In this paper, the category theorist Michael Batanin explains how the groupoid of braid groups may be viewed as a collection of ordinals, and discusses equivalences between categories of braided and other operads. As usual, an $n$-ordinal is a rooted tree with $n$ levels. Things are most interesting for small values of $n$.
For example, when $n = 2$ an ordinal is given by a string $(n_1,n_2, \cdots, n_k)$ of ordinary ordinals, as in the arguments of a generalised zeta value. The $n = 2$ case already defines a large class of polytopes, including the $1$-ordinal associahedra.
6 years ago