Centre of Monoids, Centralisers, and Localisation

Abstract

Given a monoid object A in a symmetric monoidal category (C, ⊗, 1), we associate a commutative monoid z(A): = Hom A−Bimod (A, A), which we refer to as the set of central elements of A. If A has a centre Z(A) in C, we describe an isomorphism z(A) = Hom A−Bimod (A, A) ≅ Hom Z(A)−Mod (Z(A), Z(A)). When C is closed and complete, we show that every subset S ⊆ Hom A−Mod (A, A) has a centraliser Z A (S) which is a Z(A)-algebra and under certain conditions, we show that Z A (Hom A−Mod (A, A)) = Z(A). Further, for any multiplicatively closed set S ⊆ z(A), we define the localisation M S of any right A-module M with respect to S and show that several properties of classical localisation extend to this context.