# Constructing $C^*$-Categories from Groupoids

Let $G$ be a discrete group and $\mathcal{G}$ a groupoid, that is, a small category in which every arrow is an isomorphism. Wolfgang Lück explains how we can construct a $C^*$-category from $\mathcal{G}$ this way. This construction is very useful, because if we consider a discrete group $G$ as a groupoid with one element $e$ and morphisms $Mor_G(e,e)=G$, then $Mor_{C_r(G)}(e,e)$ agrees with the reduced group $C^*$ -algebra $C_r(G)$, as the notation implies. Now, I want to find a way to extend this principle to the reduced crossed product.

Let $A$ be a separable $G$-$C^*$-algebra. I want to find a way to construct a similar functor $F$ to that described by Lück, only that if I consider $G$ or a subgroup $H$ of $G$ to be a groupoid, I actually have $Mor_{F(H)}(e,e)$ agreeing with the reduced cross product of $H$ and $A$. I think it's easy if $G$ acts trivially on $A$: I would just take the (minimal) tensor product of the construction described by Lück and A. But what if the action isn't trivial?

My overall aim is to formulate the Baum-Connes conjecture with coefficients using spectra as introduced by Lück.

Thank you