A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

$\text{DeclareMathOperator}\text{G}\mathbb{G}\text{DeclareMathOperator}\text{Rep}Rep\text{DeclareMathOperator}\text{Corr}Corr$Let $\text{G}$ be a locally compact quantum group and $(M,\alpha )$ a $\text{G}$-$W^{\ast }$-algebra. The object of study of this paper is the $W^{\ast }$-category ${\text{Rep}}^{\text{G}}(M)$ of normal, unital $\text{G}$-representations of $M$ on Hilbert spaces endowed with a unitary $\text{G}$-representation. This category has a right action of the category $\text{Rep}(\text{G})={\text{Rep}}^{\text{G}}(\mathbb{C})$ for which it becomes a right $\text{Rep}(\text{G})$-module $W^{\ast }$-category. Given another $\text{G}$-$W^{\ast }$-algebra $(N,\beta )$, we denote the category of normal $\ast$-functors ${\text{Rep}}^{\text{G}}(N)\to {\text{Rep}}^{\text{G}}(M)$ compatible with the $\text{Rep}(\text{G})$-module structure by ${\mathrm{Fun}}_{\text{Rep}(\text{G})}({\text{Rep}}^{\text{G}}(N),{\text{Rep}}^{\text{G}}(M))$ and we denote the category of $\text{G}$-$M$-$N$-correspondences by ${\mathrm{Corr}}^{\text{G}}(M,N)$. We prove that there are canonical functors $P:{\text{Corr}}^{\text{G}}(M,N)\to {\mathrm{Fun}}_{\text{Rep}(\text{G})}({\text{Rep}}^{\text{G}}(N),{\text{Rep}}^{\text{G}}(M))$ and $Q:{\mathrm{Fun}}_{\text{Rep}(\text{G})}({\text{Rep}}^{\text{G}}(N),{\text{Rep}}^{\text{G}}(M))\to {\mathrm{Corr}}^{\text{G}}(M,N)$ such that $Q\circ P\cong \mathrm{id}\mathrm{.}$ We use these functors to show that the $\text{G}$-dynamical von Neumann algebras $(M,\alpha )$ and $(N,\beta )$ are equivariantly Morita equivalent if and only if ${\text{Rep}}^{\text{G}}(N)$ and ${\text{Rep}}^{\text{G}}(M)$ are equivalent as $\text{Rep}(\text{G})$-module-$W^{\ast }$-categories. Specializing to the case where $\text{G}$ is a compact quantum group, we prove that moreover $P\circ Q\cong \mathrm{id}$, so that the categories ${\text{Corr}}^{\text{G}}(M,N)$ and ${\mathrm{Fun}}_{\text{Rep}(\text{G})}({\text{Rep}}^{\text{G}}(N),{\text{Rep}}^{\text{G}}(M))$ are equivalent. This is an equivariant version of the Eilenberg-Watts theorem for actions of compact quantum groups on von Neumann algebras.