### Liang Kong on Levin-Wen Models

#### Posted by John Baez

Liang Kong gave what was probably the first talk at the Centre for Quantum Technologies to explicitly mention tricategories:

- Liang Kong, Levin-Wen models and tensor categories. (Joint work with Alexei Kitaev.)

But as his talk shows, tricategories are a quite natural formalism for studying models of 2d condensed matter physics. Two dimensions of space, one dimension of time: a tricategory!

The relation to the work of Fjelstad–Fuchs–Runkel–Schweigert is visible, but the focus on lattice models of condensed matter physics — in particular, the so-called Levin–Wen models — gives Liang Kong’s work a somewhat different flavor.

If you’re in a big rush, see the two-part “dictionary” on pages 41 and 42. Also see the diagram on page 46 — it’s probably quite cryptic taken out of context, but it means that Kong is working with a tricategory having:

- unitary tensor categories $C, D,\dots$ as objects,
- $C-D$ bimodule categories as 1-morphisms,
- maps between $C-D$ bimodule categories as 2-morphisms,
- natural transformations between these maps as 3-morphisms.

And, each bit of this structure corresponds to something people see in the Levin–Wen models:

- Michael A. Levin and Xiao-Gang Wen, String-net condensation: a physical mechanism for topological phases.

## Re: Liang Kong on Levin-Wen Models

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.