@RuneHaugseng i think i remember talking about this with mark behrens and some of his students (one of them was trying to line up the classical and \infty cases), you might try writing to him
@ReubenStern this comes up in defining fukaya categories: depending on how much structure you can put on your lagrangians (orientation, etc.) you get to define a Z/n-graded chain complex (the hom-object) for varying values of n
i'm pretty sure you get d^2=0 even without any such structure, so you might hope to get a category enriched in differential modules (or really some flavor of A_\infty-category)
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?
