nLab
parenthesized braid operad

Contents

Idea

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

Definition

Let PaBPaB 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,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 have an obvious structure of a braided monoidal category. In fact:

Theorem

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:

Theorem

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

References

PaBPaB was originally defined in

Revised on September 16, 2012 15:06:26 by Adrien B? (94.1.220.83)