Theoretical Atlas

Writing sizeable chunks of math blog takes longer than I expected. Here are a few non-intensive things that occurred to me.

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While I was walking home from the UWO campus, I was reminded of the nature of Canada in late November: everything, from sky to plantlife to earth, is in shades of grey, brown, ochre, and the occasional desaturated greenish-whatever. Autumn leaves have pretty much stopped falling, and are on the ground turning greyish versions of whatever colours they were before. There are whole vistas of bare branches, dead underbrush, and so on.

Which seems dreary for a while, until you’re immersed in it, as I am on the particular route I walk home, along London, Ontario’s Thames River (not to be confused with the River Thames in London, England), which is lined with parks. Then, after a while, all the subtle differences in shading and texture start to jump out at you more and more, until brownish moss on a tree under overcast late-afternoon light is vibrant green, a patch of snow is glowing bluish white, the occasional flicker of sunset through the cloud cover is warm pumpkin-orange, that one particular bush’s leaves look startlingly red… and then you see something artificial, like someone’s nylon jacket, or a kid’s plastic play-structure, and their colours look implausibly oversaturated, like a badly photoshopped picture.

Which got me to thinking about fine distinctions that seem drab outside their context - the way these colours look at first. Or nitpicky, like having 30 different words for “cold” and the different qualities it can have, or recognizing 15 different types of snowflake from a distance. Coming back to Canada after several years in California, I noticed all this specialized knowledge I’d forgotten about, and seems terribly arcane outside its native habitat. It occurred to me that this is how mathematics probably seems to outsiders - like physicists, or statisticians… (I jest)

For instance: I often have the experience of using the term “categorification” in describing something I’m doing - often in scare-quotes, followed by some kind of explanation - only to have it echoed back as “categorization”, and wonder whether to risk pedantry and explain that they’re not the same thing at all. “Categorification, not to be confused with…”

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On another note, I went looking for this paper by Carter, Kauffman and Saito, on a kind of invariant of 4-manifolds which generalizes 3D Dijkgraaf-Witten invariants, on the supposition that it would be closely related to some things I’ve been thinking about, from a diagrammatic point of view I’ve not paid much attention to in the last year or so. As I was looking through seach results, I noticed a paper from about 10 years ago by Kauffman and Smolin with an interesting sounding title, A Possible Solution to the Problem of Time in Quantum Cosmology. Since Lee Smolin has written on linking topological field theory and quantum gravity, I guessed it would also be interesting to look at. Only after reading the first few pages did I notice that the first listed author was not Louis Kauffman, who studies knot theory (and things tangent thereto), but Stuart Kauffman, who studies biocomplexity and complex systems.

I happen to be interested in the work of both Kauffmans - more immediately and professionally that of Louis, but I also read a couple of Stuart’s more accessible books, “At Home in the Universe”, and “Investigations” - and since the paper was short, I finished reading it. The basic premise is that the configuration space for 4D quantum gravity may not be constructible by any finite procedure (classifying spin networks, they say, might present a problem; doing path integrals over all 4-manifold topologies certainly does). So the “problem of time”, that there’s no role for time in describing dynamics in terms of paths through a configuration space, wouldn’t make sense - at least for a constructivist. (Or indeed a constructivist, though of course they shouldn’t be confused.) One thing that threw me off in noticing which Kauffman was involved was that part of this portion of the argument was about classifying knots.

That cleared itself up when they got to the part proposing a solution - that the total space of possible states isn’t a-priori given, but time re-enters the situation as the universe evolves, at each time step having some amplitude to move into each configuration in a (newly defined!) space called the adjacent possible. Having read Stuart K.’s books, this was when I realized my mistake - he describes this concept in “Investigations” in the context of a biosphere, or an economy, where a theorist also doesn’t have an explicit description of all possible future states given in advance.

It seems like this idea has a lot in common with type theory as a solution to Russell’s paradox: the collection of all sets isn’t a set, and so to get at it, sets are generated starting with nothing in successive stages. Whether this also doubles as a solution to the problem of time, I don’t know. In any case, it’s an interesting idea. It definitely would be a problem to have to do path integrals over a space of all topologies for 4-manifolds, when these can’t be classified, so some sort of suggestions are definitely a good thing here.

A recent colloquium talk at UWO was given by Rick Jardine, who is a prominent member of the department, with a lot of graduate students. I’m not sure of all the details of what he works on, but it seems to mostly have to do with homotopy theory, category theory, and related things. His talk was called “Categories, Symmetric Groups, and Spheres”. It was rather involved for me to describe here, tying together as it did a bunch of different topics. However, I thought it was interesting, so I’ll try to give a summary of at least some of what it was about.

The last of the three topics - spheres - had to do with the fact that the end result was to show that some construction turns out to be closely related to sphere spectra. Spectra are sequences of spaces, say , such that there’s a map from the suspension of each space into the next, . A suspension is just a sort of double-cone on a space: to get , add two points, and then connect each point of to each of the two new points. For example, if you start with a circle, the result is a sphere - your original circle was the “equator”, then you added two poles, and drew in the points in between. This example generalizes, so a really simple spectrum is just the sequence of spheres of increasing dimension (then the map is just the identity).

These spectra are important in homotopy theory, and in particular, in stable homotopy theory. As I understand it, stable homotopy talks about those parts of homotopy groups that stay the same when you repeatedly take suspensions - so you pass from homotopy classes of maps from a circle into , to maps from a sphere into the suspension , to maps from a 3-sphere into the suspension , and so on… the only changes that can occur is that you might lose some distinctions, so the groups could get smaller. Eventually, they stabilize - and voila!, stable homotopy groups. So anyway, spectra are important to this subject.

In particular, the theorem Rick was explaining (in, as he said, a “modern exposition”, originally due to Barratt and Priddy) has to do with a space called , whose homotopy groups are the same as the stable homotopy groups of spheres. The theorem says that it has the same homology as the infinite symmetric group. So the idea he was presenting is a construction involving symmetric groups. The point of it is that there’s a basically combinatorial description of everything involved - that is, a description involving just finite sets (which is where the symmetric groups come from).

How does this work? Well, first of all, it uses a construction called the “category of elements” for a functor . This is a category whose objects are pairs where , and whose morphisms are morphisms such that . That is, this makes a new category from all the elements of the sets coming from objects in , where the arrows are compatible with those in - each object is multiplied, and so are the morphisms.

The category of elements we’re talking about is a functor . Here, has finite sets for objects, and 1-1 functions (”injections”, “monomorphisms”, etc.) as morphisms, and is the category of pointed sets. This functor depends on a particular choice of pointed set , or for short. The way it works is that is the set of all functions from into - which is pointed, since the function where everything goes to is distinguished - so this is just . Given an injection , you get a map from the set of functions to , which you get by extending a function so anything in not in the image of just goes to the special point (this is why we needed pointed sets). So the category of elements in question is . The point is that it gives a nice space.

Again: how? Well, this uses the idea of a “nerve”.

Any category has a nerve: this is a simplicial set related to . The way you get an -simplex is to look at any chain of arrows in . The vertices form the edges, the arrows give some of the edges, and the various ways of composing (some of) them give other edges. Each composition of two gives a triangle, and the higher simplices come from various equations. The different simplices are stuck together by various incidence relations that show the structure of the category . This nerve is called , which is a purely combinatorial object. (Ultimately, the simplicial set that’ll show up in this story is from the category of elements above). It becomes a space when you take its geometric realization: replace abstract simplices with actual triangles, tetrahedra, and so forth, taken as topological spaces living in . This space is called , and it’s a topologically nice space - a -complex (being built by gluing simplices together).

Then you have this simplicial set, which can be thought of as a space, - a so-called “gamma space”, which are what correspond to these spectra mentioned up above. In particular, if the pointed set , with 0 the distinguished point, then it turns out that , the disjoint union of the spaces obtained from all the finite symmetric groups. This is because the symmetric group acts on the category of elements .

So part of the point of this part is what was, as Rick pointed out, the first adjoint pair of functors which was seriously studied - a pair of functors going between and (simplicial sets and topological spaces). The geometric realization functor is a left adjoint to a functor , so that , giving a simplicial set for a topological space . And homotopy theory in then has an equivalent in - so there’s a completely combinatorial core of homotopy theory. (Technically - and I admittedly don’t quite grok this concept yet - these two adjoint functors are giving a Quillen equivalence). Now, homotopy doesn’t tell you everything about a space - but it tells a lot, so it’s useful to get the idea that all this information about a space from something very combinatorial, like permutations of finite sets.

I have to admit I find a lot of this stuff is a bit technical for me to fully appreciate what’s clearly a very elegant fact relating spaces and combinatorics, but I find it interesting that a correlation like that exists. The apparent dichotomy between “smooth” or “continuous” things like spaces, and discrete, combinatorial things like integers, finite sets, permutations, etc. - and the various ways this dichotomy gets resolved, overcome, or bridged - is one of the really interesting cores of mathematics to my mind.

A recent talk in the noncommutative geometry seminar here, was by Farzad Fathizadeh. He was talking about a few ideas - the main part of the talk being about how to construct the Dixmier-Douady invariant, which is related to the question of whether or not you can put a spin structure on some manifold. It’s also related to a lof other things I want to figure out anyway for longstanding reasons. Indeed, Dixmier is one of the big early names behind the theory of fields of Hilbert spaces, which are used in Crane and Yetter’s “Measurable Categories”, which are a sort of infinite dimensional analog of the 2-vector spaces I’ve been talking about. (Actually, 2-Hilbert spaces, since that structure starts to look more important there).

Since I’ve started thinking about infinite dimensional 2-Hilbert spaces again I thought I’d check it out. It turned out to be somewhat related, but not very deeply. However, precisely because it’s related to things I’ve yet to figure out, I’m going to give a superficial gloss here, and later maybe try to say something more detailed. I should be giving a talk to our group soon about various aspects of 2-Hilbert spaces, so I’ll post more when I get to that.

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The second part of the talk had a nice exposition of Morita equivalence, which was a notion that got a lot of use in the things people were talking about at Groupoidfest. I had heard about this concept before, but never quite got the hang of it until now, so here’s a quick little explanation for the record. There are two ways of describing Morita equivalence, and the content of Morita’s theorem is that the two definitions amount to the same thing.

One definition says that two algebras and are equivalent if the categories of modules over them, and are equivalent as categories. The other says that and are equivalent if there is an -bimodule, , and a -bimodule, with the properties that:

(as an -bimodule)

and

(as a -bimodule)

Where, if you’re unclear, an -bimodule is a set where the algebra acts on the left, and the algebra acts on the right, with a compatibility condition that looks like associativity: .

It shouldn’t be too hard to see that the second definition implies the first: given a (left) -module , you can turn it into a (left) module by taking , and given a -module, you turn it into an -module by similarly tensoring over on the left with . Doing both operations gives you . The assumptions on $\latex \mathcal{F}$ and $\mathcal{G}$ mean that this is equivalent to . The same (switching , \mathcal{G}$ and ) goes for a -module. So this is the equivalence from the first definition.

The hard part of Morita’s theorem is that any time you have an equivalence, you can represent it in terms of some bimodules and .

(This is related to the fact that there’s a bicategory structure for algebras in which the morphisms from to are -bimodules, and the 2-morphisms are bimodule homomorphisms. Composition works by exactly the kind of tensoring above. In fact, there’s a pseudocategory of rings, which has a horizontal bicategory like the one I just described, and a vertical category (a strict category this time) where the morphisms are ordinary ring homomorphisms. Then there are some square-shaped 2-cells, as well. When I had recently put out that paper about double bicategories of cobordisms, and ran into Peter May at the Fields Institute, he suggested I should look into this stuff more. At the time I didn’t quite see why people wanted these structures that were weak in only one direction - but it’s a little clearer now.)

Anyway, on the subject of Morita equivalence, Masoud Khalkhali pointed out that there’s a fairly easy example which makes a connection to the “bra-ket notation” which Dirac introduced to physics. This is the example that says an algebra is Morita equivalent to any of the associated algebras of square matrices . Then the bimodules that achieve the equivalence are (as row vectors, acted on by field elements on the left, matrices on the right), and (column vectors - vice versa).

You can think of a row vector as a “bra” , which is acted on by operators on the right, and a column vector as a “ket” acted on by operators on the left. In physics, a “ket” designates a state you might measure, and a “bra” a state you might set a system up in. Then you can have expressions like , denoting an inner product, or , which gives a complex number. This denotes the “transition amplitude” for a system set up in the state , which has evolved according to (or been acted on by) the operator to be measured in state . If is unitary, you tend to think of this as something like time evolution - if the operator is self-adjoint, it can be interpreted as a physical observable, in which case is the expected value of the observable.

Part of the point of the balanced Dirac notation is that the operator can be thought of as acting on either the bra or the ket - you can move across the middle. So actually is a particular representative of some element of according to our definitions above (with ). So this is equivalent to .

Similarly, expressions like denote particular linear operators (if you like, you could put a scalar multiple in the middle!) That particular one is the operator which takes a (unit) state vector and spits out , and ignores anything orthogonal to . Any linear operator is generated by ones like this, so in the notation above.

Anyway, this is a rather nice aid to grasping Morita equivalence. It’s also a special case of what was really being discussed - algebras (and modules) which look like these pointwise, but actually consist of bundles whose fibres are algebras (or modules, or what have you) that look like the above. I don’t really understand what these are good for yet, but they seem great aesthetically.

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At some later point, I’ll come back to the stuff I hinted at above about 2-Hilbert spaces. This should be important in linking the extended TQFT stuff I discussed in earlier posts with “actual” 3D quantum gravity.

So I’ve posted some slides from my talk at Groupoidfest. I also gave this talk here at Western in the Algebra seminar on Wednesday. It seemed to go over fairly well, although it was a bit of an outlier for the conference. However, I’m getting used to that consequence of trying to talk to both physicists and mathematicians. Anyway, after I got back from GFest (as they call it), it took me a few days to get caught up on lecturing and grading and so forth, but here are some slightly belated comments on what went on there. A lot of the content of the talks went over my head, as happens. However, at lunch of the first day, Arlan Ramsay gave me and a couple other beginning researchers some good advice about learning at conferences where you only grasp about 10% of what’s going on: be like a baby learning to walk. Don’t be afraid of looking stupid - just grab the 10% you understand, and then do it again. (Since I spent part the weekend watching a baby learn to walk, this was quite apropos).

So this I’ll comment a bit on some of the general themes I did manage to pick up, and in a subsequent post I may say more about some of the talks that seemed particularly relevant and/or comprehensible to me.GFest was held at the University of Iowa, in Iowa City - by happenstance, a friend from UCR, Erin Pearse, recently started there as a VIGRE postdoc, so I managed to stay with him and his family while I was in town, which was good. I was a little surprised at first that he was interested in sitting in on the talks at the conference, since his research is mostly in fractal geometry, and I didn’t initally see the relevance. However, I guess it shouldn’t have been too surprising, since part of the great thing about groupoids is their ability to represent symmetry. The kinds of fractals in question are the self-similar kind, which have various interesting types of symmetry.

In particular, Erin explained to me that the connection has something to do with shift operators. These operators, which shift a sequence of numbers and insert a new value in it, can be used iteratively to build up, for instance, the Cantor set. (Which is a set of sequences of 0’s and 2’s in ternary notation - the shift operators take you from a point in the whole set, to a point in one of its pieces, which resemble the whole.)

This was one reflection of a more general theme: since there’s a Hilbert space of sequences, namely , the shift operators can be taken as operators on a Hilbert space. So in particular, they generate an algebra of operators - a -algebra (see also some notes). The general theme is that most of the people at the GFest were interested in groupoids as a way of saying something about -algebras. I probably heard this term bandied about more than the actual term “groupoid” while I was there.

One reason my point of view was an outlier is that I was talking about finite, topologically discrete groupoids. However, this is kind of beside the point, since I’m really more interested in ones that come from Lie groups, and have some interesting topology. But I avoid getting into that so far because I’ve been postponing extending this stuff to smooth groupoids, since that leads to infinite-dimensional 2-Hilbert spaces, and gets more complicated than what I’ve been talking about so far. The theory of these does exist - Crane and Yetter develop a lot of the theory needed under the aegis of “measurable categories” - but it involves a lot more analysis.

In fact, while I’m used to thinking of groupoids as a special kind of category, a lot of the talk about them at GFest emphasized exactly this analysis a lot more. It seems to be bread-and-butter for people who work with groupoids arising in -algebras. Paul Muhly, who organized the conference, kindly gave me the current working draft of a book he’s writing on this stuff, where a lot of the important ideas people were using are collected together and explained. (Note that I’ve only started reading it, so I may be mistaking things here).

One point seems to be that these algebras coming from groupoids are related to the -algebras coming from transformation groups: situations where a (locally compact topological) group acts on a (locally compact Hausdorff) space . These can automatically be thought of as groupoids, taking objects to be points in the space, and morphisms from to to be group elements whose action takes to . Now as for -algebras, you can build them by taking algebras of compactly supported complex functions on . This becomes an algebra with the convolution product, given by integrating over the group (so we’re assuming has a nice invariant measure like Haar measure on a Lie group):

and the “star” operation is just complex conjugation.

You can do something similar for groupoids generally, since groupoids decompose into isomorphism classes, each of which looks just like a set with the action of some particular group on it. For this to really make sense, you must be talking about topological groupoids. Here, they think of groupoids as a set of all morphisms, with being the set of composable pairs. Given a topology on , this gets the subspace topology on the product. This is making use of the fact that objects of the groupoid needn’t be defined separately - they correspond to the “identity” morphisms (with ), which again gets the subspace topology automatically (which makes source and target maps continuous).

Then we’d like to again define a -algebra on using something like the above definition. But then we need to define a convolution product, and for that, we needed a Haar measure on the group. Fortunately, for topologically reasonable groups, you’re guaranteed to have one, and it’s unique (maybe up to a scalar multiple); unfortunately, you don’t have either existence or uniqueness guaranteed for groupoids. So instead you need to have a Haar system.

This is a family of measures on (the set of all morphisms), one for each object: , which we’ll use to do convolution at the which correspond to the object . The measure is supported on the component of the object . The whole system needs to have some nice properties. One is that for any function , the function taking to the integral of with respect to should be in 9. The other is that is equivariant, in the sense that if ,

(shifting which measure we use by is the same as shifting the function by ).

This is a bit obscure to me at the moment, but it’s clear enough that you need some family of measures to define a convolution. The first property just ensures that the algebra is closed under this product. The second is just the kind of property you should expect from groupoids: if you’ve defined something that’s not equivariant, you’re just asking for aggravation. So then finally, making a bunch of assumptions, such as that is locally compact, Hausdorff, and so on, we get , the set of smooth, compactly supported complex functions with a convolution product:

(where is the source object of the morphism ). The star operation is still complex conjugation.

So, while I’m running a bit long here, this is the basic setup behind most of what people were talking about at Groupoidfest. Either studying these -algebras in their own right, or using groupoids to think of already existing algebras as coming from this setup for some groupoid . The point, I suppose, is that representations of these algebras, and of the groupoids they come from, are closely related, just as representations of groups and their group algebras are.

This subject - representations of groupoids, is exactly what my talk was about, except that I ignored all the topology to simplify certain things. Right after my talk, Marius Ionescu gave one about irreducible representations of groupoid -algebras, which I’m trying to get up to speed on to see how these things are done in the case with more interesting topology. (For my purposes, it’ll also be necessary to understand infinite-dimensional 2-Hilbert spaces better, but that’s another story…) Maybe when I see what that’s about, I’ll say something further on that subject.

There were a number of other good talks - perhaps soon I’ll see if I can summarize what I gathered from some of them.

In “The Fabric of Reality”, David Deutch gives a refutation of solipsism. I’m not entirely sure it works - all he really tries to do is to show that the difference between solipsism and realism is more nearly a mere semantic distinction than is generally assumed. But in any case, along the way, there’s an anecdote about a solipsist professor lecturing his (imaginary?) class merely to help him clarify his ideas. The idea being that, even if the imaginary students don’t really exist, it helps to clarify the professor’s own ideas by lecturing to them, answering questions, and so forth. In this view, you don’t really understand your own opinions - let alone justifiably believe in them - unless you’ve argued for them against a variety of possible criticisms. (J.S. Mill gave a defense of full-fledged freedom of speech, even for grossly offensive and even “dangerous” opinion, on this ground.)

I mention this because, when I told Dan about the blog, he seemed dubious about blogging as a way of communicating math. It’s certainly more solipsistic than a usenet newsgroup, or a mailing list. Those are channels devoted to a particular subject, with many participants. A blog, comments notwithstanding, is mainly a channel devoted to one voice, on many particular subjects. It’s true that half the point of communicating ideas is to get feedback on them from other people. You make your thinking part of one of those great processes like cathedral-building - ad-hoc, gradual, and (significantly) collective. Even so, relatively solipsistic channels are not entirely pointless.

To wit: by working through my theorems about transporting 2-vectors through spans - both for this blog, and for my talk at Groupoidfest, I discovered some problems. Nobody pointed them out, but discovering them was a consequence of approaching the material again from a new angle, with an audience in mind.

The problem is a conceptually important one - mistaking an n-dimensional space for a 1-dimensional space. I’m fairly sure, for various reasons, that the theorem that there is a 2-functor is still true, but the proof I have in my thesis (in the special case where the groupoids are flat connection groupoids on spaces) has a problem. Since that affects the Part 4 of “Spans and Vector Spaces” which I was going to post, I’ll put that off for a while as I get the proof straightened out.

Here is the issue in a nutshell, however:

The proof I have involves a construction of a functor by a particular method, which I’ve been describing in the last three posts. The final step I was going to describe involved what the contstruction does for 2-morphisms - spans between spans. (There is more to the proof, but the remainder is technical enough to be fairly unenlightening - basically, to be a 2-functor, there need to be specified natural isomorphisms replacing the equations for preserving identities and composition in the definition of a functor, and these have to obey some equations which need to be checked.)

The construction given in my thesis is supposed to give a way to take a span of spans of groupoids, and give a natural transformation between a pair of 2-linear maps. But a 2-linear map can be written as a matrix of vector spaces, and a natural transformation is then written as a matrix of linear operators which act componentwise. So one way to look at the problem is to construct a linear map between vector spaces from a span of groupoids.

That is, we have spans and . Picking basis objects for and (namely, objects and , plus representations of their automorphism groups) gives a subgroupoid of of , consisting of those objects which are sent to and under the maps in the span. It also gives a vector space which is built as a colimit of some vector spaces associated to these objects. Assuming is skeletal, this works out (as I described before) to for each of the in question. The same holds for .

Now suppose we have a span-of-spans making the obvious diagram commute. Then because of that commutation, we also have a span of groupoids over each of the choices of objects, and so then the question becomes, partly, how to get a linear map between the vector spaces we just constructed. If you have bases for all the vector spaces here, it’s not too bad: vectors can be seen as complex-valued functions on the basis. We can push these through the span just as we’ve been talking about in the last few posts here: first pull back a function along one leg by composition, then push forward along the other leg. The push-forward will involve a sum over some objects, and some normalizing factors having to do with the groupoid cardinalities of the groupoids in the span.

However, I won’t go too far into detail about this, because the construction I actually outlined doesn’t adequately specify the basis to use. In fact, it will really only work if all the vector spaces is one-dimensional. Then there is a basis for the combined space which just consists of all the objects . I’d hoped that Schur’s lemma (that intertwiners from to itself, or from to itself, have to be multiples of the identity) would get out of this problem, but I’m not sure it does. So there is a problem with the construction I was trying to use.

As I say, I’m fairly sure the theorem remains true - it’s just the proof needs fixing, which I don’t expect to be too hard. However, I’ll refrain from getting sidetracked until I know I have it worked out.

Instead, next time I’ll describe some of the things I learned at Groupoidfest 07 when I presented a talk on this stuff. (At first I was nervous, having discovered this flaw while preparing the talk - but then, a lot of people were talking about work-in-progress, so I don’t feel too bad now. Plus, the meeting was a lot of fun.)