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.)
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.
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.
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.)