Cohen-Montgomery duality for bimodules and singular equivalences of Morita type

Let $G$ be a group and $\Bbbk$ a commutative ring. All categories and functors are assumed to be $\Bbbk$-linear. We define a $G$-invariant bimodule ${}_SM_R$ over $G$-categories $R, S$ and a $G$-graded bimodule ${}_BN_A$ over $G$-graded categories $A, B$, and introduce the orbit bimodule $M/G$ and the smash product bimodule $N\# G$. We will show that these constructions are inverses to each other. This will be applied to Morita equivalences, stable equivalences of Morita type, singular equivalences of Morita type, and singular equivalences of Morita type with level to show that the orbit (resp. smash product) bimodule construction transforms an equivalent pair of $G$-categories (resp. $G$-graded categories) of each type to an equivalent pair of $G$-graded categories (resp. $G$-categories) of the same type.