update


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

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

Quantales

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.)

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.

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 \mathbf{2-Cat} (the strict form) and \mathbf{Bicat} (the weak form). Whereas there is no triequivalence between (strict) \mathbf{3-Cat} and (weak) \mathbf{Tricat}. There is a triequivalence between \mathbf{Tricat} and \mathbf{Gray} – 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 \mathbf{Bicat} and \mathbf{2-Cat} factors through pairs of adjoint functors between each of these and \mathbf{Graph}). 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 \omega-category recursively: generate faces of dimension n 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 \omega-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 \omega-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 U/G, where U is an open set in \mathbb{R}^n, and G 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 X/L where X is a manifold and L 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 G and H turns out to be the same as having a nice enough SPAN of groupoids

G \leftarrow K \rightarrow H

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.

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 \mathbb{R}^3 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 r, the radius of curvature.  Then you have \kappa = \frac{1}{r} 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 K.  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 \mathbb{R}^3, 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.

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