The Cocktail Party Version

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.)

 

New Paper(s)

So one of the things I’ve been doing recently is finishing up a version, and talking about, this paper which I’ve now put on the arXiv. While at it, I figured I should update a previous paper – the current version cuts out part of the original subject (cobordism categories) and expands on the category-theory side of things, giving more detailed proofs, etc. That part will then be out of the way when the topology side shows up in another paper, yet to appear, which will also use the stuff about 2-vector spaces and groupoids from the “new” paper.

Ironically, although I fixed the “issue” which arose when I was posting on the subject – and I’ll come back to that – I’ve already talked about most of what’s in the “new” paper, whereas I never got around to talking about what’s in the “old” one, updated version or not. That’s the one called “Double Bicategories and Double Cospans”, which is the most strictly category-theoretic thing I’ve produced: all the motivation from physics has been abstracted away.  So when I have some time, I’ll write something about that one.

For now, I just wanted to link to this new stuff.

 

Octoberfest 08

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.

 

Update. Curvature. 2-monads.

First off, a nice recent XKCD comic about height.

…

I’ve been busy of late starting up classes, working on a paper which should appear on the archive in a week or so on the groupoid/2-vector space stuff I wrote about last year.  I resolved the issue I mentioned in a previous post on the subject, which isn’t fundamentally that complicated, but I had to disentangle some notation and learn some representation theory to get it figured out.  I’ll maybe say something about that later, but right now I felt like making a little update.  In the last few days I’ve also put together a little talk to give at Octoberfest in Montreal, where I’ll be this weekend.  Montreal is a lovely city to visit, so that should be enjoyable.

A little while ago I had a talk with Dan’s new grad student – something for a class, I think – about classical and modern differential geometry, and the different ideas of curvature in the two settings.  So the Gaussian curvature of a surface embedded in has a very multivariable-calculus feel to it: you think of curves passing through a point, parametrized by arclength.  The have a moving orthogonal frame attached: unit tangent vector, its derivative, and their cross-product.  The derivative of the unit tangent is always orthogonal (it’s not changing length), so you can imagine it to be the radius of a circle, with length , the radius of curvature.  Then you have curvature along that path.  At any given point on a surface, you get two degrees of freedom – locally, the curve looks like a hyperboloid or an ellipse, or whatever, so there’s actually a curvature form.  The determinant gives the Gaussian curvature .  So it’s a “second derivative” of the surface itself (if you think of it as ).  The Gaussian curvature, unlike the curvature in particular directions, is intrinsic – preserved by isometry of the surface, so it’s not really dependent on the embedding.  But this fact takes a little thinking to get to.  Then there’s the trace – the scalar curvature.

In a Riemannian manifold, you  need to have a connection to see what the curvature is about.  Given a metric, there’s the associated Levi-Civita connection, and of course you’d get a metric on a surface embedded in , inherited from the ambient space.  But the modern point of view is that the connection is the important object: the ambient space goes away entirely.  Then you have to think of what the curvature represents differenly, since there’s no normal vector to the surface any more.  So now we’re assuming we want an intrinsic version of the “second derivative of the surface” (or n-manifold) from the get-go.  Here you look at the second derivative of the connection in any given coordinate system.  You’re finding the infinitesimal noncommutativity of parallel transport w.r.t two coordinate directions: take a given vector, and transport it two ways around an infinitesimal square, and take the difference, get a new vector.  This all is written as a (3,1)-form, the Riemann tensor.  Then you can contract it down and get a matrix again, and then contract on the last two indices (a trace!) and you get back the scalar curvature again – but this is all in terms of the connection (the coordinate dependence all disappears once you take the trace).

I hadn’t thought about this stuff in coordinates for a while, so it was interesting to go back and work through it again.

In the noncommutative geometry seminar, we’ve been talking about classical mechanics – the Lagrangian and Hamiltonian formulation.  So it reminded me of the intuition that curvature – a kind of second derivative – often shows up in Lagrangians for field theories using connections because it’s analogous to kinetic energy.  A typical mechanics Lagrangian is something like (kinetic energy) – (potential energy), but this doesn’t appear much in the topological field theories I’ve been thinking about because their curvature is, by definition, zero.  Topological field theory is kind of like statics, as opposed to mechanics, that way.  But that’s a handy simplification for the program of trying to categorify everything.  Since the whole space of connections is infinite dimensional, worrying about categorified action principles opens up a can of worms anyway.

So it’s also been interesting to remember some of that stuff and discuss it in the seminar – and it was inially suprising that it’s the introduction to “noncommutative geometry”.  It does make sense, though, since that’s related to the formalism of quantum mechanics: operator algebras on Hilbert spaces.

Finally, I was looking for something on 2-monads for various reasons, and found a paper by Steve Lack which I wanted to link to here so I don’t forget it.

The reason I was looking was that (a) Enxin Wu, after talking about deformation theory of algebras, was asking after monads and the bar construction, which we talked about at the UCR “quantum gravity” seminar, so at some point we’ll take a look at that stuff.  But it reminded me that I was interested in the higher-categorical version of monads for a different reason. Namely, I’d been talking to Jamie Vicary about his categorical description of the harmonic oscillator, which is based on having a monad in a nice kind of monoidal category.  Since my own category-theoretic look at the harmonic oscillator fits better with this groupoid/2-vector space program I’ll be talking about at Octoberfest (and posting about a little later), it seemed reasonable to look at a categorified version of the same picture.

But first things first: figuring out what the heck a 2-monad is supposed to be.  So I’ll eventually read up on that, and maybe post a little blurb here, at some point.

Anyway, that update turned out to be longer than I thought it would be.