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Functoriality of the center of an algebra

The notion of the center of an algebra over a field k has a far reaching generalization to algebras in monoidal categories. The center then lives in the monoidal center of the original category. This generalization plays an important role in the study of bulk-boundary duality of rational conformal field theories. In this paper, we study functorial properties of the center. We show that it gives rise to a 2-functor from the bicategory of semisimple indecomposable module categories over a fusion category to the bicategory of commutative algebras in the monoidal center of this fusion category. Morphism spaces of the latter bicategory are extended from algebra homomorphisms to certain categories of cospans. We conjecture that the above 2-functor arises from a lax 3-functor between tricategories, and that in this setting one can relax the conditions from fusion categories to finite tensor categories. We briefly outline how one is naturally lead to the above 2-functor when studying rational conformal field theory with defects of all codimensions. For example, the cospans of the target bicategory correspond to spaces of defect fields and to the bulk-defect operator product expansions.