nLab parenthesized braid operad

Idea

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

Definition

Let $PaB_n$ denote 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 $PaB$ of the $PaB_n$‘s is a braided operad?.

The composition

$$\circ_i:PaB_n \times PaB_m \rightarrow PaB_{m+n-1}$$

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

$PaB$ also carries an obvious structure of a braided monoidal category. In fact:

Theorem

$PaB$ 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 $CPaB_n$ 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 $p:B_n\rightarrow S_n$ 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:

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

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

A topological interpretation of $CPaB$ is as follows:

Theorem

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

References

$PaB$ was originally defined in

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

The operad structure was pointed out in

• Dmitry Tamarkin, Formality of Chain Operad of Little Discs, Letters in Mathematical Physics 66:65–72, 2003.