Correspondences and Spans

In the past couple of weeks, Masoud Khalkhali and I have been reading and discussing this paper by Marcolli and Al-Yasry. Along the way, I’ve been explaining some things I know about bicategories, spans, cospans and cobordisms, and so on, while Masoud has been explaining to me some of the basic ideas of noncommutative geometry, and (today) K-theory and cyclic cohomology. I find the paper pretty interesting, especially with a bit of that background help to identify and understand the main points. Noncommutative geometry is fairly new to me, but a lot of the material that goes into it turns out to be familiar stuff bearing unfamiliar names, or looked at in a somewhat different way than the one I’m accustomed to. For example, as I mentioned when I went to the Groupoidfest conference, there’s a theme in NCG involving groupoids, and algebras of -linear combinations of “elements” in a groupoid. But these “elements” are actually morphisms, and this picture is commonly drawn without objects at all. I’ve mentioned before some ideas for how to deal with this (roughly: is easy to confuse with the algebra of matrices over ), but anything special I have to say about that is something I’ll hide under my hat for the moment.

I must say that, though some aspects of how people talk about it, like the one I just mentioned, seem a bit off, to my mind, I like NCG in many respects. One is the way it ties in to ideas I know a bit about from the physics end of things, such as algebras of operators on Hilbert spaces. People talk about Hamiltonians, concepts of time-evolution, creation and annihilation operators, and so on in the algebras that are supposed to represent spaces. I don’t yet understand how this all fits together, but it’s definitely appealing.

Another good thing about NCG is the clever elegance of Connes’ original idea of yet another way to generalize the concept “space”. Namely, there was already a duality between spaces (in the usual sense) and commutative algebras (of functions on spaces), so generalizing to noncommutative algebras should give corresponding concepts of “spaces” which are different from all the usual ones in fairly profound ways. I’m assured, though I don’t really know how it all works, that one can do all sorts of things with these “spaces”, such as finding their volumes, defining derivatives of functions on them, and so on. They do lack some qualities traditionally associated with space - for instance, many of them don’t have many, or in some cases any, points. But then, “point” is a dubious concept to begin with, if you want a framework for physics - nobody’s ever seen one, physically, and it’s not clear to me what seeing one would consist of…

(As an aside - this is different from other versions of “pointless” topology, such as the passage from ordinary topologies to, sites in the sense of Grothendieck. The notion of “space” went through some fairly serious mutations during the 20th century: from Einstein’s two theories of relativity, to these and other mathematicians’ generalizations, the concept of “space” has turned out to be either very problematic, or wonderfully flexible. A neat book is Max Jammer’s “Concepts of Space“: though it focuses on physics and stops in the 1930’s, you get to appreciate how this concept gradually came together out of folk concepts, went through several very different stages, and in the 20th century started to be warped out of all recognition. It’s as if - to adapt Dan Dennett - “their word for milk became our word for health”.I would like to see a comparable history of mathematicians’ more various concepts, covering more of the 20th century. Plus, one could probably write a less Eurocentric genealogy nowadays than Jammer did in 1954.)

Anyway, what I’d like to say about the Marcolli and Al-Yasry paper at the moment has to do with the setup, rather than the later parts, which are also interesting. This has to do with the idea of a correspondence between noncommutative spaces. Masoud explained to me that, related to the matter of not having many points, such “spaces” also tend to be short on honest-to-goodness maps between them. Instead, it seems that people often use correspondences. Using that duality to replace spaces with algebras, a recurring idea is to think of a category where morphism from algebra to algebra is not a map, but a left-right -bimodule, . This is similar to the business of making categories of spans.

Let me describe briefly what Marcolli and Al-Yasry describe in the paper. They actually have a 2-category. It has:

Objects: An object is a copy of the 3-sphere with an embedded graph .

Morphisms: A morphism is a span of branched covers of 3-manifolds over :

such that each of the maps is branched over a graph containing (perhaps strictly). In fact, as they point out, there’s a theorem (due to Alexander) proving that ANY 3-manifold can be realized as a branched cover over the 3-sphere, branched at some graph (though perhaps not including a given , and certainly not uniquely).

2-Morphisms: A 2-morphism between morphisms and (together with their maps) is a cobordism , in a way that’s compatible with the structure of the $lateux M_i$ as branched covers of the 3-sphere. The are being included as components of the boundary - I’m writing it this way to emphasize that a cobordism is a kind of cospan. Here, it’s a cospan between spans.

This is somewhat familiar to me, though I’d been thinking mostly about examples of cospans between cospans - in fact, thinking of both as cobordisms. From a categorical point of view, this is very similar, except that with spans you compose not by gluing along a shared boundary, but taking a fibred product over one of the objects (in this case, one of the spheres). Abstractly, these are dual - one is a pushout, and the other is a pullback - but in practice, they look quite different.

However, this higher-categorical stuff can be put aside temporarily - they get back to it later, but to start with, they just collapse all the -categories into -sets by taking morphisms to be connected components of the categories. That is, they think about taking morphisms to be cobordism classes of manifolds (in a setting where both manifolds and cobordisms have some branched-covering information hanging around that needs to be respected - they’re supposed to be morphisms, after all).

So the result is a category. Because they’re writing for noncommutative geometry people, who are happy with the word “groupoid” but not “category”, they actually call it a “semigroupoid” - but as they point out, “semigroupoid” is essentially a synonym for (small) “category”.

Apparently it’s quite common in NCG to do certain things with groupoids - like taking the groupoid algebra of -linear combinations of morphisms, with a product that comes from multiplying coefficients and composing morphisms whenever possible. The corresponding general thing is a categorical algebra. There are several quantum-mechanical-flavoured things that can be done with it. One is to let it act as an algebra of operators on a Hilbert space.

This is, again, a fairly standard business. The way it works is to define a Hilbert space at each object of the category, which has a basis consisting of all morphisms whose source is . Then the algebra acts on this, since any morphism which can be post-composed with one starting at acts (by composition) to give a new morphism starting at - that is, it acts on basis elements of to give new ones. Extending linearly, algebra elements (combinations of morphisms) also act on .

So this gives, at each object , an algebra of operators acting on a Hilbert space - the main components of a noncommutative space (actually, these need to be defined by a spectral triple: the missing ingredient in this description is a special Dirac operator). Furthermore, the morphisms (which in this case are, remember, given by those spans of branched covers) give correspondences between these.

Anyway, I don’t really grasp the big picture this fits into, but reading this paper with Masoud is interesting. It ties into a number of things I’ve already thought about, but also suggests all sorts of connections with other topics and opportunities to learn some new ideas. That’s nice, because although I still have plenty of work to do getting papers written up on work already done, I was starting to feel a little bit narrowly focused.

 

A Problem Arises

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.)

 

Spans and Vector Spaces - pt 3

Well, I was out of town for a weekend, and then had a miserable cold that went away but only after sleeping about 4 extra hours per day for a few days. So it’s been a while since I continued the story here.

To recap: I first explained how to turn a span of sets into a linear operator between the free vector spaces on those sets. Then I described the “free” 2-vector space on a groupoid - namely, the category of functors from to . So now the problem is to describe how to turn a span of groupoids into a 2-linear map. Here’s a span of groupoids:

Here we have a span , of groupoids. In fact, they’re skeletal groupoids: there’s only one object in each isomorphism class, so they’re completely described, up to isomorphism, by the automorphism groups of each object. The object , for instance, has automorphism group , and the object has automorphism group . This diagram shows the object maps of the “source” and “target” functors and explicitly, but note that with each arrow indicated in the diagram, there is a group homomorphism. So, since the object map for sends to , that strand must be labelled with a group homomorphism . (We’re leaving these out of the diagram for clarity).

So, we want to know how to transport a -valued functor - along this span. We know that such a functor attaches to each a representation of on some vector space . As with spans of sets, the first stage is easy: we have the composable pair of functors , so “pulling back” to gives .

What about the other leg of the span? Remember back in Part 1 what happened when we pushed down a function (not a functor) along the second leg of a span. To find the value of the pushed-forward function on an element , we took a sum of the complex values on every element of the preimage . For vector-space-valued functors, we expect to use a direct sum of some terms. Since we’re dealing with functors, things are a little more complex than before, but there should still be a contribution from each object in the preimage (or, if we’re not talking about skeletal groupoids, the essential preimage) of the object we look at.

However, we have to deal with the fact that there are morphisms. Instead of adding scalars, we have to combine vector spaces using the fact that they are given as representation spaces for some particular groups.

To see what needs to be done, consider the situation of groupoids with just one object, so the only important information is the homomorphism of groups. These can be seen as one-object groupoids, which we can just call and . A functor between them is given by the single group homomorphism .

Now suppose we have a representation of the group on (so that and ). Then somehow we need to get a representation of which is “induced” by the homomorphism , :

This diagram shows “the answer” - but how does it work? Essentially, we use the fact that there’s a nice, convenient representation of any group , namely the regular representation of on the group algebra . Elements of are just complex linear combinations of elemenst of , which are acted on by by left multiplication. The group also has regular representation, on . These are the most easily available building blocks with which to build the “push-forward” of onto .

To see how, we use the fact that has a right-action of , and hence , by way of . An element acts on by right-multiplication by - and this extends linearly to . So we can combine this with the left action of on (also extended linearly from ) by taking a tensor product of with over . This lets us “mod out” by the actions of which are not detected in . The result, called the induced representation of , in turn gives us back a left-action of on . I’ll call this .

(Note that usually this name refers to the situation where is a subgroup of , but in fact this can be defined for any homomorphism.)

This tells us what to do for single-object groupoids. As we remarked earlier, if more than one object is sent to the same , we should get a direct sum of all their contributions. So I want to describe the 2-linear map, which I’ll now call which we get from the span above, thought of as in . Here and (where I’m now being more explicit that this whole process is a functor in some reasonable sense).

I have to say what does to a given 2-vector (what it does to morphisms between 2-vectors is straightforward to work out, since every operation we do is a tensor product or direct sum). Suppose we have is one. Then . We can now say what this works out to. At some object , we get (still assuming everything is skeletal for simplicity):

And this is a direct sum of a bunch of such expressions where is a basis 2-vector - i.e. assigns an irreducible representation to some one object, and the trivial rep on the zero vector space to every other. That allows this to be written as a matrix with vector-space components, just like any 2-linear map.

So the 2-linear map has a matrix representation. The indices of the matrix are the simple objects in and , which consist of a choice of (a) object in or (which we assume are skeletal - otherwise it’s a choice of isomorphism class), and (b) irreducible representation of the automorphism group of that object. Given a choice of index on each side, the corresponding coefficient in the matrix is a vector space. Namely the direct sum, over all the objects that restrict down to our chosen pair, of a bunch of terms like . This is just a quotient space of the one group algebra by the image of the other.

Next up: a quick finisher about what happens at the 2-morphism level, then back to TQFT and gravity!

 

Spans and Vector Spaces - pt 1

In the last couple of posts, I described how an extended TQFT gives a 2-vector space, with generators corresponding to particular states of matter, for each boundary of space (mostly talking about 1-D uboundaries of 2D space in 3D spacetime). I was starting to build up to talking about how cobordisms give rise to “spin network states” on space with given boundary conditions. Before I can do that, it’s probably helpful to talk about something a little more general. Since the general thing in question is something I’m developing a talk on to give in Iowa, this is helpful for me anyway.

A slightly more general thing has to do with spans of groupoids, and how to get 2-linear maps from them. A span in a category is a diagram like this:

Now, as for spans, let me first give a couple of link-outs (the blathyspherian version of a shout-out) to a couple of guys named John… Given a category with pullbacks, there is a (bi)category , where spans are composed using pullbacks. John Armstrong recently posted about spans, describing , which has the same objects as , and morphisms which are spans in .

In fact, it also has 2-morphisms, which are span maps - given two spans with central objects and , a span map is a map from to which makes the resulting diagram commute. It turns out these make into a bicategory - one of the classic examples, in fact, which goes back to Jean Benabou’s “Introduction to Bicategories” (1967) in which the concept was introduced. However, one can ignore these, and just think of it as a category, by taking spans only up to isomorphism.

John Baez recently posted some slides for a talk about spans in quantum mechanics, which gives a nice overview of the context that makes this stuff relevant to this discussion of TQFT. A key concept is summarized in the abstract:

Many features of quantum theory — quantum teleportation, violations of Bell’s inequality, the no-cloning theorem and so on — become less puzzling when we realize that quantum processes more closely resemble pieces of spacetime than functions between sets.

And the point both of them make is that cobordisms can be seen as spans (actually, cospans, although a cospan in is by definition a span in ). This is an important idea when thinking of TQFTs as functors, since and (or ) are symmetric monoidal categories with duals. A TQFT is a functor , which respects exactly this structure. So it’s important that quantum processes are “like” these “pieces of spacetime”. And “pieces of spacetime” (cobordisms) have these properties is that, any time you start off with a cartesian category with pullbacks, like , then taking spans in it gives you a symmetric monoidal category with duals.

What we’re really talking about are properties of (a) spans, and (b) certain free functors. In particular, free functors taking sets to vector spaces, groupoids to 2-vector spaces, and (potentially) so on. Both of these have something to do with how to go from a cartesian category like , or (really a 2-category), to a monoidal category with duals (”dagger compact”), like , or (also a 2-category) - but also like or … I’ll describe what happens for sets, to keep things simple for this installment.

One example of going from a cartesian category to a dagger compact one is by the “free vector space” functor , taking a set to , the free vector space on , and set maps to linear maps that just permute basis elements. Another is the process of taking and building $\mathbf{Span(C)}$. The point is that these two can be related in a rather interesting way. In particular, there’s a functor

which acts on the objects of $\mathbf{Span(Sets)}$ (which are sets) just like the free-vector-space functor. That is, given a set , it gives , the space of functions from into . (For simplicity, I’ll assume all my sets are finite).

But it does something rather special on morphisms in . These are spans of sets and therefore they have two morphisms in them. If we think of the span as a morphism in , then the two arrows in the span are distinguished as first a “backwards” arrow, then a “forwards” arrow. The point is to take a vector in - a complex-valued function on , through the span.

So the question is, if I have a complex-valued function , how do I get a complex-valued function on ? Well, first, of course, I have to get one on . Since I have a function , the obvious candidate is . Each element of just gets the same complex number as its image down in . That’s easy: we’ve “pulled back” the function along .

Now we have to transport this function down to , which is a little less obvious. A given object in may have several different objects in which map down to it, and no reason why they should all have the same function value under . What can we do with a bunch of complex numbers? The two things which are most obvious are: add them up, or multiply them. The one we pick is to add them up (it may help to remember that the preimage of some object in is the union, or coproduct, of a bunch of elements - and coproducts are like sums, just as products are like… well… products). The result is that we’ve “pushed forward” the function along , and the result is called .

How do I know the process of taking a function - that is, a vector in , and finding the vector in is a linear map? Well, it’s not too hard to check that it’s represented by a matrix, and the summation over the preimage of an object in was the sum in the matrix multiplication. (Go ahead!) This works out very nicely because is cartesian, so any span between and factors through the product . In fact, corresponds to an integer matrix, whose component is the number of elements of that project down to both and . (To get a general matrix, you’d have to give labels to the elements of , which is something I talk about in this paper - the thing I like about which is that it gives lots of pictures which make “matrix mechanics” seem pretty natural - to me, anyway.)

It turns out this gives you a functor which represents inside $\latex \mathbf{Vect}$. In fact, to really get the bigger picture, instead of in everything I’ve said here, you should replace , and for you should replace . I’ll say something about that in the next installment - but “morally speaking” it’s much the same as what I’ve talked about here.