symmetric monoidal (∞,1)-category of spectra
The parenthesized braid operad is an operad in Grpd modelled on the braid group.
Let $PaB$ 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 $PaB$ of the $PaB_n$‘s is a braided operad?. The composition
is given by replacing the $i$th strand of the first braid, by the second braid made very thin.
$PaB$ also have an obvious structure of a braided monoidal category. In fact:
$PaB$ is the free braided monoidal category on one object. As a consequence, it is an initial object in the category of braided monoidal categories.
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:
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:
$CPaB$ may be identified with a full sub-operad of the fundamental groupoid of the little 2-disk operad.
$PaB$ was originally defined in