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 $\mathbb{C}$-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: $\mathbb{C}$ is easy to confuse with the algebra of $1 \times 1$ matrices over $\mathbb{C}$), 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 $A$ to algebra $B$ is not a map, but a left-right $(A,B)$-bimodule, $_AM_B$. 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 $S^3$ with an embedded graph $G$.

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

$G_1 \subset S^3 \stackrel{\pi_1}{\longleftarrow} M \stackrel{\pi_2}{\longrightarrow} S^3 \supset G_2$

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

2-Morphisms: A 2-morphism between morphisms $M_1$ and $M_2$ (together with their $\pi$ maps) is a cobordism $M_1 \rightarrow W \leftarrow M_2$, in a way that’s compatible with the structure of the $lateux M_i$ as branched covers of the 3-sphere. The $M_i$ are being included as components of the boundary $\partial W$ – 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 $hom$-categories into $hom$-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 $\mathcal{G}$ – like taking the groupoid algebra $\mathbb{C}[\mathcal{G}]$ of $\mathbb{C}$-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 $\mathcal{H}(G)$ at each object $G$ of the category, which has a basis consisting of all morphisms whose source is $G$. Then the algebra acts on this, since any morphism $M'$ which can be post-composed with one $M$ starting at $G$ acts (by composition) to give a new morphism $M' \circ M$ starting at $G$ – that is, it acts on basis elements of $\mathcal{H}(G)$ to give new ones. Extending linearly, algebra elements (combinations of morphisms) also act on $\mathcal{H}(G)$.

So this gives, at each object $G$, an algebra of operators acting on a Hilbert space $\mathcal{H}(G)$ – 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.