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 $S\_{\infty}$.



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> 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 $B\_{\infty}?$
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> Ditto for the pure braids $P\_{\infty}$, the kernal of $B\_{\infty} \to S\_{\infty}$.
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Loop spaces and infinite braids 
Yes, $B \beta\_\infty$ is homology equivalent to $\Omega^2\_0 S^2$, the zero component of the double loop space of $S^2$. The map $B\_\infty \to S\_\infty$ induces the obvious stablisation map $\Omega^2\_0 S^2 \to Q\_0S^0$.


