# Cohomology of colored braid groupoids

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.