nLab parenthesized braid operad




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


Let PaB nPaB_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,ts,t is given by the braid group B nB_n whenever ss and tt are words of the same legnth nn, and is empty otherwise.

Then the collection PaBPaB of the PaB nPaB_n‘s is a braided operad?.

The composition

i:PaB n×PaB mPaB m+n1\circ_i:PaB_n \times PaB_m \rightarrow PaB_{m+n-1}

is given by replacing the iith strand of the first braid, by the second braid made very thin.

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


PaBPaB 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 nCPaB_n be the groupoid defined as follows:

  • it set objects are parenthesized permutations of {1,,n}\{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,ts,t are braids connecting each letter in ss to the same letter in tt. In other words, let p:B nS np: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,ts,t as permutations:

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

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

A topological interpretation of CPaBCPaB is as follows:


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


PaBPaB was originally defined in

The operad structure was pointed out in

Last revised on September 20, 2022 at 09:27:53. See the history of this page for a list of all contributions to it.