Morita equivalence of certain crossed products

We introduce an alternative criterion for Morita equivalence over graded tensor categories using equivariant centers and equivariantizations. While Morita equivalence has been extensively studied in the context of fusion categories, primarily through the examination of their centers, recent advancements have broadened its scope to encompass graded tensor categories. This paper presents a novel criterion for characterizing Morita equivalence in graded tensor categories by leveraging equivariant centers and equivariantizations. Notably, the identification of Morita equivalence can be expedited when the Brauer Picard groups are known, offering an efficient approach to establishing the equivalence relationship.To generalize the properties of fusion categories to finite tensor categories, we utilize theconcept of an exact module category, which was introduced by Etingof and Ostrik. Exact module categories offer an intermediary restriction between the semisimplemodule categories of a fusion category and more general cases that may not be semisimpleor finite.