So there’s a lot of preparations going on for the workshop HGTQGR coming up next week at IST, and the program(me) is much more developed – many of the talks are now listed, though the schedule has yet to be finalized.  This week we’ll be having a “pre-school school” to introduce the local mathematicans to some of the physics viewpoints that will be discussed at the workshop – Aleksandar Mikovic will be introducing Quantum Gravity (from the point of view of the loop/spin-foam approach), and Sebastian Guttenberg will be giving a mathematician’s introduction to String theory.

These are by no means the only approaches physicists have taken to the problem of finding a theory that incorporates both General Relativity and Quantum Field Theory.  They are, however, two approaches where lots of work has been done, and which appear to be amenable to using the mathematical tools of (higher) category theory which we’re going to be talking about at the workshop.  These are “higher gauge theory”, which very roughly is the analog of gauge theory (which includes both GR and QFT) using categorical groups, and TQFT, which is a very simple type of quantum field theory that has a natural description in terms of categories, which can be generalized to higher categories.

I’ll probably take a few posts after the workshop to write up these, and the many other talks and mini-courses we’ll be having, but right now, I’d like to say a little bit about another talk we had here recently.  Actually, the talk was in Porto, but several of us at IST in Lisbon attended by a videoconference.  This was the first time I’ve seen this for a colloquium-style talk, though I did once take a course in General Relativity from Eric Poisson that was split between U of Waterloo and U of Guelph.  I thought it was a great idea then, and it worked quite well this time, too.  This is the way of the future – and unfortunately it probably will be for some time to come…

Anyway, the talk in question was by Thomasz Brzezinski, about “Synthetic Non-Commutative Geometry” (link points to the slides).  The point here is to take two different approaches to extending differential geometry (DG) and combine the two insights.  The “Synthetic” part refers to synthetic differential geometry (SDG), which is a program for doing DG in a general topos.  One aspect of this is that in a topos where the Law of the Excluded Middle doesn’t apply, it’s possible for the real-numbers object to have infinitesimals: that is, elements which are smaller than any positive element, but bigger than zero.  This lets one take things which have to be treated in a roundabout way in ordinary DG, like dx, and take them at face value – as an infinitesimal change in x.  It also means doing geometry in a completely constructive way.

However, these aspects aren’t so important here.  The important fact about it here is that it’s based on building a theory that was originally defined in terms of sets, or topological spaces – that is, in the toposes Sets, or Top  – and transplanting it to another category.  This is because Brzezinski’s goal was to do something similar for a different extension of DG, namely non-commutative geometry (NCG).  This is a generalisation of DG which is based on the equivalence CommAlg^{op} \simeq lCptHaus between the categories of commutative C^{\star}-algebras (and algebra maps, read “backward” as morphisms in CommAlg^{op}), and that of locally compact Hausdorff spaces (which, for objects, equates a space X with the algebra C(X) of continuous functions on it, and an algebra A with its spectrum Spec(A), the space of maximal ideals).  The generalization of NCG is to take structures defined for lCptHaus that create DG, and make similar definitions in the category Alg^{op}, of not-necessarily-commutative C^{\star}-algebras.

This category is the one which plays the role of the topos Top.  It isn’t a topos, though: it’s some sort of monoidal category.  And this is what “synthetic NCG” is about: taking the definitions used in NCG and reproducing them in a generic monoidal category (to be clear, a braided monoidal category).

The way he illustrated this is by explaining what a principal bundle would be in such a generic category.

To begin with, we can start by giving a slightly nonstandard definition of the concept in ordinary DG: a principal G-bundle P is a manifold with a free action of a (compact Lie) group G on it.  The point is that this always looks like a “base space” manifold B, with a projection \pi : P \rightarrow B so that the fibre at each point of B looks like G.  This amounts to saying that \pi is an equalizer:

P \times G \stackrel{\longrightarrow}{\rightarrow} P \stackrel{\pi}{\rightarrow} B

where the maps from G\times P to P are (a) the action, and (b) the projection onto P.  (Being an equalizer means that \pi makes this diagram commute – has the same composite with both maps – and any other map \phi that does the same factors uniquely through \pi.)  Another equivalent way to say this is that since P \times G has two maps into P, then it has a map into the pullback P \times_B P (the pullback of two copies of P \stackrel{\pi}{\rightarrow} B), and the claim is that it’s actually ismorphic.

The main points here are that (a) we take this definition in terms of diagrams and abstract it out of the category Top, and (b) when we do so, in general the products will be tensor products.

In particular, this means we need to have a general definition of a group object G in any braided monoidal category (to know what G is supposed to be like).  We reproduce the usual definition of a group objects so that G must come equipped with a “multiplication” map m : G \otimes G \rightarrow G, an “inverse” map \iota : G \rightarrow G and a “unit” map u : I \rightarrow G, where I is the monoidal unit (which takes the role of the terminal object in a topos like Top, the unit for \times).  These need to satisfy the usual properties, such as the monoid property for multiplication:

m \circ (m \otimes id_G) = m \circ (id_G \otimes m) : G \otimes G \otimes G \rightarrow G

(usually given as a diagram, but I’m being lazy).

The big “however” is this: in Sets or Top, any object X is always a comonoid in a canonical way, and we use this implictly in defining some of the properties we need.  In particular, there’s always the diagonal map \Delta : X \rightarrow X \times X which satisfies the dual of the monoid property:

(id_X \times \Delta) \circ \Delta = (\Delta \times id_X) \circ \Delta

There’s also a unique counit \epsilon \rightarrow \star, the map into the terminal object, which makes (X,\Delta,\epsilon) a counital comonoid automatically.  But in a general braided monoidal category, we have to impose as a condition that our group object also be equipped with \Delta : G \rightarrow G \otimes G and \epsilon : G \rightarrow I making it a counital comonoid.  We need this property to even be able to make sense of the inverse axiom (which this time I’ll do as a diagram):

This diagram uses not only \Delta but also the braiding map \sigma_{G,G} : G \otimes G  \rightarrow G \otimes G (part of the structure of the braided monoidal category which, in Top or Sets is just the “switch” map).  Now, in fact, since any object in Set or Top is automatically a comonoid, we’ll require that this structure be given for anything we look at: the analog of spaces (like P above), or our group object G.  For the group object, we also must, in general, require something which comes for free in the topos world and therefore generally isn’t mentioned in the definition of a group.  Namely, the comonoid and monoid aspects of G must get along.  (This comes for free in a topos essentially because the comonoid structure is given canonically for all objects.)  This means:

For a group in Sets or Top, this essentially just says that the two ways we can go from (x,y) to (xy,xy) (duplicate, swap, then multiply, or on the other hand multiply then duplicate) are the same.

All these considerations about how honest-to-goodness groups are secretly also comonoids does explain why corresponding structures in noncommutative geometry seem to have more elaborate definitions: they have to explicitly say things that come for free in a topos.  So, for instance, a group object in the above sense in the braided monoidal category Vect = (Vect_{\mathbb{F}}, \otimes_{\mathbb{F}}, \mathbb{F}, flip) is a Hopf algebra.  This is a nice canonical choice of category.  Another is the opposite category Vect^{op} – this is a standard choice in NCG, since spaces are supposed to be algebras – this would be given the comonoid structure we demanded.

So now once we know all this, we can reproduce the diagrammatic definition of a principal G-bundle above: just replace the product \times with the monoidal operation \otimes, the terminal object by I, and so forth.  The diagrams are understood to be diagrams of comonoids in our braided monoidal category.  In particular, we have an action \rho : P \otimes G \rightarrow P,which is compatible with the \Delta maps – so in Vect we would say that a noncommutative principal G-bundle P is a right-module coalgebra over the Hopf algebra G.  We can likewise take this (in a suitably abstract sense of “algebra” or “module”) to be the definition in any braided monoidal category.

To have the “freeness” of the action, there needs to be an equalizer of:

\rho, (id_P \otimes \epsilon) : P \otimes G \stackrel{\longrightarrow}{\rightarrow} P \stackrel{\pi}{\rightarrow} B

The “freeness” condition for the action is likewise defined using a monoidal-category version of the pullback (fibre product) P \times_B P.

This was as far as Brzezinski took the idea of synthetic NCG in this particular talk, but the basic idea seems quite nice.  In SDG, one can define all sorts of differential geometric structures synthetically, that is, for a general topos: for example, Gonzalo Reyes has gone and defined the Einstein field equations synthetically.  Presumably, a lot of what’s done in NCG could also be done in this synthetic framework, and transplanted to other categories than the usual choices.

Brzezinski said he was mainly interested in the “usual” choices of category, Vect and Vect^{op} – so for instance in Vect^{op}, a “principal G-bundle” is what’s called a Hopf-Galois extension.  Roger Picken did, however, ask an interesting question about other possible candidates for the category to work in.  Given that one wants a braided monoidal category, a natural one to look at is the category whose morphisms are braids.  This one, as a matter of fact, isn’t quite enough (there’s no braid m : n \otimes n \rightarrow n, because this would be a braid with 2n strands in and n strands out – which is impossible.  But some sort of category of tangles might make an interestingly abstract setting in which to see what NCG looks like.  So far, this doesn’t seem to have been done as far as I can see.