Some remarks on morita theory, Azumaya algebras and center of an algebra in braided monoidal categories

Abstract

We study the left and the right center of an algebra in a braided monoidal category C. We characterize them by specializing a general result and point out why in symmetric monoidal categories one does not distinguish the two centers. We review Morita theorems for a monoidal category D and analyze necessary conditions for the associativity of the tensor product over an algebra in D. The central part of the Morita theorems in braided categories is also discussed. We review the notion of Azumaya algebra in C distinguishing left and right faithfully projective objects and study the relation between the two.