The n-Category Café

Mike Shulman said he’d spy on this talk and report back to us.

As usual at Chicago, Chris first gave a `pre-talk’ with background material before the main talk. The pretalk was about Clifford algebras, which you all know about (or can read about in TWF). To summarize: we have a sequence of Clifford algebras $Cl_n$ which are generated by $n$ anticommuting square roots of $\pm 1$. The sequence is periodic up to Morita equivalence; $Cl_8$ is $\mathbb{R}(16)$, the algebra of $16\times 16$ real matrices, which is Morita equivalent to $\mathbb{R}$, and from then on it repeats every 8 with extra matrix dimensions thrown in.

By the way, Chris remarked on something which I’ve never thought about before: it’s also true that $Cl_6$ is $\mathbb{R}(8)$, so why don’t we get a period of 6 instead of 8? The answer is that the Clifford algebras are really best thought of as $\mathbb{Z}/2$-graded algebras, and $Cl_6$ is not Morita equivalent to $\mathbb{R}$ as a graded algebra.

All of this has relevance to K-theory, because it turns out that $K^n(X)$ can be represented geometrically by `bundles of Clifford modules’ over $X$. Let’s start with $K^0$; we know that elements of $K^0(X)$ are `formal differences’ $V-W$ of vector bundles over $X$. We can model the formal difference $V-W$ with an honest geometric object by using the $\mathbb{Z}/2$-graded vector bundle $V\oplus W$, where $V$ is even and $W$ is odd. Such a thing should represent the zero class in K-theory just when $V$ and $W$ are isomorphic; this can be rephrased as saying that there exists an odd operator $e$ on $V\oplus W$ (hence, taking $V$ to $W$ and vice versa) such that $e^2=1$. But this just says that $V\oplus W$ has an action of the first Clifford algebra $Cl_1$.

More generally, Karoubi proved that for any $n$, $K^{-n}(X)$ can be represented by $Cl_n$-module bundles on $X$ modulo those such that the $Cl_n$-action extends to a $Cl_{n+1}$-action. When $n=0$ this is what we had above, since a $Cl_0$-module is just a vector space. This allows us to deduce Bott periodicity for K-groups from the algebraic periodicity (up to Morita equivalence) of Clifford algebras.

K-theory tells us about bundles of C-modules for a Clifford algebra C, so it cares about the category C-mod of C-modules. A vector bundle can be thought of as a functor from the category of paths in X to Vect, and similarly a C-module bundle is a functor to C-mod.

Now, Clifford algebras live naturally as the 0-cells of a symmetric monoidal bicategory CL2, whose 1-cells are bimodules and whose 2-cells are bimodule maps. Internal equivalence in this bicategory gives us the notion of Morita equivalence. We find the category C-mod inside CL2 as the hom-category from C to the unit 1. Moreover, the endomorphism category CL2(1,1) of the unit object is just the category Vect that we originally thought $K^0$ was telling us about.

What we want to do is find a `higher’ version of all this that applies to elliptic cohomology. We can no longer get away with using finite-dimensional things, so we have to replace our ordinary vector spaces with Hilbert spaces and our Clifford algebras with von Neumann algebras (algebras that embed into operators on a Hilbert space and are complete in an induced topology).

We also move up one level, so that instead of ordinary bundles, we’re talking about functors defined on the 2-category of points, paths, and surfaces in X. Instead of Vect, these 2-functors should land in the 2-category of von Neumann algebras, bimodules, and maps. Thus, this looks like a 2-dimensional QFT, rather than a 1-dimensional one. Stolz and Teichner had the idea that this should tell us something about $Ell^0(X)$, the 0-degree part of the elliptic cohomology of $X$.

The goal is now to get information about $Ell^n(X)$ for higher $n$. By analogy, we hope to find some object $?_n$ like $Cl_n$, so that elements of $Ell^n(X)$ will be represented by functors into $?_n$-mod. We hope to find some sort of 3-category $??3$ such that $??3(?_n,1)$ is the 2-category $?_n$-mod and $??3(1,1)$ is the 2-category of von Neumann algebras, bimodules, and maps that we started with for $Ell^0$.

The solution proposed by Chris, Arthur Bartels, and Andre Henriques (DBH), is a 3-categorical structure whose objects are conformal nets. A conformal net is a `cosheaf of von Neumann algebras on the category of intervals’. The idea is that you have a von Neumann algebra for each subinterval of an interval, with inclusions for inclusions of intervals, and gluing conditions for unions of subintervals. Chris told us that conformal nets have been around for a while in other contexts, with slightly different definitions. There is a conformal net $Ff$ called the net of free fermions which plays the role of $Cl_1$, and the three of them have proven that its order is divisible by 24 (this would be the periodicity, analogous to the 8-periodicity of Clifford algebras). They don’t know what the order actually is yet, but they’re betting on 24 or 48.

The thing I haven’t said much about is the type of `3-categorical structure’ they’re using. It’s not a 3-category or tricategory, but something more like a 3-dimensional version of a double category (or a framed bicategory), provided you interpret that correctly. A double category can be defined to be an internal category object in the 2-category Cat; what they’re looking at is an `internal bicategory object’ in the 2-category of symmetric monoidal categories. This notion is something I thought about myself a while ago as a natural generalization of framed bicategories, but never developed much, so it makes me happy to see that the same idea has occurred to other people independently.

It’s easy to define an internal strict $n$-category object in any category with pullbacks: you have objects $C_k$ of $k$-cells for all $0\le k\le n$, source and target maps $C_k\to C_{k-1}$, compositions $C_k\times_{C_\ell} C_k\to C_k$, and so on, satisfying the obvious associativity, unit, and interchange axioms. Given this, it’s not too hard to see what you mean by a `pseudo’ such object (which DBH call a `coherent $n$-category object’) in a 2-category with pullbacks: you weaken the associativity, unit, and interchange axioms to be up to coherent isomorphism in your 2-category. For instance, a pseudo 1-category object in the 2-category of categories is just a pseudo double category. A pseudo 1-category object in the 2-category of symmetric monoidal categories is a symmetric monoidal pseudo double category, and so on. What I’m calling `pseudo $n$-category objects in Cat’ were actually defined by Michael Batanin, who called them `monoidal $n$-globular categories’.

The really neat thing, from my point of view, is that they also impose fibrational conditions on the source and target maps, just like in my definition of framed bicategory. As in the case of monoidal framed bicategories, which I mentioned in the appendix to my paper, here fibrational conditions imply that if you look only at the objects of $C_0$, the objects of $C_1$, and the objects and morphisms of $C_2$, you get an ordinary tricategory—and often with a lot less work than would be involved in checking the definition by hand, since here the only coherence involved is `2-categorical’.

I’ve thought some about this myself, including the example (related to one of theirs) of rings, algebras, and bimodules, but didn’t get around to doing much with it. I also didn’t come up with a good name for these things. The best I’ve thought of so far is `3-level category’. This fits the way I tend to think about them, generalizes nicely to `$n$-level category’, and, like `$n$-fold category’, suggests something $n$-dimensional which isn’t quite the same as an $n$-category.

Actually, DBH end up weakening this notion further, although it’s not clear whether this extra weakness is essential. You can actually define an internal bicategory object in any category with pullbacks, by equipping yourself with maps such as $C_1\times_{C_0} C_1\times_{C_0} C_1 \to C_2$ that picks out an associator for each composable triple. And you can then `pseudoify’ that in a 2-category, in an appropriate way. DBH call this a `coherent bicategory object’. If what you pseudoify is the notion of `bicategory with strict associativity’, they call it a `coherent semi-strict bicategory object’. Fibrational conditions on a coherent 2-category object should allow you to `lift’ the coherence to make it a coherent bicategory object as well.

Chris has given me permission to link to his precise definitions of coherent semi-strict bicategory object and coherent 2-category object.

Mike

Clifford modules are those super Lie modules for the super Lie algebra $$\mathbb{R}^{1|q}$$ on which $\mathbb{R}^{1|0}$ acts as a multiple of the identity.

Another way to say this is that the Clifford algebra $Cl_q$ is obtained from the universal enveloping algebra of the super Lie algebra $\mathbb{R}^{1|q}$ by identifying the generator of $\mathbb{R}^{1|0}$ with the identity in the algebra.

Here $\mathbb{R}^{1|q}$ is the super Lie algebra spanned by a single even element $I$ and $q$ odd elements $\{\theta_i\}$ with the only non-trivial super Lie bracket being

$$[\theta_i,\theta_j] = \delta_{ij} \cdot I \,.$$

So possibly it would be fruitful to look for super Lie 2-algebra extensions of $\mathbb{R}^{1|q}$.