Conference: MakkaiFest ‘09 (Models, Logic, Higher Categories)

It’s taken me a while to write this up, since I’ve been in the process of moving house – packing and unpacking and all the rest. However, a bit over a week ago, I was in Montreal, attending MakkaiFest ‘09 at the Centre de Recherches Mathematiques at the University of Montréal (and a pre-conference workshop hosted at McGill, which I’m including in the talks I mention here). This was in honour of the 70th birthday of Mihaly (Michael) Makkai, of McGill University. Makkai has done a lot of important foundational work in logic, model theory, and category theory, and a great many of the talks were from former students who’d gone on and been inspired by him, so one got sense of the range of things he’s worked on through his life.

The broad picture of Makkai’s work was explained to us by J.P. Marquis, from the Philosophy department at U of M. He is interested in philosophy of mathematics, and described Makkai’s project by contrast with the program of axiomatization of the early 20th century, along the lines suggested by Hilbert. This program provided a formal language for concrete structures – the problem, which category theory is part of a solution to, is to do the same for abstract structures. Contrast, for instance, the concrete description of a group as a (particular) set with some (particular) operation, with the abstract definition of a group object in a category. Makkai’s work in categorical logic, said Marquis, is about formalizing the process of abstraction that example illustrates.

Model Theory/Logic

This matter – of the relation between abstract theories and concrete models of the theories – is really what model theory is about, and this is one of the major areas Makkai has worked on. Roughly, a theory is most basically a schema with symbols for types, members of types, and some function symbols – and a collection of sentences built using these symbols (usually generated from some axioms by rules of logical inference). A model is (intuitively), an interpretation of the terms: a way of assigning concrete data to the symbols – say, a symbol for a type is assigned the set of all entities of that type, and a function symbol is assigned an actual function between sets, and so on – making all propositions true. A morphism of models is a map that preserves all the properties of the model that can be stated using first order logic.

This is an older way to say things – Victor Harnik gave an expository talk called “Model Theory vs. Categorical Logic” in which he compared two ways of adding an equivalence relation to a theory. The model theory way (invented by Shelah) involves taking the theory (list of sentences) and extending it to a new theory . This has, for instance, some new types – if we had a type for “element of group”, for example, we might then get a new type “equivalence class of elements of group”, and so on. Now, this extension is “tight” in the sense that the categories of all models of and of are equivalent (by a forgetful functor ) – but one can prove new theorems in the extended theory. To make this clear, he described work (due to Makkai and Reyes) about pretopos completion. Here, one has the concept of a “Boolean logical category” – is an example, as is, for any theory, a certain category whose objects are the formulas of the theory. This is related to Lawvere theories (see below). There are logical functors between such categories – functors into are models, but there are also logical functors between theories. The point is that a theory embeds into (abusing notation here – these are now the boolean logical categories). Then the point is that arises as a kind of completion of – namely, it’s a boolean pretopos (not just category). Moreover, it has some nice universal properties, making this point of view a bit more natural than the model-theoretic construction.

Bradd Hart’s talk, “Conceptual Completeness for Cantinuous Logic”, was a bit over my head, but made some use of this kind of extension of a theory to . The basic point seems to be to add some kind of continuous structure to logic. One example comes from a metric structure – defining a metric space of terms, where the metric function is some sum , where the are formulas with two variables, either true or false – where true gives a , and false gives a in this sum. This defines a distance from to associated to the given list of formulas . A continuous logic is one with a structure like this. The business about equivalence relations arises if we say two things are equivalent when the distance between them is 0 – this leads to a concept of completion, and again there’s a notion that the categories of models are equivalent (though proving it here involves some notion of approximating terms to arbitrary epsilon, which doesn’t appear in standard logic).

Anand Pillay gave a talk which used model theory to describe some properties of the free group on n generators. This involved a “theory of the free group” which applies to any free group, and regard each such group as a model of the theory – in fact a submodel of some large model, and using model-theoretic methods to examine “stability” properties, in some sense which amounts to a notion of defining “generic” subsets of the group.

Logic and Higher Categories

A number of talks specifically addressed the ground where logic meets higher dimensional categories, since Makkai has worked with both.

In one talk, Robert Paré described a way of thinking about first-order theories as examples of “double Lawvere theories”. Lawvere’s way of formalizing “theories and models” was to say that the theory is a category itself (which has just the objects needed to describe the kind of structure it’s a theory of) – and a model is a functor into (or some other category – a model of the theory of groups in topological spaces, say, is a topological group). For example, the theory of groups includes an object and powers of it, multiplication and inverse maps, and expresses the axioms by the fact that certain diagrams commute. A model is a functor , assigning to the “group object” a set of elements, which then get the group structure from the maps. Instead of a category, this uses a double category. There are two kinds of morphisms – horizontal and vertical – and these are used to represent two kinds of symbols: function symbols, and relation symbols. (For example, one can talk about the theory of an ordered field – so one needs symbols for multiplication and addition and so forth, but also for the order relation ). Then a model of such a theory is a double functor into the double category whose objects are sets, and whose horizontal and vertical morphisms are respectively functions and relations.

André Joyal gave a talk about the first order logic of higher structures. He started by commenting on some fields which began life close together, and are now gradually re-merging: logic and category theory; category theory and homotopy theory (via higher categories); homotopy theory and algebraic geometry. The higher categories Joyal was thinking of are quasicategories, or “-categories, which are simplicial sets satisfying a weak version of a horn-filling condition (the “strict” version of this, a Kan complex, includes as example , the nerve of a category – there’s an n-simplex for each sequence of n composable morphisms, whose other edges are the various composites, and whose faces are “compositors”, “associators”, and so on – which for are identities). The point of this is that one can reproduce most of category theory for quasicategories – in particular, he mentioned limits and colimits, factorization systems, pretoposes, and model theory.

Moving to quasicategories on one side of the parallel between category theory and logic has a corresponding move on the other side – on the logic side, one aspect is that the usual notion of a language is replaced by what’s called Martin-Löf type theory. This, in fact, was the subject of Michael Warren’s talk, “Martin-Löf complexes” (I reported on a similar talk he gave at Octoberfest last year). The idea here is to start by defining a globular set, given a theory and type – a complex whose n-cells have two faces, of dimension (n-1). The 0-cells are just terms of some type . The 1-cells are terms of types like , where and are variables of type – the type has an interpretation as a proposition that “extensionally” (i.e. not via a proof – but as for instance when two programs with non-equivalent code happen to always produce the same output). This kind of operation can be repeated to give higher cells, like , and so on. Given a globular set , one gets a theory by an adjoint construction. Putting the two together, one has a monad on the category of globular sets – algebras for the monad are Martin-Löf complexes. Throwing in syntactic rules to truncate higher cells (I suppose by declaring all cells to be identities) gives n-truncated versions of these complexes, . Then there is some interesting homotopy theory, in that the category of n-truncated Martin-Löf complexes is expected to be a model for homotopy n-types. For example, is equivalent to , and there is an adjunction (in fact, a Quillen equivalence – that is, a kind of “homotopy” equivalence) between and .

Category Theory/Higher Categories

There were a number of talks that just dealt with categories – including higher categories – in their own right. Makkai has worked, for example, on computads, which were touched on by Marek Zawadowski in one of his two talks (one in the pre-conference workshop, the other in the conference). The first was about categories of “many-to-one shapes”, which are important to computads – these are a notion of higher-category, where every cell takes many “input” faces to one “output” face. Zawadowski described a “shape” of an n-cell as an initial object in a certain category built from the category of computads with specified faces. Then there’s a category of shapes, and an abstract description of “shape” in terms of a graded tensor theory (graded for dimension, and tensor because there’s a notion of composition, I believe). Zawadowski’s second talk, “Opetopic Sets in Lax Monoidal Fibrations”, dealt with a similar topic from a different point of view. A lax monoidal fibration (LMF) is a kind of gadget for dealing with multi-level structures (categories, multicategories, quasicategories, etc). There’s a lot of stuff here I didn’t entirely follow, but just to illustrate: categories arise as LMF, by the fibration , where is the category with two objects , and two arrows from to . An object in the functor category consists of a “set of morphisms and set of objects” with maps – making this a category involves the monoidal structure, and how composition is defined, and the real point is that this is quite general machinery.

Joachim Lambek and Gonzalo Reyez, both longtime collaborators and friends of Makkai, also both gave talks that touched on physics and categories, though in very different ways. Lambek talked about the “Lorentz category” and its appearance in special relativity.  This involves a reformulation of SR in terms of biquaternions: like complex numbers, these are of the form , but and are quaternions.  They have various conjugation operations, and the geometry of SR can be described in terms of their algebra (just as, say, rotations in 3D can be described in terms of quaternions).  The Lorentz category is a way of organizing this – its two objects correspond to “unconjugated” and “conjugated” states.

Gonzalo Reyez gave a derivation of General Relativity in the context of synthetic differential geometry.  The substance of this derivation is not so different from the usual one, but with one exception.  Einstein’s field equations can be derived in terms of the motions of small regions full of of freely falling test particles – synthetic differential geometry makes it possible to do the same analysis using infinitesimals rigorously all the way through.  The basic point here is that in SDG one replaces the real line as usually conceived, with a “real line with infinitesimals” (think of the ring , which is like the reals, but has the infinitesimal , whose square is zero).

Among other talks: John Power talked about the correspondence between Lawvere theories in universal algebra and finitary tree monads on sets – and asked about what happens to the left hand side of this correspondence when we replace “sets” with other categories on the righ hand side. Jeff Egger talked about measure theory from a categorical point of view – namely, the correspondence of NCG between C*-algebras and “noncommutative” topological spaces, and between W*-algebras and “noncommutative” measure spaces, thought of in terms of locales. Hongde Hu talked about the “codensity theorem”, and a way to classify certain kinds of categories – he commented on how it was inspired by Makkai’s approach to mathematics: 1) Find new proofs of old theorems, (2) standardize the concepts used in them, and (3) prove new theorems with those concepts. Fred Linton gave a talk describing Heath’s “V-space”, which is a half-plane with a funny topology whose open sets are “V” shapes, and described how the topos of locally finite sheaves over it has surprising properties having to do with nonexistence of global sections. Manoush Sadrzadeh, whom I met recently at CQC (see the bottom of the previous post) was again talking about linguistics using monoidal categories – she described some rules for “clitic movement” and changes in word order, and what these rules look like in categorical terms.

Other

A few other talks are a little harder for me to fit into the broad classification above.  There was Charles Steinhorn’s talk about ordered “o-minimal” structures, which touched on a bit of economics – essentially, a lot of economics is based on the assumption that preference orders can be made into real-valued functions, but in fact in many cases one has (variants on) “lexicographic order”, involving ranked priorities.  He talked about how typically one has a space of possibilities which can be cut up into cells, with one sort of order in each cell.  There was Julia Knight, talking about computable structures of “high Scott rank” – in particular, this is about infinite structures that can still be dealt with computably – for example, infinitary logical formulas involving an infinite number of “OR” statements where all the terms being joined are of some common form.  This ends up with an analysis of certain infinite trees.  Hal Kierstead gave a talk about Ramsey theory which I found notable because it used the kind of construction based on a game: to prove that any colouring of a graph (or hypergraph) has some property, one devises a game where one player tries to build a graph, and the other tries to colour it, and proves a winning strategy for one player.  Finally, Michael Barr gave a talk about a duality between certain categories of modules over commutative rings.

All in all, an interesting conference, with plenty of food for thought.

Barr, Kierstead, Knight, Steinhorn

 

Conference: Categories, Quanta, Concepts – Pt. 2

Continuing from the previous post…

I realized I accidentally omitted Klaas Lansdman’s  talk on the Kochen-Specker theorem, in light of topos theory.  This overlaps a lot with the talk by Andreas Doring, although there are some significant differences.  (Having heard only what Andreas had to say about the differences, I won’t attempt to summarize them).  Again, the point of the Kochen-Specker theorem is that there isn’t a “state space” model for a quantum system – in this talk, we heard the version saying that there are no “locally sigma-Boolean” maps, from operators on a Hilbert space, to .  (This is referring to sigma-algebas (of measurable sets on a space), and Boolean algebras of subsets – if there were such a map, it would be representing the system in terms of a lattice equivalent to some space).  As with the Isham/Doring approach, they then try to construct something like a state space – internal to some topos.  The main difference is that the toposes are both categories of functors into sets from some locale – but here the functors are covariant, rather than contravariant.

Now, roughly speaking, the remaining talks could be grouped into two kinds:

Quantum Foundations

Many people came to this conference from a physics-oriented point of view.  So for instance Rafael Sorkin gave a talk asking “what is a quantum reality?”. He was speaking from a “histories” interpretation of quantum systems. So, by contrast, a “classical reality” would mean one worldline: out of some space of histories, one of them happens. In quantum theory, you typically use the same space of histories, but have some kind of “path integral” or “sum over histories” when you go to compute the probabilities of given events happening. In this context, “event” means “a subset of all histories” (e.g. the subset specified by a statement like “it rained today”). So his answer to the question is: a reality should be a way of answering all questions about all events.  This is called a “coevent”.  Sorkin’s answer to “what is a quantum reality?” is: “a primitive, preclusive coevent”.

In particular, it’s a measure .  For a classical system, “answering” questions means yes/no, whether the one history is in a named event – for a quantum system, it means specifying a path integral over all events – i.e. a measure on the space of events.  This measure needs some nice properties, but it’s not, for instance, a probability measure (it’s complex valued, so there can be interference effects).  Preclusion has to do with the fact that the measure of an event being zero means that it doesn’t happen – so one can make logical inferences about which events can happen.

Other talks addressing foundational problems in physics included Lucien Hardy’s: he talked about how to base predictive theories on operational structures – and put to the audience the question of whether the structures he was talking about can be represented categorically or not.  The basic idea is an “operational structure” is some collection of operations that represents a physical experiment whose outcome we might want to predict.  They have some parameters (”knob settings”), outcomes (classical “readouts”), and inputs and outputs for the things they study and affect (e.g. a machine takes in and spit out an electron, doing something in the middle).  This sort of thing can be set up as a monoidal category – but the next idea, “object-oriented operationalism”, involved components having “connections” (given relations between their inputs) and “coincidences” (predictable correlations in output).  The result was a different kind of diagram language for describing experiments, which can be put together using a “causaloid product” (he referred us to this paper, or a similar one, on this).

Robert Spekkens gave a talk about quantum theory as a probability theory – there are many parallels, though the complex amplitudes give QM phenomena like interference.  Instead of a “random variable” , one has a Hilbert space ; instead of a (positive) function of , one has a positive operator on ; standard things in probability have analogs in the quantum world.  What Robert Spekkens’ talk dealt with was how to think about conditional probabilities and Bayesian inference in QM.  One of the basic points is that when calculating conditional probabilities, you generally have to divide by some probability, which encounters difficulties translating into QM.  He described how to construct a “conditional density operator” along similar lines – replacing “division” by a “distortion” operation with an analogous meaning.  The whole thing deeply uses the Choi-Jamiolkowski isomorphism, a duality between “states and channels”.  In terms of the string diagrams Bob Coecke et. al. are keen on, this isomorphism can be seen as taking a special cup which creates entangled states into an ordinary cup, with an operator on one side.  (I.e. it allows the operation to be “slid off” the cup).  The talk carried this through, and ended up defining a quantum version of the probabilistic concept of “conditional independence” (i.e. events A and C are independent, given that B occurred).

A more categorical look at foundational questions was given by Rick Blute’s talk on “Categorical Structures in AQFT”, i.e. Algebraic Quantum Field Theory.  This is a formalism for QFT which takes into account the causal structure it lives on – for example, on Minkowski space, one has a causal order for points, with if there is a future-directed null or timelike curve from to .  Then there’s an “interval” (more literally, a double cone) , and these cones form a poset under inclusion (so this is a version of the poset of subspaces of a space which keeps track of the causal structure).  Then an AQFT is a functor from this poset into C*-algebras (taking inclusions to inclusions): the idea is that each local region of space has its own algebra of observables relevant to what’s found there.  Of course, these algebras can all be pieced together (i.e. one can take a colimit of the diagram of inclusions coming from all regions on spacetime.  The result is .  Then, one finds a category of certain representations of it on a hilbert space (namely, “DHR” representations).  It turns out that this category is always equivalent to the representations of some group , the gauge group of the AQFT.  Rick talked about these results, and suggested various ways to improve it – for example, by improving how one represents spacetime.

The last talk I’d attempt to shoehorn into this category was by Daniel Lehmann.  He was making an analysis of the operation “tensor product”, that is, the monoidal operation in .  For such a fundamental operation – physically, it represents taking two systems and looking at the combined system containing both – it doesn’t have a very clear abstract definition.  Lehmann presented a way of characterizing it by a universal property analogous to the universal definitions for products and coproducts.  This definition makes sense whenever there is an idea of a “bimorphism” – a thing which abstracts the properties of a “bilinear map” for vector spaces.  This seems to be closely related to the link between multicategories and monoidal categories (discussed in, for example, Tom Leinster’s book).

Categories and Logic

Some less physics-oriented and more categorical talks rounded out the part of the program that I saw.  One I might note was Mike Stay’s talk about the Rosetta Stone paper he wrote with John Baez.  The Rosetta Stone, of course, was a major archaeological find from the Ptolemaic period in Egypt – by that point, Egypt had been conquered by Alexander of Macedon and had a Greek speaking elite, but the language wasn’t widespread.  So the stone is an official pronouncement with a message in Greek, and in two written forms of Egyptian (heiroglyphic and demotic), neither of which had been readable to moderns until the stone was uncovered and correspondences could be deduced between the same message in a known language and two unknown ones.  The idea of their paper, and Mike’s talk, is to collect together analogs between four subjects: physics, topology, computation, and logic.  The idea is that each can be represented in terms of monoidal categories.  In physics, there is the category of Hilbert spaces; in topology one can look at the category of manifolds and cobordisms; in computation, there’s a monoidal category whose objects are data types, and whose morphisms are (equivalence classes) of programs taking data of one type in and returning data of another type; in logic, one has objects being propositions and morphisms being (classes) of proofs of one proposition from another.  The paper has a pretty extensive list of analogs between these domains, so go ahead and look in there for more!

Peter Selinger gave a talk about “Higher-Order Quantum Computation”.  This had to do with interesting phenomena that show up when dealing with “higher-order types” in quantum computers.  These are “data types”, as I just described – the “higher-order” types can be interpreted by blurring the distinction between a “system” and a “process”.  A data type describing a sytem we might act on might be or .  A higher order type like describes a process which takes something of type and returns something of type .  One could interpret this as a black box – and performing processes on a type is like studying that black box as a system itself.  This type is like an “internal hom” – and so one might like to say, “well, it’s dual to tensor – so it amounts to taking , since we’re in the category of Hilbert spaces”.  The trouble is, for physical computation, we’re not quite in the category where that works.  Because not all operators are significant: only some class of totally positive operators are physical.  So we don’t have the hom-tensor duality to use (equivalently, don’t have a well-behaved dual), and these types have to be considered in their own right.  And, because computations might not halt, operations studying a black box might not halt.  So in particular, a “co-co-qubit” isn’t the same as a qubit.  A co-qubit is a black box which eats a qubit and terminates with some halting probability.  A co-co-qubit eats a co-qubit and does the same.  If not for the halting probability, one could equally well see a qubit “eating” a co-co-qubit as the reverse.  But in fact they’re different.  A key fact in Peter’s talk is that quantum computation has new logical phenomena happening with types of every higher order.  Quantifying this (an open problem, apparently) would involve finding some equivalent of Bell inequalities that apply to every higher order of type.  It’s interesting to see how different quantum computing is, in not-so-obvious ways, from the classical kind.

Manoush Sadrzadeh gave a talk describing how “string diagrams” from monoidal categories, and representations of them, have been used in linguistics.  The idea is that the grammatical structure of a sentence can be build by “composing” structures associated to words – for example, a verb can be composed on left and right with subject and object to build a phrase.  She described some of the syntactic analysis that went into coming up with such a formalism.  But the interesting bit was to compare putting semantics on that syntax to taking a representation.  In particular, she described the notion of a semantic space in linguistics: this is a large-dimensional vector space that compares the meanings of words.  A rough but surprisingly effective way to clump words together by meaning just uses the statistics on a big sample of text, measuring how often they co-occur in the same context. Then there is a functor that “adds semantics” by mapping a category of string diagrams representing the syntax of sentences into one of vector spaces like this.  Applying the kind of categorical analysis usually used in logic to natural language seemed like a pretty neat idea – though it’s clear one has to make many more simplifying assumptions.

…

On the whole, it was a great conference with a great many interesting people to talk to – as you might guess from the fact that it took me three posts to comment on everything I wanted.

 

Conference: Categories, Quanta, Concepts – Pt. 1

So as I mentioned in my previous post, I attended 80% of the conference “Categories, Quanta, Concepts”, hosted by the Perimeter Institute.  Videos of many of the talks are online, but on the assumption that not everyone will watch them all, I’ll comment anyway…

It dealt with various takes on the uses of category theory in fundamental physics, and quantum physics particularly. One basic theme is that the language of categories can organize and clarify the concepts that show up here. Since there doesn’t seem to be a really universal agreement on what “fundamental” physics is, or what the concepts involved might be, this is probably a good thing.

There were a lot of talks, so I’ll split this into a couple of posts – this first one dealing with two obvious category-related themes – monoidal categories and toposes.  The next post will cover most of the others – roughly, focused on fundamentals of quantum mechanics, and on categories for logic and language.

Monoidal Categories

So a large contingent came from Oxford’s Comlab, many of them looking at ideas that I first saw popularized by Abramsky and Coecke about describing the features of quantum mechanics that appear in any dagger-compact category. This yields a “string diagram” notation for quantum systems. (An explanation of this system is given by Abramsky and Coecke – http://arxiv.org/abs/0808.1023 – or more concisely by Coecke – http://arxiv.org/abs/quant-ph/0510032).

Samson Abramsky talked about diagonal arguments. This is a broad class of arguments including Cantor’s theorem (that the real line is uncountable), Russell’s paradox in set theory (about the “set” of non-self-membered sets), Godel’s incompleteness theorem, and others. Abramsky’s talk was based on Bill Lawvere’s analysis of these arguments in general cartesian closed categories (CCC’s). The relevance to quantum theory has to do with “no-cloning” theorems – that quantum states can’t be duplicated. Diagonal arguments involve two capabilitiess: the ability to duplicate objects, and the ability to represent predicates (think of Godel numbering, for instance) which is related to a fixed point property. Generalizing to other monoidal categories, one still has representability: linear functionals on Hilbert spaces can be represented by vectors. But diagonal arguments fail since there is no diagonal .

Bob Coecke and Ross Duncan both spoke about “complementary observables”. Part of this comes from their notion of an “observable structure”, or “classical structure” for a quantum system. The intuition here is that this is some collection of observables which we can simultaneously observe, and such that, if we restrict to those observables, and states which are eigenstates for them, we can treat the whole system as if it were classical. In particular, this gives us “copy” and “destroy” operations for states – these maps and their duals actually turn out to define a Frobenius algebra. In finite-dimensional Hilbert spaces, this is equivalent to choosing an orthonormal basis.

Complementary observables is related to the concept of mutually unbiased bases. So the bases and are unbiased if all the inner products have the same magnitude. If these bases are associated to observables (say, they form a basis of eigenvectors), then knowing a classical value of one observable gives no information about the other – all eigenstates are equally likely. For a visual image, think of two sets of bases for the plane, rotated 45 degrees relative to each other. Each basis vector in one has a projection of equal length onto both basis vectors of the other.

Thinking of the orthonormal bases as “observable structures”, the mutually unbiased ones correspond to “complementary” observables: a state which is classical for one observable (i.e. is an eigenstate for that operator) is unbiased (i.e. has equal probablities of having any value) for the other observable. Labelling the different structures with colours (red and green, usually), they could diagrammatically represent states being classical or unbiased in particular systems.

This is where “phase groups” come into play. The setup is that we’re given some system – the toy model they often referred to was a spinning particle in 3D – and an observable system (say, just containing the observable “spin in the X direction”). Then there’s a group of symmetries of the system which leave that observable untouched (in that example, the symmetries are rotation about the X axis). This is the “phase group” for that observable.

Bill Edwards talked about phase groups and how they can be used to classify systems. He gave an example of a couple of toy models with six states each. One was based on spin (the six states describe spins about each axis in 3-space in each direction). The other, due to Robert Spekkens, is a “hidden variable” theory, where there are four possible “ontic” states (the “hidden” variable), but the six “epistemic” states only register whether the state lies in of six possible PAIRS of ontic states. The two toy models resemble each other at the level of states, but the phase groups are different: the truly “quantum” one has a cyclic group (for the X-spin observable, it’s generated by a right-angled rotation about the X axis); the “hidden variable” model, which has some quantum-mechanics-like features, but not all, has phase group . The suggestion of the talk was that this phase group distinguishes “local” from “nonlocal” systems (i.e. ones with hidden variable models and ones without).

Marni Sheppard also gave a talk about Mutually Unbiased Bases, p-adic arithmetic, and algebraic geometry over finite fields, which I find hard to summarize because I don’t understand all those fields very well. Roughly, her talk made a link between quantum mechanics and an axiomatic version of projective geometry (Hilbert spaces in QM ought to be projective, after all, so this makes sense).  There was also a connection between mutually unbiased bases and finite fields, but again, this sort of escaped me.

Also in this group was Jamie Vicary, whom I’ve been working with on a project about the categorified harmonic oscillator.  His talk, however, was about n-Hilbert spaces, and n-categorical extended TQFT.  The basic point is that a TQFT assigns a number to a closed n-manifold, and a Hilbert space to each (n-1)-manifold (such as a boundary between two parts of a closed one), and if the TQFT is fully local (i.e. can be derived from, say, a triangulation), this can be continued to have it assign k-Hilbert spaces to (n-k)-manifolds for all k up to n.  He described the structure of 2-Hilbert spaces, and also monoidal ones (as many interesting cases are), and how they can all be realized (in finite dimensions, at least) as categories of representations of supergroupoids.  Part of the point of this talk was to suggest how not just dagger-compact categories, but general n-categories should be useful for quantum theory.

Toposes

The monoidal category setting is popular for dealing with quantum theories, since it abstracts some properties of Hilbert spaces, which they’re usually modelled in.  Topos theory is usually thought of as a generalization of the category of sets, and in particular they model intuitionistic classical, not quantum, logic.  So the talk by Andreas Döring (based on work with Christopher Isham – see many of Andreas’ recent papers) called “Why Topos Theory in the Foundations of Physics?” is surprising if you haven’t heard this idea before.  One motivation could be described in terms of the Kochen-Specker theorem, which, roughly, says that a quantum theory – involving observables which are operators on a Hilbert space of dimension at least three – can’t be modeled by a “state space”.  That is, it’s not the case that you can simultaneously give definite values to all the observables in a consistent way – in ANY state!  (That is, it’s not just the generic state: there is no state at all which corresponds to the classical picture of a “point” in some space parametrized by the observables.)

Now, part of the point is that there’s no “state space” in the category of sets – but maybe there is in some other topos!  And sure enough, the equivalent of a state space turns out to be a thing they call the “spectral presheaf” for the theory.  It’s an object in some topos.  The KS theorem becomes a statement that it has no “global points”.  To see what this means, you have to know what the spectral presheaf is.

This is based on the assumption that one has a (noncommutative) von Neumann algebra of operators on a Hilbert space – among them, the observables we might be interested in.  The structure of this algebra is supposed to describe some system.  Now you might want to look for subalgebras of it which are abelian.  Why?  Because a system of commuting operators, should they be observables, are ones which we CAN assign values to simultaneously – there’s no issue of which order we do measurements in.  Call this a “context” – a choice of subalgebra making the system look classical.  So maybe we can describe a “state space” in a context: so what?

Well, the collection of all such contexts forms a poset – in fact, lattice – in fact, a complete Heyting algebra.  These objects are just the same (object-wise) as “locales” (a generalization from topological spaces, and their lattice of open sets).  The topos in question is the category of presheaves on this locale, which is to say, of contravariant functors to Set.  Which is to say… a way of assigning a set (the “state space” I mentioned), with a way of restricting sets along inclusion maps.  This restriction can be a bit rough (in fact, the fact that restriction can be quite approximate is just where uncertainty principles and the like come from).  The main point is that this “spectral presheaf” (the assignment of local state spaces to each context) supports a concept of logic, for reasoning about the system it describes.  It’s a lot like the logic of sets, but operations happen “context-by-context”.  A proposition has a truth value which is a “downset” in the lattice of contexts – the collection of contexts where the proposition is true.  A proposition just amounts to a subobject of the spectral presheaf by what they call “daseinization” – it’s the equivalent of a proposition being a subset of a configuration space (where the statement is true).

One could say a lot more, but this is a blog post, after all.

There are philosophical issues that this subject seems to provoke – the sign of an interesting theory is that it gets people arguing, I suppose.  One is the characterization of this as a “neo-realist interpretation” of quantum theory.  A “naive realist” interpretation would be one that says a “state” is just a way of saying what all the values of all the observable quantities is – to put it another way, of giving definite truth values to all definite “yes/no” questions.  This is just what the KS theorem says can’t happen.  The spectral presheaf is supposedly “neo-realist” because it does almost these things, but in an exotic topos (of presheaves on the locale of all classical contexts).  As you might expect, this is a bit of a head-scratcher.

 

My Talk at “Categories, Quanta, Concepts”

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.

 

Conference: Connections in Geometry and Physics

As promised in the previous post, here is a little writeup of the second conference I was at recently…

Connections in Geometry and Physics

The conference at PI was an interestingly varied cross-section of talks, with a good many of them about geometry which, to be honest, is a little over my head.  Ostensibly about “connections”, the talks actually ranged quite widely, which was interesting, and reminded me I have a lot af geometry to catch up on.  A lot of talks had to do with structures at various places along the heirarchy: (1) symplectic manifolds, (2) Kähler manifolds, and (3) Calabi-Yau manifolds.  These last are interesting to string theorists and others, in part because they satisfy a form of Einstein’s equations, while also carrying a bunch of extra structure.

Now, at least I know what all the above things are: Symplectic manifolds have the “symplectic form” , a non-degenerate exact 2-form (a canonical example being in the cotangent space to , which happens to be the configuration space for a particle moving in – symplectic forms often show up on configuration spaces).  A Kähler manifold is symplectic, but also has a complex structure (i.e. a way to multiply tangent vectors by ), which preserves the symplectic form, and a metric, which gets along with both of the above.  If the metric satisfies Einstein’s equations and is flat (this really amounts to the connection to “connections”, since this is the same as there being some flat connections, namely the Levi-Civita connection), then is a Calabi-Yau manifold.

Anyway, this sets up the kind of geometry a lot of people were talking about, and while I didn’t exactly have the background to follow everything, I got a sense of what kinds of questions people are interested in, which was good.  A lot of questions have to do with Lagrangian submanifolds of any of the above (from symplectic through Calabi-Yau).  These are submanifolds where the symplectic form gives zero when applied to any tangent, and which have the highest possible dimension consistent with this property (namely , if the original thing is -dimensional).  Another theme which came up several times – for example, in the talk by Denis Auroux – has to do with “mirror symmetry” for Kähler manifolds (and Calabi-Yaus), which has to do with finding a “mirror” for the manifold , called where the complex geometry on the mirror corresponds to the symplectic geometry on , and vice versa.

There were some talks in the direction of physics.  One of the most obviously physical was Niky Kamran’s, talking about a project he’s worked on with F. Finster, J. Smoller, and S-T. Yau, about long-time dynamics of particles satisfying the Dirac equation, living on a background geometry described by the Kerr metric – which describes a rotating black hole.  Since I worked with Niky on a related project for my M.Sc (my thesis was basically a summary putting together a bunch of results by these same four people), I followed this talk better than many of the others.

Working on this project, I got a strong sense of how important symmetry is in studying a lot of real-world problems.  One of the essential facts about the Kerr metric is that it’s very symmetric: it’s stable in time, and rotationally symmetric.  Actually, all the black-hole solutions to Einstein’s equations are quite symmetric – there is only a small family of solutions, parametrized by mass and angular momentum (and electrical charge).  The symmetry makes differential equations written in terms of this metric much nicer – you can split things into the radial and angular parts, for example – and in particular, the wave equations Niky was talking about are integrable just because of this symmetry, so it’s possible to get exact analytic results.  (Other approaches to this kind of problem get results only numerically and approximately, but can deal with much more general backgrounds.)  The starting point (which basically is what my thesis summarizes) is to show that there are no “bound states” for the Dirac equation.  Fermions (which is what it describes) are most familiar to us in bound states: in shells orbiting the nucleus of an atom.  But if the attractive force pulling on them is gravity, rather than electical charge, this situation isn’t stable.  The work Niky was talking about deals with what happens instead: what are the long-term dynamics of a fermion near a rotating black hole?

They use spectral methods – basically, Fourier analysis – to find out.  The Dirac equation is a wave equation (for a spinor field), and you can look at the different frequencies, and get an estimate of how fast they decay.  (Since there aren’t stable orbits, the strength of the spinor field has to decay over time.)  In fact, they get a sharp estimate of the order (namely ).  Basically, one should imagine that the wave is a superposition of “ripples” – some radiating outward from the event horizon, and some converging toward it.  Put in terms of a particle – an electron, say, or a neutrino – this says it will either fall into the black hole, or (if it has enough energy) escape off to infinity.

There were some other physics-ish talks, such as that by James Sparks, on the geometry of the “AdS/CFT” correspondence.  This correspondence has to do with two kinds of quantum field theory.  The “AdS” stands for “Anti de Sitter”, which is a sort of geometric structure for a manifold which resembles a hyperboloid – actually, all the unit vectors in where the metric has signature (4,2): that is, the metric is something like .  This hyperboloid is 5-dimensional, and has a metric with one timelike dimension.  Plain old “de Sitter” space is a similar thing, but using a metric with signature (5,1).  It’s possible to define some field theory on AdS space, called supersymmetric supergravity.  This theory turns out to have exactly the same algebra of observables as a different theory, “CFT” or conformal field theory, on the (conformal) boundary of Anti de Sitter space.  Sparks told us about a geometric interpretation of this.

Then there was Sergei Gukov, with a talk called “Brane Quantization”, based on this work with Ed Witten.  He was a little reticent to actually describe how this “brane quantization” actually works, preferring to refer us to that paper, but gave us a very nice, and relatively comprehensible overview of different approaches to quantizing a symplectic manifold.  (As I said, they tend to show up as configuration spaces in classical physics. A basic problem of quantization is how to turn the algebra of functions on a symplectic manifold into an algebra of operators on a Hilbert space .)  In particular, he contrasted their method with geometric quantization (which needs to make some arbitrary choices, then takes to be a space of sections of some line bundle on with a connection whose curvature is ), and with deformation quantization (which needs no special choices, but only constructs an algebra of operators by algebraic deformation, and not actually itself, which some people, but not Sergei Gukov, find satisfactory).  The basic idea of Brane quantization seems to be that gets complexified (somehow – it might be either impossible, or non-unique), and then studying something called an A-model of the result.  This is apparently related to, for example, Gromov-Witten theory, which I’ve written about here recently.

Finally, I’ll mention a few other talks which stood out as rather different from the rest.  Veronique Godin talked about “Relative String Topology” – string topology being a way of studying space by looking at embeddings of the circle (or of paths) into it – that is, its loop space (or path space).  Usually, invariants that come from path spaces only detect the homotopy type of the original spaces – in particular, they’re not helpful as knot invariants.  Godin talked about a clever way to detect more structure by means of an -coalgebra structure on the cohomology groups of the path space.  The “relative” part means one’s looking at a manifold with embedded submanifold (for example, is a knot in ), and considering only paths starting and ending on .  (This is how one can get a coalgebra structure – turning one path into two paths if it crosses through again is a comultiplication – this extends to chains in the cohomology).

Chris Brav gave a talk about how braid groups act on derived categories, which I didn’t entirely follow, but subsequently he did explain to me in a pretty comprehensible way what people are trying to accomplish when they look at derived categories.  At some point I’ll have to think about this more carefully and maybe post about it.  But roughly, it’s the same sort of “nice categorical properties” I mentioned in the previous post, about smooth spaces.  Looking at derived categories of sheaves on a space, makes the objects seem more complicated, but it also makes them behave better with respect to taking things like limits and colimits.

Benjamin Young prefaced his talk, “Combinatorics Inspired by Donaldson-Thomas Theory” by pointing out that he’s a combinatorialist, not a geometer.  But Donaldson-Thomas invariants are apparently a kind of “signed count” of some geometric structures (as are a lot of invariants – the same kind of “weighted count” invariants appear in Gromov-Witten and Dijkgraaf-Witten theory, just for instance).  So he described some geometry relating to “brane tilings” – basically, embedding certain kinds of graphis in a torus – and how they give rise to structures that correspond to certain kinds of Young diagrams (”not the same Young”, he added, perhaps unnecessarily, but it got a chuckle anyway).  So the counts can be turned into a combinatorial problem of counting those Young diagrams with the appropriate sign, which can be done using a generating function.

So in any case, this conference had a whole range of talks, from several different fields.  While I found myself lost in a number of talks, I was also quite fascinating with how wide a range of topics were embraced under its umbrella – “connections” indeed!  So in the end this was one of those conferences which opened my eyes to a wider view of the field, which was certainly a good reason to go!

 

Conference: Smooth Structures in Ottawa

I’ve been to two conferences in the past two weeks, and seen a lot of interesting talks. A couple of weekends ago, I was in Ottawa at the Fields Institute workshop on “Smooth Structures in Logic, Category Theory, and Physics“. There were quite a few interesting talks, on a fairly wide range of points of view, and I had some interesting conversations as well. A good workshop overall. Another report on it by Alex Hoffnung is here on the n-Category Cafe. Then this past weekend, I attended the conference “Connections in Geometry and Physics“, at the Perimeter Institute in Waterloo (also jointly sponsored by the U of Waterloo math department, and the Fields Institute), where I gave a version of my Extended TQFT talk (if you’ve seen a previous version, this is similar, but references a few more recent variations, but is mainly distinguished as the first time I caved in and decided to use Beamer – frankly I find the graphical wingdings distracting and unhelpful, but it made sense given the facilities in the venue).

One personally interesting thing was that one of the talks in Ottawa was given by John Baez, who was my advisor for my Ph.D at UCR, and one of the talks at PI was given by Niky Kamran, who was my advisor for my M.Sc. at McGill, so I got to touch base with both of them in the space of a week. It also reminded me that I’ve worked on a fairly eclectic sampling of things, since John was talking about cartesian closed categories of smooth spaces, and Niky was talking about the long-time dynamics of the Dirac equation in the neighborhood of black holes.

In the near future I’ll make a post on the Connections conference but the following was getting long enough already…

Smooth Structures

So, as the title suggests, the Fields workshop addressed the topic of “smoothness” from several points of view, with the three mentioned being only the most obvious. To begin with, “smooth” carries the connotation of “infinitely differentiable”. So for example the space of smooth functions on the real line has the property that, if is any function in it, you can take the derivative , and you get another smooth function, hence can take the derivative again, and so on. So one way to characterize the space is that it has a differential operator (which satisfies some algebraic properties like the product rule etc.), and is closed under .

One theme explored at the workshop has to do with finding a nice general notion of “smooth space”. The smooth structure on that makes this possible is the model for smooth structures on other spaces. The most familiar way goes: (1) first extend the concept to via partial derivatives, so we know what a smooth map is (or between open subsets of these), and (2) define a “smooth manifold” as a (topological) space equipped with “charts” for an open subset of some , satisfying a bunch of conditions. Then we can tell smooth functions on by pulling them back to and using the familiar concept there. We can tell smooth maps by composing with charts and their inverses and seeing that the resulting map is smooth. This concept is great, and underlies, just to name a couple of examples, General Relativity and gauge theory, which are the basis of 20th century physics. It’s rather brittle, though, because simple operations like taking function spaces , or subspaces , or quotients , give objects which are not manifolds.

John Baez talked about some of these issues in categorical language. Those three operations illustrate the general categorical constructions of exponentials, equalizers and coequalizers (or, generally, limits and colimits). The category of differentiable manifolds has important objects, but since it lacks these constructions in general, it’s not a nice category. The idea is to find a nice category – a cartesian closed one – which contains it. John described roughly how some approaches to this problem work – there were more detailed talks by Alex Hoffnung about the Diffeological spaces of Souriau, and by Andrew Stacey summarizing how the different categories are related and making a case for Frölicher spaces – but mostly focused on examples where these categories would be useful.

One interesting case deals with orbifolds (John suggested checking out this paper of Eugene Lerman, “Orbifolds as Stacks”), which were also the subject of Dorette Pronk’s talk. Dorette described some orbifolds – sometimes they arise by taking quotients of manifolds under the action of finite groups, and sometimes they only look locally as if they did. She also talked about the right way to think about maps between orbifolds, which is basically in terms of spans. That is, the right kind of “map” between orbifolds and consists of (certain) maps into each of them from some common orbifold .

Under “logic” (and overlapping with “categories”), the first two days started off with talks by Anders Kock about Kähler differentials and synthetic differential geometry (regarding which see e.g. Mike Schulman’s intro, or the book by Anders Kock himself). SDG is a generalization of differential geometry to be internal to some topos – in particular, getting rid of the assumption that the geometry is based on some space which has an underlying set of points (which is special to the topos ), and doing everything in the internal language of the topos. Kock introduced some work with Eduardo Dubuc – describing a “Fermat Theory” (an abstract characterization of a ring with a concept of partial derivatives) and showed how it fits into SDG.

Some other talks in the “logic” world included those of Rick Blute, Robin Cockett, and Thomas Erhard, about linear logic and differential categories. The basic idea is that a category can be associated to any logic, by taking formulas as objects, and (equivalence classes under rewriting rules) of proofs as morphisms. Linear logic, which is topical to quantum computation, is interesting from this point of view because it has some standard logical facts (like the deMorgan laws, which the intuitionistic logic of toposes do not entirely have) and also good categorical properties (one has a nice monoidal category). It’s a little strange if you’re used to thinking of classical logic: “propositions” are replaced by “resources” whose truth can be consumed (think of a quantum computer with a bit stored somewhere – you can move the bit, but not read and copy it); there are both “additive” and “multiplicative” versions of connectives which in classical logic go by the names “AND” and “OR” (correspondingly, there are additive and multiplicative versions of “TRUE” and “FALSE”). The relation to smoothness is that a new variation on this adds an operator to the category which behaves like differentiation. This is formally very interesting, though I haven’t really grokked what it’s good for, but apparently the derivative acts like a quantifier.  Really!?

Among other talks, Konrad Waldorf spoke about parallel transport for extended objects – basically, this is a roundabout way of studying nonabelian gerbes (a kind of categorification of bundles), not by looking at the gerbes themselves but by directly looking at parallel transport for connections on them. Kristine Bauer gave two talks about “Functor Calculus” (in particular the Goodwillie calculus), which has to do with constructing something like a “Taylor series” for a functor into topological spaces, approximating the functor by “polynomials”, and which shows up in homotopy theory.  I also have the sense that the Goodwillie calculus generalizes to topological spaces a lot of what Joyal’s species (related to “analytic” functors in a similar sense of being representable by “polynomials”), but I don’t understand this well enough at the moment to say just how.

 

Moduli Spaces of Connections – Two Recent Talks

In the last couple of weeks of the winter term, there were two series of talks here at UWO, by different speakers, from very different points of view, which bear on the subject of moduli spaces of connections.

There seem to be several schools of thought approaching the subject of moduli spaces, and in particular how to handle the reduction by symmetries without losing too much – three approaches I know of are the symplectic point of view (thinking of the moduli space as a symplectic space, or perhaps orbifold, and reduction by taking whole “leaves” to points), the algebraic-geometric (describing them using Deligne-Mumford stacks), and the groupoid point of view (which is the one I’m most familiar with). I suppose, in light of my previous note, that there must be a noncommutative-geometry view of the subject, though if anyone is using NCG to look at these moduli spaces in particular I don’t know who. Before talking about reducing the moduli spaces, there’s already a lot to say about them which people have studied in some detail.

The first speaker here who touched on this was Fred Cohen, who gave a series of three talks about special subspaces of products (and talked a lot about about stable homotopy theory). The second was Eduardo Gonzalez, who gave a seminar and a colloquium talk on equivariant Gromov-Witten theory. I’ll try to briefly give an overview of what they each had to say, mainly focusing on this common element.

Part 1 – Talks by Fred Cohen

Fred Cohen was speaking about various subspaces of products. He was summarizing a number of different projects, including for example this (on loop spaces of configuration spaces) and this (about spaces of homomorphisms). The first talk dealt with the seemingly simple space of distinct n-tuples of points in a space , and the related natural space (the action of the symmetric group makes the points unlabeled). In the case , a point in is a list of n distinct points. So a loop in this space is a motion of the n points which returns them to their original locations – considered up to homotopy, this is just a braid. In fact, , the n-strand pure braid group; and , the full braid group (points needn’t end up in their original positions). In fact, the configuration spaces are spaces – that is, they are classifying spaces of these groups, and have no higher homotopy groups above .

Replacing here with , a surface, the same sort of thing defines the n-strand “surface braid group” for , which is . We heard how this decomposes in terms of the “Borromean” braid group – the subgroup of braids which become disconnected when you remove one strand (this is the kernel of a map induced by the projections into ).

There was more about the homotopy type of these spaces, and a second talk covered “moment-angle complexes”, but here I’m interested in Cohen’s third talk about subspaces of products. This was on “representations” of a discrete group, which in this context means – almost – homomorphisms into a chosen group . (If , these are the more famous linear representations.) This is related to subspaces of the product , which arise from looking at the moduli space , where is a discrete group and a topological group.

In particular, if , for a space , such a representation can be thought of as a -connection. In this picture, a connection is just a way of assigning an element of the gauge group to each path in .) Actually, I mentioned this is “almost” the space of representations, which is actually – the moduli space of flat connections modulo gauge transformations. A gauge transformation (assuming is connected) acts by conjugation: , for a class of loops in .

This is the usual way of looking at this moduli space of geometric structures – I’ve mentioned here the alternative view that a flat connection is a functor , and a gauge transformation is a natural transformation. Then the moduli space becomes a moduli stack, which as mentioned above I tend to think of as a groupoid. But the moduli spaces of homomorphisms (the objects) and representations (isomorphism classes of objects) carry a lot of information. Particular cases which Cohen discussed were , the free group on generators, and , the free abelian group on generators. These are fundamental groups of, respectively, the -punctured plane and the genus- torus. Now , and the map induces an inclusion – in fact it’s a subvariety – so this is a subset of a product, and techniques for dealing with these were Cohen’s real subject.

One that he discussed (described in the paper linked above by Adem, Cohen and Torres-Giese) uses the “descending central series” of . This is a sequence of subgroups generated by the -fold commutators . In particular, one looks at the groups , and in fact their spaces of homomorphisms:

So there’s a filtration of spaces associated to and .

Now it’s pretty standard that there are maps (by dropping the generator), and (sending the extra generator to the identity). These, thought of as face and degeneracy maps, turn the collection of spaces (for all ) into a simplicial space. This has a geometric realization, which is the classifying space (or, shifting which set is considered to be the -simplices, , where , and there’s a bundle ). BUT, each of the has the filtration above – so it turns out there’s a filtration of simplicial spaces, and in fact of bundles. The paper above uses this to find the cohomology, fundamental group, and so on of the spaces I just mentioned – including the moduli space of connections.

(Then Cohen talked about a generalization of this to arbitrary “transitively commutative” groups, but that takes us away from the geometry I started off talking about).

Part 2 – Talks by Eduardo Gonzalez

The second set of talks which touched on moduli spaces of connections was by Eduardo Gonzalez, related to stuff in this paper by Gonzalez and Chris Woodward speaking about gauged (or equivariant) Gromov-Witten invariants. These are discussed in this paper by Givental, and Gonzalez referenced several other people who’ve worked on related things, including Chen and Ruan (see this on GW theory for orbifolds), and Abramovich, Graber and Vistoli (see this, on GW theory for stacks). Strictly speaking, this doesn’t address just the moduli space of flat connections, but actually a more complex moduli space for a theory involving a choice of connection (on a bundle), and also a section of the bundle. It is called the moduli space of symplectic vortices, and is very much involved with symplectic geometry as you might expect.

The usual Gromov-Witten invariants, roughly, count the number of holomorphic curves on a -dimensional symplectic manifold . (That is, has an exact symplectic form – i.e. and is nondegenerate – and there’s an almost-complex structure - that is ; these give a metric ). This determines a complex derivative in a natural way.

A curve is a map , where is a Riemann surface (i.e. complex curve), which is holomorphic if . The moduli space of these holomorphic curves – which is also the space of sections of suitable bundles over – each one amounts to a choice of a particular bundle over , and a connection and holomorphic section of the bundle. This is where the Gromov-Witten invariants come from. Actually, it comes from a compactification of the space of maps from with “marked” (distinguished) points (so here actually we start to circle back around to the configuration space Fred Cohen talked about).

Given a cohomology class (that is, 2-cocycles), one gets a form which can be integrated over . The Gromov-Witten invariant, for that choice of form, is just the total “volume” of the moduli space with respect to that form, (the form is pulled back under the map evaluating it at the marked points). This is sometimes described (rather roughly) as “counting” the pseudoholomorphic maps.

One thing people seem to be quite interested in is how this is related to so-called “quantum cohomology” for the space . Since the GW invariants take some forms and give numbers, the idea is that they can be used to define a “three point function” on cohomology classes (by taking all but three of the cocycles to be the fixed ), which in turn can be taken to be the structure coefficients for a deformation of the cup product for cohomology. (Take the cohomology ring, take its tensor product with a ring of power series, and write the new product as a power series whose first terms give the usual cup product).

However, what Gonzalez was talking about was “gauged” Gromov-Witten invariants, where spaces are replaced by stack – in particular, stacks that come from an action of a group on the space (which, since is a symplectic manifold, should preserve the form ). The symplectic geometry way to talk about this is one I’m not very familiar with, but Gonzalez referred to as the “categorical quotient” (i.e. the transformation groupoid, in the language I’m more used to) or the “symplectic reduction” (here’s a brief note on the subject, and here a long paper on the relevance to physics which I’m linking so I can find it later). Roughly, this is a two-step process, the second stage being a reduction to a quotient by a group action. The result, in general, will be a symplectic orbifold (if the action is free on orbits, it’ll be a manifold – otherwise, some orbits have extra symmetry, which give the special points of the orbifold).

In particular – and here we really get to the point of contact with the groupoid picture I’m more familiar with, the gauged GW invariants are associated to a space , where is a space of connections on some bundle , and is the group of gauge transformations. Now, these aren’t the space of flat connections, which I’ve thought more about, but rather connections satisfying another equation, namely that the curvature plus a certain volume form should be zero (defining the volume form takes a while and I don’t get it in enough detail to try to sort it out here). Connections satisfying this equation are called vortices, for reasons which escape me.

But in any case, the invariants amount to some geometry-aware generalization of the groupoid cardinality of this orbifold, thought of as an (equivalence class of) groupoid(s), defined by the integral above. There is much more to say here, but it’s taken me long enough to write this up as is, so maybe I’ll return to those things in a separate post some time.

 

A Note on Quantum Groupoids

I’ve been looking over the last little bit at quantum groupoids, and how they can be used to deform the 2-linearization 2-functor (or into ) which I’ve discussed in here.

First a little motivation: that functor was part of the way I constructed extended TQFT’s. The inclusion realized cobordisms (with corners) in terms of spans of manifolds. Looking at fundamental groupoids using the 2-functor allows us to think about these in terms of the bicategory , and then applying gave 2-Hilbert spaces and 2-linear maps for manifolds and cobordisms (and then natural transformations for cobordisms with corners). Since I made the claim that, with gauge group – and a suitably infinitary version of , the extended TQFT gives a theory equivalent to the Ponzano-Regge model of quantum gravity, a reasonable question is: what about the Turaev-Viro model? The PR model is based on labelling edges of a triangulation with representations of , and the TV model, with representations of .

Now, the groupoids that show up in the above – groupoids of -connections on a manifold, modulo gauge transformations – are quite closely related to this. In particular, the groupoid of connections for a circle (the basic 1-dimensional manifold that the 3-dimensional theory builds from) is , the transformation groupoid produced from the action of on itself by conjugation. (That is: the objects are elements of , and the morphisms are all the conjugacy relations.) Applying gives the representation category of this, namely , so in particular, at the identity of , one has as a sub-2-vector space. (The “states” in the 2-Hilbert space for the circle in the ETQFT are labelled by “masses and spins” – the mass=0 case is what gives the representations of , and for nonzero mass, one has .)

More broadly: one can describe the state space of a gauge theory – or many other kinds of theory, in terms of transformation groupoids given by symmetries (gauge transformations, say) acting on states (connections, in that case). Is there a way of doing the same for systems whose symmetries are described by quantum groups? If so, then instead of getting 2-vector spaces which are representation categories of groupoids, we should get some which are representation categories of quantum groupoids.

This paper by Ping Xu describes quantum groupoids – or rather, quantum universal enveloping algebras. They’re described here as a “unification of quantum groups and star products” (star products being the partially-defined composition found in groupoids). This paper by Nikshych and Vainerman describes finite quantum groupoids and some applications – in particular, quantum transformation groupoids, which is the immediately relevant application.

First off, quantum groups: these are Hopf algebras, which in particular are bialgebras – they have both a product

and “coproduct”

.

This is because the point here is that we’re following the pattern in which spaces are replaced by algebras: in some simple examples, these are the algebras of functions on a space. The point of noncommutative geometry is that there’s a (contravariant) equivalence between the category of locally compact Hausdorff spaces and the category of commutative algebras, so generalizing to noncommutative algebras (and taking the opposite category) gives a generalization of “locally compact Hausdorff space”. Topological groups like Lie groups are group objects in this category of spaces – and quantum groups are group objects in . So in particular, the group operation shows up as the coproduct , and the inverse operation is the antipode

.

Of course there are also the unit

and co-unit

(where is the base field, say ). The co-unit is of course the “unit” map for the group object. These maps all satisfy some obvious relations.

Now what about quantum groupoids? These are “groupoid objects” – or rather, models of the theory of groupoids – in . We can’t quite say “groupoid objects”, since a groupoid internal to a category consists of two objects in . For example, a Lie groupoid is a groupoid in , the category of manifolds. It has a base manifold and a total manifold , and two maps , and so forth. The interpretation is that there is a set (or manifold, or what-have-you) of objects, and a set (etc.) of morphisms. There is a (partially-defined) composition operation allowing morphisms to be composed if the source of one is the target of the other, and so forth.

So (a slightly tweaked version of) the definition of a quantum groupoid given by Xu has it consisting of . These unpack in pretty natural ways: it helps to compare to both the definition of, say, a Lie groupoid, and a quantum group. is the “total algebra$ and the “base algebra”, and they correspond to the “noncommutative spaces” of morphisms and objects of a groupoid, respectively. Just as a group can be seen as a groupoid with just one object, a quantum group would be a quantum groupoid where the base algebra is just the base field .

But then, if is not , we need some nontrivial – the source and target maps respectively, which replace the unit map to . Notice they go from the base to the total algebra , not the other way around, because everything works as usual in . The other maps are likewise dual to those in the definition of a groupoid. The major difference is that we need the equivalent of a partially defined multiplication/composition and the dual “co-multiplication”/”co-composition” . This works because using and , we get left and right actions of the base on , which is thus an -bimodule, hence we can form the bimodule product , and thus:

and

The obvious analog of the unit we had for quantum groups is hidden in Xu’s definition (it seems like it should take the place of the requirement that be unital), but the co-unit

is the dual way of describing the “identity” function .

The antipode plays the role of the inverse map for morphisms in groupoids.

All these maps have to satisfy various identities which are implied by saying this is a model of the theory of groupoids – check out either of the above papers to see them all explicitly.

(A final observation about the definition: a groupoid is a category which has an inverse map from morphisms to morphisms. If we relax the assumption that we have an antipode , we end up with just the definition of a bialgebroid (having makes it a “Hopf” algebroid). So “bialgebroid” would seem to be the natural “quantum” version of the concept of a general category…)

So how might one construct such a “quantum action groupoid”? This is addressed (at least in the finite case) in the paper by Nikshych and Vainerman, in their section 2.6. This is generalizing the action groupoid arising from a group acting on a set. The set is replaced by an algebra (which must be separable, for them – the equivalent of a finite set – and thought of as a “quantum space”). The group is replaced by a quantum group (or, generally, Hopf algebra) . The equivalent of having action of the group on the set is that is a (right) -module.

Now, the action groupoid for a action on has for objects the elements of , and for morphisms, all relations , which we can write as morphisms , with source and target . The action quantum groupoid associated to the -module is the double crossed product , with multiplication, co-multiplication, etc. defined in fairly natural ways. (Note: those triangles should be semidirect products, but I can’t seem to make that symbol appear here.)

So finally, I seem to be claiming that a such a quantum groupoid, let’s call it is the right “classical” state space (if that’s not too blatant a contradiction in terminology) for a theory having quantum-group symmetry – at least in the categorified picture. No doubt in many cases there is additional structure, capturing the equivalent of, say, symplectic structure, that should also be included (such things certainly can be found in NCG, but I’m still absorbing how exactly).

Then the 2-vector space for the quantized version of such a theory is the category , and a “2-state” just an object in here – a representation of .

One thing that’s not quite clear to me just now is how this relates to the usual idea of “state” in NCG – a state for a “quantum space” (which is an algebra) being a linear functional on that algebra. Not necessarily a character (i.e. a homomorphism into ), mind you – that would be a 1-dimensional representation, but just a functional.

 

Recent Talks – Wade Cherrington’s Ph.D. Defense

Last week was Wade Cherrington’s Ph.D. defense – he is (or, rather, WAS) a student of Dan Christensen. The title was “Dual Computational Methods for Lattice Gauge Theory”. The point of which is to describe some methods for doing numerical computations of various physical systems governed by gauge theories. This would include electromagnetism, Yang-Mills theory (which covers the Standard Model and other quantum field theories), as well as gravity. In any gauge theory, the fundamental objects being studied are fields described by -connections, for some (Lie) group . To some degree of approximation, a connection gives a group element for any path in space: . Then the dynamics of these fields are described by a Lagrangian, where the action for a field is the integral of the curvature over the whole space : (plus possibly some other terms to couple the field to sources).

Now, the point here is to get non-perturbative ways to study these theories: rather than, say, getting differential equations for the fields and finding solutions by expanding a power series. The approach in question is to take a discrete version of this continuum theory, which is finite and can be dealt with exactly, and then take a limit.

So in lattice gauge theory, continuous space is replaced by a – well, a lattice , say , for definiteness (then eventually take the continuum limit as the spacing of the lattice goes to zero). The lattice also include edges joining adjacent points – say the set of edges is . Paths in the lattice are built from these edges. (Furthermore, since an infinite lattice can’t be represented in the computer, the actual computations use a quotient of this – a lattice in a 3-torus, or equivalently, one considers only periodic fields.) Then it’s enough to say that a connection assigns a group element to each edge of the lattice, .

Of course, to back up, describing connections as functions , or , often provokes various objections from people used to differential geometry. One is that the group elements assigned don’t have any direct physical meaning – since a physical state is only defined up to gauge equivalence. So if an edge joins lattice points and , a gauge transformation acts on to give with . Clearly, for any given edge, there are gauge-equivalent connections assigning any group element you want. As Wade pointed out, one benefit of the dual models he was describing is that their states can be given a definite physical meaning – there are no gauge choices. Another, helpfully, is that they’re easier to calculate with (sometimes). A more physical motivation Wade suggested is that these methods can deal with spin-foam models of quantum gravity, and also matter fields: a realistic look at a theory of gravity should have some matter to gravitate, so this gives a way to simulate them together.

So what are these dual methods? This is described in some detail in this paper by Wade, Dan, and Igor Khavkine. The first step is to find a discrete version of curvature: instead of the action , we want a sum of face amplitudes. Curvature is described by the holonomy around a contractible loop, so the basic element is a face in the lattice (say is the set of faces). Given a square face with edges whose holonomies are through (assuming all faces are oriented in a consistent direction), the holonomy around the face is . From this, one defines an amplitude for the face, for some function (there are various possibilities – Wade’s example used the heat kernel action mentioned in the paper above), and then the total action is the sum over all faces. Then instead of integrating over an infinite-dimensional, and generally intractable, space of smooth connections, one itegrates over , the space of discrete connections.

The duality here is the expansion of this in terms of group characters: a function can be written as a combination of irreducible characters: . Then one can pull this sum over characters outside the integral over (so that local quantities are inside).

There are many nice images on Wade’s homepage (above) showing visualizations of the resulting calculations – one finds sums over certain labellings of the lattice, namely those which can be described by having certain surfaces. In particular, closed (boundaryless), branched (possibly self-intersecting) surfaces with face and edge labels given by representations of and intertwining operators between them… that is, spin foams. These dual spin foam configurations have the advantage of having a physical interpretation (though I confess I don’t have a good intuition about it) which doesn’t depend on gauge choices.

A variant on this comes about when the action is changed to include a term coupling the Yang-Mills field to fermions (one thinks of quarks and gluons, for example). In this case, the fermion part is described by “polymers” (closed, possibly self-intersecting paths, rather than surfaces), and the coupled system allows the surfaces used in the YM calculations to have boundaries – but only on these polymers. (Again, Wade has some nice images of this on his site. Personally, I find a lot of the details here remain obscure, though I’ve seen a few versions of this talk and related ones, but the pictures give a framework to hang the rest of it on.)

Wade identified two “key” ingredients for doing calculations with these dual spin foams:

1. Recoupling moves for the graphs (as described, for instance, by Carter, Flath, and Saito) which simplify the calculation of amplitudes, and
2. A set of local moves (changes of configuration) which are ergodic – that is, between them they can take any configuration to any other. (The point here is to allow a reasonably random sampling – the algorithm is stochastic – of the configuration space, while making only local changes, requiring a minimum of recomputation, at each step.)

Finally, Wade summed up by pointing out that the results obtained so far agree with the usual methods, and in some cases are faster. Then he told us about some future projects. Some involve optimizing code and adapting it to run on clusters. Others were more theoretical matters: doing for what has been done for (which will involve developing much of the recoupling theory for 3j- and 6j-symbols); finding and computing observables for these configurations (such as Wilson loops); and modelling supersymmetry and other notions about particle physics.

 

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.)