The L-theoretic Farrell–Jones conjecture with additive-category coefficients

Let $G$ be a group, let $E_{\mathcal{VCY}}(G)$ be the classifying $G$-space for the family of virtually cyclic subgroups, and let $\mathcal{A}$ be an additive $G$-category with involution. The equivariant homology theory with coefficients in the $L$-theory spectrum is denoted $H_*^G(-;\mathbf{L}_{\mathcal{A}}^{\langle-\infty\rangle})$. The L-theoretic Farrell–Jones conjecture with additive-category coefficients. For every additive $G$-category with involution $\mathcal{A}$ and every $n\in\mathbb{Z}$, the assembly map induced by $E_{\mathcal{VCY}}(G)\to G/G$,

$$H_n^G(E_{\mathcal{VCY}}(G);\mathbf{L}_{\mathcal{A}}^{\langle-\infty\rangle})\longrightarrow H_n^G(G/G;\mathbf{L}_{\mathcal{A}}^{\langle-\infty\rangle})=L_n^{\langle-\infty\rangle}(\mathcal{A}[G]),$$

is bijective. This is the coefficient-enriched $L$-theoretic Farrell–Jones conjecture; the source gives no resolution status.