Consider braids on $n$ strands and pick $n$ distinct labels $1, \dots, n$. There is a groupoid $\mathcal P_n$ whose objects are tuples $(l_1, \dots, l_n)$ of labels and whose morphisms are braids, with the braids acting on the labels by permutations. The endomorphism group of the object $(1, \dots, n)$ is the pure braid group on $n$ strands, and $\mathcal P_n$ is really just a more convenient way of working with pure braids.

I am interested in more complicated colorings. Suppose we have a quandle $(Q, \triangleright)$. Then we can define a category $\mathcal B_n^Q$ with objects tuples $(x_1, \dots, x_n)$ of quandle elements and morphisms braids, with the action on labels given by $$ \sigma : (x_1, x_2) \to (x_1 \triangleright x_2, x_1) $$ where $\sigma$ is a braid generator.

Specifically, I care about the case where $Q$ is a conjugation quandle of a Lie group $G$, in particular $G = \operatorname{SL}_2(\mathbb C)$, so that $x_1 \triangleright x_2 = x_1 x_2 x_1^{-1}$. The closures of such braids are links $L$ with a representation $\rho : \pi_1(S^3 \setminus L) \to G$.

Is anything known about the cohomology of $\mathcal B_n^G$ for $G$ the conjugation quandle of a Lie group? In particular I want to know if $\mathcal B_n^G$ has nontrivial extensions, which are measured by $H^2$. $G$ has a nontrivial topology, so I think I want to only consider continuous cochains in some appropraite sense.

If $Q$ is the trivial quandle $T$, so that $\mathcal B_n^T = \mathcal P_n$, then this was computed by Arnold.


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