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. 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. Our construction is motivated by two-dimensional conformal field theory enriched by defects of all codimensions.