nLab
parenthesized braid operad

Idea

The parenthesized braid operad is an operad in Grpd modelled on the braid group.

Definition

Let $\mathrm{PaB}$ be the category defined as follows:

• its set of objects is the free magma? on one generator, or equivalently the set of rooted binary tree?s.
• the set of morphisms between two objects $s,t$ is given by the braid group $B_n$ whenever $s$ and $t$ are words of the same legnth $n$, and is empty otherwise.

Then the collection $\mathrm{PaB}$ of the ${\mathrm{PaB}}_n$’s is a braided operad?. The composition

$${\circ }_i:{\mathrm{PaB}}_n\times {\mathrm{PaB}}_m\to {\mathrm{PaB}}_{m+n-1}$$\circ_i:PaB_n \times PaB_m \rightarrow PaB_{m+n-1}

is given by replacing the $i$th strand of the first braid, by the second braid made very thin.

$\mathrm{PaB}$ also have an obvious structure of a braided monoidal category. In fact:

Theorem

$\mathrm{PaB}$ is the free braided monoidal category on one object. As a consequence, it is an initial object in the category of braided monoidal categories.

Colored/ordered version

let ${\mathrm{CPaB}}_n$ be the groupoid defined as follows:

• it set objects are parenthesized permutations of $\{1,\dots ,n\}$, that is non-associative, non-commutative monomials on this set in which every letter appears exactly once.
• morphisms between two objects $s,t$ are braids connecting each letter in $s$ to the same letter in $t$. In other words, let $p:B_n\to S_n$ be the canonical projection from the braid group to the symmetric group whose kernel is the pure braid group. Then, forgetting the parenthesization and viewing $s,t$ as permutations:

$$\mathrm{Hom}(s,t)=p^{-1}(\{s^{-1}t\})$$Hom(s,t)=p^{-1}(\{s^{-1}t\})

Then $\mathrm{CPaB}$ is an (ordinary) operad, the operadic structure being the same as for the non-colored version.

A topological interpretation of $\mathrm{CPaB}$ is as follows:

Theorem

$\mathrm{CPaB}$ may be identified with a full sub-operad of the fundamental groupoid of the little 2-disk operad.

References

$\mathrm{PaB}$ was originally defined in

• Dror Bar-Natan On Associators and the Grothendieck-Teichmuller Group I, arXiv:q-alg/9606021