The Artin braid groups $B_n$ and the symmetric groups $S_n$ are closely related by the maps $1 \to P_n \to B_n \to S_n \to 1$.  The infinite symmetric group has interesting interactions with homotopy theory, due to a result of Barratt-Priddy(-Segal(-Quillen(-others))) that "identifies" the sphere spectrum $QS^0$ with $S_{\infty}$.

> In light of the short exact sequence above, are there any Barrat-Priddy-*esque* results on the infinite braid group $B_{\infty}?$ Is there even a loop space structure on $B_{\infty}?$

> Ditto for the pure braids $P_{\infty}$, the kernal of $B_{\infty} \to S_{\infty}$.