M Theory Lesson 179

Recall that Bar-Natan's 1998 paper on the Grothendieck Teichmuller group discusses associated braids. On forgetting the crossing information, a braid in $B_{n}$ becomes a permutation on $n$ letters. So with bracketed endpoint sets, it marks a vertex of the $(n - 1)$ dimensional permutoassociahedron for $S_{n}$.

As well as the category of bracketed permutations, Bar-Natan considers the category of bracketed braids with linearised morphisms of the form $\sum \alpha_{j} B^{j}$, where the $B^{j}$ are allowable braids corresponding to a given element $P \in S_{n}$ and the $\alpha_{j}$ are numerical coefficients. For example, the Pauli permutation $\sigma_{x}$ gives morphisms of the form where $a$ and $b$ are usually rational numbers. The functor to the permutation category that forgets the braid structure is an example of a fibration of a very nice kind. The GT group is a certain group of endofunctors (from bracketed braids to bracketed braids) which fix $\sigma_{x}$ (with a choice of crossing). Bar-Natan shows that the bracketed braids are generated by the two basic diagrams shown above (ie. associator and $\sigma_{x}$ and inverses).