The n-Category Café

That tricategory looks like it’s secretly what I would call a $(1\times 2)$-category. A lot of the tricategories that I see people studying nowadays are secretly either $(1\times 2)$-categories or $(2\times 1)$-categories. Chris Douglas also likes to talk about this distinction; I think he draws rectangular grids of dots to indicate how the cells are arranged.

In general an $(n\times k)$-category is an $n$-category (weakly) internal to $k$-categories. If it is sufficiently “fibrant,” then it has an underlying $(n+k)$-category, which is sometimes all one needs, but sometimes it’s better to work with $(n\times k)$-categories. In particular, they can be easier to construct, and they have different and sometimes better-behaved higher transfors.

For instance, a $(1\times 1)$-category is an internal category in $Cat$, which is a double category, and any double category has an underlying 2-category obtained by discarding the vertical (or the horizontal) arrows. So in that case no fibrancy is required.

A $(1\times 2)$-category is an internal category in 2-Cat. So it has a 2-category of objects, and a 2-category of 1-cells. In Kong’s case, the 2-category of objects would be the 2-category of unitary tensor categories with an appropriate notion of functor and natural transformation. The 2-category of 1-cells would be the 2-category of bimodule categories, with morphisms being functors that are equivariant relative to a couple of tensor functors on either side, and similarly for transformations. By forgetting the morphisms and 2-morphisms in the 2-category of objects, and relabeling those in the 2-category of 1-cells as “2-cells” and “3-cells,” one obtains a tricategory; no fibrancy is required here either (that seems to be the case with $(1\times k)$-categories for any $k$).

By contrast, a $(2\times 1)$-category is an internal 2-category in Cat, so it has a category of objects, a category of 1-cells, and a category of 2-cells. In the prototypical example here, the objects are commutative rings, the 1-cells $R\to S$ are $(R\otimes S^{op})$-algebras, and the 2-cells are bimodules for such algebras; each with a suitable notion of morphism. By forgetting the ring and algebra morphisms and relabeling the bimodule morphisms as “3-cells” one obtains a tricategory, now using the “fibrancy” fact that any ring morphism gives rise to a bimodule.

Hi, John, thank you for being interested in my joint work with Kitaev!

Let me add a few remarks:

Levin-Wen’s original work is about the bulk theory. Alexei and I worked out the boundary and defect theories. This generalization is quite simple. It is, however, a good example of the physical meaning behind extended topological field theories.

In the special case of Kitaev’s Toric code model, the boundary theories has already been worked out by Sergey B. Bravyi and Alexei Yu. Kitaev in quant-ph/9811052.

This work can be viewed as a categorified version of Fjelstad-Fuchs-Runkel-Schweigert’s theory of defects in RCFT.

Very cool stuff - thanks for posting the notes! Fascinating to see the condensed matter story taken so far in this direction. I recently heard a very interesting talk by Chris Schommer-Pries on joint work with Chris Douglas and Noah Snyder, relating these kinds of TFTs to the cobordism hypothesis – more precisely they show that fusion categories are fully dualizable objects in the 3-category of tensor categories, bimodule categories etc that is discussed here (hence define framed extended 3d TFTs). Moreover spherical categories form $SO(3)$-invariant objects, so define oriented extended 3d TFTs.

There is another interesting thing.

If we forget about extended TQFT, just ask a naive algebraic question: is the notion of (monoidal) center functorial? Then you realize that one need at least 4 layers of structures to do it if it is indeed possible. This gives a pure algebraic support of the role of n-category played in an n-dimensional extended TQFT.

Of course, one can check even simpler cases. Is the notion of the center of an algebra functorial? A naive guess is negative because an algebra map $f: A \to B$ does not give an algebra map $f: Z(A) \to Z(B)$ between centers. But if we change the 1-morphisms in the target category (a category of commutative algebras) to cospans, together with proper 2-cells, then the center is functorial (a lax functor). Alexei Davydov, Ingo Runkel and myself are writing another paper on this topic. In particular, we show that the notion of center for algebras in tensor category is functorial (a functor between two bicategories). The physics behind it is 2-dimensional rational conformal field theories with defects of codimension 1,2. That is why one need bicategories. In that case, it is also the content of the so-called open-closed duality. So maybe a better name for such functor is a boundary-to-bulk functor or Holographic functor.

It also suggests that boundary-to-bulk functor (by taking center) is an dispensable ingredients of extended TQFTs. I believe that this phenomenon is universal. One can check it in other contexts such as A-infinity algebras, tensor-infinity categories, etc. It is desirable to understand it in more general contexts.

The idea assembling the possible defects of an $d$-dimensional TQFT into a $d+1$-dimensional TQFT is developed in Section 6.7 of my recent paper with Scott Morrison (math.AT/1009.5025, but a more frequently updated version can be found on my web page). In that paper we call them “sphere modules” rather than “defects”, but the idea is the same. Turaev-Viro type TQFTs (a.k.a. Levin-Wen models) were one of the motivating examples.

The main construction is as follows. (For simplicity, assume that all $n$-categories mentioned are an $n$-dimensional version strict pivotal). Given an $n$-category $C$, we can form an $n-k$-category $C(S^k)$ whose $j$-morphisms are shaped like $B^j\times S^k$. ($B^j$ is a $j$-dimensional ball, $S^k$ is a $k$-dimensional sphere.) A representation of $C(S^k)$ is (by definition) a family of $(n-k-1)$-categories with some additional structure. We call such a representation a $k$-sphere module, but “codimension $k+1$ defect” would also be an appropriate name. More generally, one can replace the unadorned sphere $S^k$ with a pair $(S^k, K)$, where $K$ is a cell complex embedded in $S^k$, and the codimension-$m$ cells of $K$ are labeled by $m-1$-sphere modules. One can again form an $n-k$ category $C(S^k, K)$, and representations of such a $n-k$-category are also called $k$-sphere modules.

We can now form an $n+1$-category whose objects are $n$-categories and whose $k$-morphisms are $k-1$-sphere modules. The $n+1$-morphisms are ordinary intertwiners between representations of 1-categories.

We can also restrict to various subcategories of this big category of all sphere modules. For example, given a pivotal 2-category $A$, we can consider the sub-3-category of sphere modules with

• only a single object, $A$
• only a single 1-morphism, $_A A_A$ (i.e. $A$ though of as the trivial 0-sphere module (categorified bimodule) over itself)
• 2-morphisms are all possible representations of the “annular” 1-category A(S^1)
• 3-morphsims are all possible intertwiners

It’s not hard to see that this 3-category is isomorphic to the Drinfeld center of $A$. Variations on this construction give various higher centers of $n$-categories.