M Theory Lesson 113

Another new paper of interest to M theorists is Eugenia Cheng's Iterated Distributive Laws. Recall that a distributive law is a map (natural transformation) between two monads $ST \Rightarrow TS$ such that, in particular, $TS$ is again a monad. Cheng points out that whether or not $S(TU)$ is a monad, given monads $S$, $T$ and $U$, depends on the hexagon rule, aka the Yang-Baxter equation. Later this is associated to a hexagon for the Gray tensor product.

One example of a distributive law is for the category of 2-globular sets, which are two dimensional globule diagrams of sources and targets $S_2 \Rightarrow S_1 \Rightarrow S_0$ in Set such that $ss = st$ and $ts = tt$. One monad is vertical composition, and the other is horizontal composition. The distributive law is just the interchange rule for bicategories. Iterating this idea, Cheng considers composition in n-categories using n-globular sets, a la Batanin.