1:03 AM
Let $B_n^0$ be the kernel of the abelianization map of the braid group $B_n\to \mathbb{Z}$. Then the geometric realization of the groupoid $\coprod_{n>0} BB_n^0$ of classifying spaces has the homotopy type of $\Omega^2(S^3\langle 3\rangle)$, a double loop space. However, that groupoid is not braided monoidal (it's only monoidal, as far as I know). Anyone have an explanation for this?