Framed version of braided monoidal category

Question

The operad $\mathcal{D}_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB_n,1)$ where $PB_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}_2$ have a categorical analogue called braided monoidal category. More precisely, there is an operad in groupoids $\mathcal{P}aB$ whose algebras in the category of categories are braided monoidal category and which is such that a levelwise application of the nerve to $\mathcal{P}aB$ yields an operad equivalent to $\mathcal{D}_2$.

The operad $\mathcal{D}_2$ has a variant called the framed little $2$-disks operad and usually denoted $f\mathcal{D}_2$. The operad $f\mathcal{D}_2$ also has the property that each of its spaces are $K(\pi,1)$'s.

My question : Is there a known framed analogue of a braided monoidal category ? More precisely, is there an operad $\mathcal{X}$ in groupoids whose algebras in the category of categories have been defined somewhere in the literature and with the property that levelwise application of the nerve to $\mathcal{X}$ yields an operad equivalent to $f\mathcal{D}_2$.

Note that the question is not about the existence of $\mathcal{X}$ (which is straightforward) but really about the existence of an $\mathcal{X}$ whose algebras have been defined somewhere.

Answer 1

First of all let me correct you: it is not true that the groups $PB_n$ assemble into an operad in groups.

The point is that $PB_n$ is the fundamental group of $D_2(n)$, which requires the choice of a base point. But it is impossible to choose basepoints on the spaces $D_2(n)$ simultaneously in a way that is compatible with the operad composition and the $S_n$-action. What you can do is consider the fundamental groupoids $\Pi_1(D_2(n))$. These carry natural $S_n$-actions and are compatible with composition. Since $\Pi_1(D_2(n))$ is a bit unwieldy one can try to find a "small" full subgroupoid of $\Pi_1(D_2(n))$ which is closed under the $S_n$-action and composition. This is exactly the combinatorially defined operad $PaB$ which you mention: $PaB$ is isomorphic to the suboperad generated by a single object of $\Pi_1(D_2(2))$.

Now the fundamental group of $fD_2(n)$ is $PB_n \times \mathbf Z^n$, and one can play exactly the same game. One is naturally led to an operad $PaFB$ of parenthesized framed braids which is defined in the same way as $PaB$ except the braids are framed, that is, instead of isotopy classes of embedded intervalls we are considering isotopy classes of embedded ribbons. (I hope it's clear what I mean.) The algebras over $PaFB$ are exactly balanced monoidal categories. There is in fact a standard graphical calculus for balanced monoidal categories which is based on braids whose strands are embedded ribbons.