms (which is what I usually understand “weak” to mean).  The strong ones are the ones which give the equivalences we’re looking for, though.

It turns out that an action of the categorical on an additive category gives: (1) a way to split up for integers , and (2) the action of the generators and with and , such that (3) there are commutation isomorphisms analogous to the commutator identities for regular .  I note that algebraic geometers prefer to use additive categories – where the -sets are abelian groups, rather than vector spaces, which is what they would be in a 2-vector space.  In fact, later in the talk we heard about generalizations to triangulated categories – even a weaker condition.  In the special case where the additive category happened to be a 2-vector space, we’d have a “2-linear representation of a 2-algebra”.

Now, the main example was similar to the one above involving Grassman varieties.  The difference is that one doesn’t of cooking up a vector space from from the cohomology of its cotangent bundle, one cooks up an abelian category.  This is where, again, , for .  This is the derived category of coherent sheaves on the cotangent bundle.  There seems to be some analogy between the two: cohomology involves maps into $\mathbb{C}$ (and the exterior algebra of forms), while coherent sheaves might be thought of as (algebraic) vector-space valued functions, a categorified version of functions.  Also, while the cohomology is a chain complex, the objects of the derived category are themselves chain complexes.  Exactly how the analogy works is something I can’t explain just now.

Anyway, the key result, due to Chuang and Rouqier, says that from a “strong” categorical action (in the sense above) and the and are exact functors (in 2-vector spaces, they’d be “2-linear maps”), then there is an equivalence (given in terms of the and ) between the categories of complexes on and .  This isn’t quite what was wanted (we wanted an equivalence ), so for the remainder of the talk we heard about work directed at this question: cases where it works, counterexamples when it doesn’t, some generalizations, and so on.

Since coming back from Montreal, I’ve given an exam for a very large linear algebra class, but before I forget, I’d like to make a few notes about some of the talks.

The first day, Saturday, October 4, was a long day of mostly half-hour talks, and some 20-min talks, including my late-registering contribution. It was about the 2-linearization of spans of groupoids which I’ve talked about before, but with a problem fixed. I’ll say more about that soon.

It was interesting to see the range of talks – category theory spans a few areas of mathematics, after all. To start off the day, there was a session in which Michael Makkai and Victor Harnik both gave talks about higher-dimensional categories in one form or another.

Makkai’s was about “revisiting coherence in bicategories and tricategories”. Coherence is an issue that comes up once you get into higher categories – that is, looking at things bearing more complicated relationships than “equal” and “not-equal”, such as “isomorphic”, or “equivalent”. Or “biequivalent”, I suppose – Makkai covered some work of Nick Gurski and Steve Lack about how bicategories and tricategories are (or are not) equivalent to strict versions of themselves. More precisely, that there’s a biequivalence between (the strict form) and (the weak form). Whereas there is no triequivalence between (strict) and (weak) . There is a triequivalence between and – something intermediate between strict and weak. He also explained how these equivalences pass through a relationship with the category of graphs. (An equivalence is a pair of adjoint functors – the equivalence between and factors through pairs of adjoint functors between each of these and ). There was more to the talk, but it was somewhat over my head.

Harnik’s talk, “Placed composition in higher dimensional categories”, was about a recursive way of defining partial composition operations in higher dimensions. Here, the point is that it’s easy and obvious how to compose one-dimensional arrows: you stick them tip-to-tail. Higher-dimensional morphisms need more complicated rules telling how to stick them together along various numbers of shared faces. (A line-segment arrow has only two faces, both points with no sub-faces). Harnik described how to generate an -category recursively: generate faces of dimension by freely adjoining some indeterminate cells, which need all these operations telling how they can be stuck together. Then you have to impose some algebraic relations – certain composites are the same. This is like a problem of presenting groups in terms of generators and relations: it can be hard to tell whether two elements are equal or not – two elements being declared equal if they can be proved so in some algebraic system (not an easy question to test, usually).

In fact, questions about computability came up a lot, since there is a lot of interaction between category theory and computer science. We saw several talks that touched on that in the afternoon: B. Redmond gave a talk, “Safe Recursion Revisited”, about a categorical point of view on defining recursion “safely” (i.e. keeping algorithms in polynomial time); G. Lukacs described “A cartesian closed category that might be useful for higher-type computation” – higher types being apparently the type-theory correlate of higher categories. We had heard about this earlier – M. Warren talked on “types and groupoids”, showing how to use -groupoids to look at types, variables of those types (objects), and terms or “elements of proofs” (as morphisms), and so on for “higher types”. A different take on the intersection between computing and categories was N. Yanofsky’s talk “On the algorithmic informational content of categories”, which applied Kolmogorov complexity (the size of a turing machine required to produce a given output) to productions describing categories. Productions like the one that takes a simpler description – of the category of topological spaces, say – and turns it into a more complex one, like the category of pointed topological spaces. Or from vector spaces to Banach spaces, or what-have-you. He described a little language that can be used to specify (some, not all) categories by such operations, starting with a few building blocks – which then allows you to ask about the Kolmogorov complexity of the category itself.

On a different vein, there was also a reasonable cross-section of topological ideas going around. Certainly any time -groupoids come up, it also comes up that they classify homotopy types of spaces. But much more detailed geometric pictures also come up. Walter Tholen talked about the Gromov metric on the category of metric spaces: the distance between two metric spaces is defined as a minimum over all possible isometric embeddings into a common space, of a certain maximum separation between the spaces. One can then talk about Cauchy sequences of metric spaces, and the fact that (for example), the category of complete metric spaces is itself complete.

Dorette Pronk also brought in some geometry when she talked about “Transformation groupoids and orbifolt homotopy theory”. I’m quite interested in transformation groupoids, which show up when a set is acted on by a group. The example I’ve talked about is from gauge theory, where there is a group of gauge transformations acting on the moduli space of configurations (i.e. connections). This was one of the examples she gave for where these sorts of things come from. Then she got into the connections between these sorts of groupoids and the homotopy theory of orbifolds. Orbifolds are like manifolds, except that their neighborhoods have isomorphisms to , where is an open set in , and is a finite group (a nontrivial group action distinguishes orbifolds from mere manifolds). Most can be said in the case where the orbifold is just where is a manifold and is a Lie group, acting globally. Orbifolds like this are called representable.

Now, orbifolds have groupoids associated to them (in various ways), and Dorette Pronk’s talk dealt with the fact that the orbifolds being representable (i.e. arising from a global group action) is equivalent to the associated groupoid being Morita equivalent to a transformation groupoid (i.e. one arising from a global group action). Morita equivalence for groupoids and turns out to be the same as having a nice enough SPAN of groupoids

So in fact here are spans of groupoids again – just the sort of thing I was there to talk about, and should have more to say on here shortly. So that was interesting. This situation of having a span of groupoids seems to show up in several different guises.

There were some other talks I’ve missed, but it’s taken me a while to get to this, and some of them have faded a bit, so I’ll cut this short there.

Well, I was out of town for a bit, but classes are now underway here at UWO. This term I’m teaching an introductory Linear Algebra course, which is, I believe, the largest class I’ve taught so far, with on the order of a couple of hundred students. That should be a change of pace: last year, both courses I taught had just seven students each.

Meanwhile, I’ll carry on from the last post. I described structure types (a.k.a. species) as functors , which take a finite set , and give the set of all “-structures on the underlying set “. A lot of combinatorial enumeration uses the generating functions for such structure types, which are power series whose coefficients count the number of structures for a set of size (the fact that structure types are functorial is what allows us to ignore everything but the isomorphism class of the underlying set ). Now, there is a notion of generalized species, described in this paper by Fiore, Gambino, Hyland and Winskel, which I’ll link here because I think it’s a great point of view on the setup I discussed before. But right now, I’ll go in a somewhat different direction. (Whether there’s a connection between the two is left as an exercise)

Stuff Types

To start with, there is a dual way to look at structure types (a.k.a. species). The “structures” identified by a structure type form a category . It’s a concrete category in fact: each object has an underlying set. The morphisms of are “structure-preserving” maps (the meaning of which depends, obviously, on ) of -structured sets. These correspond exactly (by fiat) to the isomorphisms of underlying sets (i.e. relabellings). These are all invertible, so this is a groupoid.

So is the category of “underlying sets”, so the forgetful functor from -structured sets into is a functor between groupoids. This functor is a sort of “dual” way to look at the structure type – the original functor . In fact, for any structure type , this dual will always be a faithful functor. That is, the morphism map is one-to-one, so morphisms in are uniquely determined by the corresponding map of underlying sets. In other words, there are no additional symmetries in but those determined by set bijections.

But this is an artifact! I declared the union of all the sets to be the objects of a category and then added morphisms by hand. That makes sense if you think of the “-structured sets built on underlying set ” as derived entirely from and . But the dual view, focusing on , tends to make us think of as given, and as observing some property – underlying sets and maps for objects and morphisms. This may throw away information about both, in principle. Faithfulness of suggests that the objects of just consists of sets with some inflexible extra decorations with no local symmetries of their own to complicate the morphisms.

So let’s treat as real and as some kind of synopsis or measurement of . If doesn’t need to be faithful, it may not correspond to a structure type, but it will be what Baez and Dolan call a stuff type, which is actually just any groupoid equipped with underlying set functor . Maybe it’s surprising that these can still be treated like power series, by taking the coefficient at to be the (real-valued) groupoid cardinality of the preimage of . (The groupoid cardinality, described here, is related to the “Leinster measure” for categories. Regular readers of the n-Category Cafe will know that there has been some discussion over there about this – some links from here, and discussion about applying it to “configuration categories” of physical systems here.)

Stuff types can be used to deal with seemingly straightforward “structures” which structure types have a hard time with. For instance, letting be the structure type “a set”, and so should be the type “a set of sets” (where the underlying set operation is the union of elements). This can be represented by a stuff type, but not a structure type.

Groupoidification

Stuff types fit into a more general pattern, which has to do with the 2-category of spans of groupoids. I really cleared up just how this works in conversation with Jamie Vicary.

Groupoidification is the program of looking for analogs of linear algebra (whose native habitat is the monoidal category ) in a different monoidal category (in fact, bicategory) of spans of groupoids, which I’ve talked about quite a bit before. Very briefly, we have a bicategory where the objects are groupoids, and the morphisms are spans like: , composed by (weak) pullback. Given spans , a 2-morphism is a map which makes the resulting diagram commute.

So the key thing now is the fact that a stuff type can be regarded as a span of groupoids in two ways:

and

Here, is the trivial groupoid consisting of just one object and its identity morphism. Any groupoid has just one functor into , so a stuff type automatically has these two incarnations as a span. One is a morphism (in ) from $\mathbf{1}$ to , and the other is its dual, going the other way. One can call these a “state” and a “costate”. Why these terms?

One important fact is that $atex \mathbf{Span(Gpd)}$ is not just a bicategory, it’s a monoidal bicategory, whose monoidal operation on objects is the (cartesian) product of groupoids. (Which also tells you what it is for morphisms, by the way). It should be clear, then, that is the monoidal unit, since .

So another way of describing a stuff type is that it’s a morphism from (or to) the monoidal unit in a certain monoidal (bi)category with duals. In the category of Hilbert spaces, if is the space associated to a quantum system, a map would be called a “state” (and the dual would be a “costate”). Stuff types provide a 2-categorical version of the same thing, where the object taking the place of is .

There is, as I’ve discussed here previously, a 2-vector space (indeed, a 2-Hilbert space) associated with this groupoid. But the point of view I’m adopting right now is based on discussion I had with Jamie Vicary about this paper. In it, he gives an abstract (i.e. categorical) description of what’s going on in the quantum mechanics of the harmonic oscillator in terms of an adjunction of categories. This can then be transplanted into various monoidal categories with duals. Here, Jamie gives a more general discussion of quantum algebras, with the same sort of flavour.

So as to the question of how species relate to QFT, this suggests one way to look at how. The harmonic oscillator is the physical system of interest when we look at “states” as spans into . Up to isomorphism, the important features of the groupoid are: its objects correspond to nonnegative integers, which label the energy levels for the oscillator (they “count photons”); each object has automorphisms corresponding to permutations of those photons (they’re indistinguishable – in particular, “bosons”). This is fairly simple, but for a more elaborate QFT picture, look at “states” for other groupoids in .  One complication is that typically these groupoids are going to have some smooth structure…

Perhaps more on this later.

In the past couple of weeks, Masoud Khalkhali and I have been reading and discussing this paper by Marcolli and Al-Yasry. Along the way, I’ve been explaining some things I know about bicategories, spans, cospans and cobordisms, and so on, while Masoud has been explaining to me some of the basic ideas of noncommutative geometry, and (today) K-theory and cyclic cohomology. I find the paper pretty interesting, especially with a bit of that background help to identify and understand the main points. Noncommutative geometry is fairly new to me, but a lot of the material that goes into it turns out to be familiar stuff bearing unfamiliar names, or looked at in a somewhat different way than the one I’m accustomed to. For example, as I mentioned when I went to the Groupoidfest conference, there’s a theme in NCG involving groupoids, and algebras of -linear combinations of “elements” in a groupoid. But these “elements” are actually morphisms, and this picture is commonly drawn without objects at all. I’ve mentioned before some ideas for how to deal with this (roughly: is easy to confuse with the algebra of matrices over ), but anything special I have to say about that is something I’ll hide under my hat for the moment.

I must say that, though some aspects of how people talk about it, like the one I just mentioned, seem a bit off, to my mind, I like NCG in many respects. One is the way it ties in to ideas I know a bit about from the physics end of things, such as algebras of operators on Hilbert spaces. People talk about Hamiltonians, concepts of time-evolution, creation and annihilation operators, and so on in the algebras that are supposed to represent spaces. I don’t yet understand how this all fits together, but it’s definitely appealing.

Another good thing about NCG is the clever elegance of Connes’ original idea of yet another way to generalize the concept “space”. Namely, there was already a duality between spaces (in the usual sense) and commutative algebras (of functions on spaces), so generalizing to noncommutative algebras should give corresponding concepts of “spaces” which are different from all the usual ones in fairly profound ways. I’m assured, though I don’t really know how it all works, that one can do all sorts of things with these “spaces”, such as finding their volumes, defining derivatives of functions on them, and so on. They do lack some qualities traditionally associated with space – for instance, many of them don’t have many, or in some cases any, points. But then, “point” is a dubious concept to begin with, if you want a framework for physics – nobody’s ever seen one, physically, and it’s not clear to me what seeing one would consist of…

(As an aside – this is different from other versions of “pointless” topology, such as the passage from ordinary topologies to, sites in the sense of Grothendieck. The notion of “space” went through some fairly serious mutations during the 20th century: from Einstein’s two theories of relativity, to these and other mathematicians’ generalizations, the concept of “space” has turned out to be either very problematic, or wonderfully flexible. A neat book is Max Jammer’s “Concepts of Space“: though it focuses on physics and stops in the 1930’s, you get to appreciate how this concept gradually came together out of folk concepts, went through several very different stages, and in the 20th century started to be warped out of all recognition. It’s as if – to adapt Dan Dennett – “their word for milk became our word for health”.I would like to see a comparable history of mathematicians’ more various concepts, covering more of the 20th century. Plus, one could probably write a less Eurocentric genealogy nowadays than Jammer did in 1954.)

Anyway, what I’d like to say about the Marcolli and Al-Yasry paper at the moment has to do with the setup, rather than the later parts, which are also interesting. This has to do with the idea of a correspondence between noncommutative spaces. Masoud explained to me that, related to the matter of not having many points, such “spaces” also tend to be short on honest-to-goodness maps between them. Instead, it seems that people often use correspondences. Using that duality to replace spaces with algebras, a recurring idea is to think of a category where morphism from algebra to algebra is not a map, but a left-right -bimodule, . This is similar to the business of making categories of spans.

Let me describe briefly what Marcolli and Al-Yasry describe in the paper. They actually have a 2-category. It has:

Objects: An object is a copy of the 3-sphere with an embedded graph .

Morphisms: A morphism is a span of branched covers of 3-manifolds over :

such that each of the maps is branched over a graph containing (perhaps strictly). In fact, as they point out, there’s a theorem (due to Alexander) proving that ANY 3-manifold can be realized as a branched cover over the 3-sphere, branched at some graph (though perhaps not including a given , and certainly not uniquely).

2-Morphisms: A 2-morphism between morphisms and (together with their maps) is a cobordism , in a way that’s compatible with the structure of the $lateux M_i$ as branched covers of the 3-sphere. The are being included as components of the boundary – I’m writing it this way to emphasize that a cobordism is a kind of cospan. Here, it’s a cospan between spans.

This is somewhat familiar to me, though I’d been thinking mostly about examples of cospans between cospans – in fact, thinking of both as cobordisms. From a categorical point of view, this is very similar, except that with spans you compose not by gluing along a shared boundary, but taking a fibred product over one of the objects (in this case, one of the spheres). Abstractly, these are dual – one is a pushout, and the other is a pullback – but in practice, they look quite different.

However, this higher-categorical stuff can be put aside temporarily – they get back to it later, but to start with, they just collapse all the -categories into -sets by taking morphisms to be connected components of the categories. That is, they think about taking morphisms to be cobordism classes of manifolds (in a setting where both manifolds and cobordisms have some branched-covering information hanging around that needs to be respected – they’re supposed to be morphisms, after all).

So the result is a category. Because they’re writing for noncommutative geometry people, who are happy with the word “groupoid” but not “category”, they actually call it a “semigroupoid” – but as they point out, “semigroupoid” is essentially a synonym for (small) “category”.

Apparently it’s quite common in NCG to do certain things with groupoids – like taking the groupoid algebra of -linear combinations of morphisms, with a product that comes from multiplying coefficients and composing morphisms whenever possible. The corresponding general thing is a categorical algebra. There are several quantum-mechanical-flavoured things that can be done with it. One is to let it act as an algebra of operators on a Hilbert space.

This is, again, a fairly standard business. The way it works is to define a Hilbert space at each object of the category, which has a basis consisting of all morphisms whose source is . Then the algebra acts on this, since any morphism which can be post-composed with one starting at acts (by composition) to give a new morphism starting at – that is, it acts on basis elements of to give new ones. Extending linearly, algebra elements (combinations of morphisms) also act on .

So this gives, at each object , an algebra of operators acting on a Hilbert space – the main components of a noncommutative space (actually, these need to be defined by a spectral triple: the missing ingredient in this description is a special Dirac operator). Furthermore, the morphisms (which in this case are, remember, given by those spans of branched covers) give correspondences between these.

Anyway, I don’t really grasp the big picture this fits into, but reading this paper with Masoud is interesting. It ties into a number of things I’ve already thought about, but also suggests all sorts of connections with other topics and opportunities to learn some new ideas. That’s nice, because although I still have plenty of work to do getting papers written up on work already done, I was starting to feel a little bit narrowly focused.

In “The Fabric of Reality”, David Deutch gives a refutation of solipsism. I’m not entirely sure it works – all he really tries to do is to show that the difference between solipsism and realism is more nearly a mere semantic distinction than is generally assumed. But in any case, along the way, there’s an anecdote about a solipsist professor lecturing his (imaginary?) class merely to help him clarify his ideas. The idea being that, even if the imaginary students don’t really exist, it helps to clarify the professor’s own ideas by lecturing to them, answering questions, and so forth. In this view, you don’t really understand your own opinions – let alone justifiably believe in them – unless you’ve argued for them against a variety of possible criticisms. (J.S. Mill gave a defense of full-fledged freedom of speech, even for grossly offensive and even “dangerous” opinion, on this ground.)

I mention this because, when I told Dan about the blog, he seemed dubious about blogging as a way of communicating math. It’s certainly more solipsistic than a usenet newsgroup, or a mailing list. Those are channels devoted to a particular subject, with many participants. A blog, comments notwithstanding, is mainly a channel devoted to one voice, on many particular subjects. It’s true that half the point of communicating ideas is to get feedback on them from other people. You make your thinking part of one of those great processes like cathedral-building – ad-hoc, gradual, and (significantly) collective. Even so, relatively solipsistic channels are not entirely pointless.

To wit: by working through my theorems about transporting 2-vectors through spans – both for this blog, and for my talk at Groupoidfest, I discovered some problems. Nobody pointed them out, but discovering them was a consequence of approaching the material again from a new angle, with an audience in mind.

The problem is a conceptually important one – mistaking an n-dimensional space for a 1-dimensional space. I’m fairly sure, for various reasons, that the theorem that there is a 2-functor is still true, but the proof I have in my thesis (in the special case where the groupoids are flat connection groupoids on spaces) has a problem. Since that affects the Part 4 of “Spans and Vector Spaces” which I was going to post, I’ll put that off for a while as I get the proof straightened out.

Here is the issue in a nutshell, however:

The proof I have involves a construction of a functor by a particular method, which I’ve been describing in the last three posts. The final step I was going to describe involved what the contstruction does for 2-morphisms – spans between spans. (There is more to the proof, but the remainder is technical enough to be fairly unenlightening – basically, to be a 2-functor, there need to be specified natural isomorphisms replacing the equations for preserving identities and composition in the definition of a functor, and these have to obey some equations which need to be checked.)

The construction given in my thesis is supposed to give a way to take a span of spans of groupoids, and give a natural transformation between a pair of 2-linear maps. But a 2-linear map can be written as a matrix of vector spaces, and a natural transformation is then written as a matrix of linear operators which act componentwise. So one way to look at the problem is to construct a linear map between vector spaces from a span of groupoids.

That is, we have spans and . Picking basis objects for and (namely, objects and , plus representations of their automorphism groups) gives a subgroupoid of of , consisting of those objects which are sent to and under the maps in the span. It also gives a vector space which is built as a colimit of some vector spaces associated to these objects. Assuming is skeletal, this works out (as I described before) to for each of the in question. The same holds for .

Now suppose we have a span-of-spans making the obvious diagram commute. Then because of that commutation, we also have a span of groupoids over each of the choices of objects, and so then the question becomes, partly, how to get a linear map between the vector spaces we just constructed. If you have bases for all the vector spaces here, it’s not too bad: vectors can be seen as complex-valued functions on the basis. We can push these through the span just as we’ve been talking about in the last few posts here: first pull back a function along one leg by composition, then push forward along the other leg. The push-forward will involve a sum over some objects, and some normalizing factors having to do with the groupoid cardinalities of the groupoids in the span.

However, I won’t go too far into detail about this, because the construction I actually outlined doesn’t adequately specify the basis to use. In fact, it will really only work if all the vector spaces is one-dimensional. Then there is a basis for the combined space which just consists of all the objects . I’d hoped that Schur’s lemma (that intertwiners from to itself, or from to itself, have to be multiples of the identity) would get out of this problem, but I’m not sure it does. So there is a problem with the construction I was trying to use.

As I say, I’m fairly sure the theorem remains true – it’s just the proof needs fixing, which I don’t expect to be too hard. However, I’ll refrain from getting sidetracked until I know I have it worked out.

Instead, next time I’ll describe some of the things I learned at Groupoidfest 07 when I presented a talk on this stuff. (At first I was nervous, having discovered this flaw while preparing the talk – but then, a lot of people were talking about work-in-progress, so I don’t feel too bad now. Plus, the meeting was a lot of fun.)