Sch\"utzenberger Products in a Category

The Sch"utzenberger product of monoids is a key tool for the algebraic treatment of language concatenation. In this paper we generalize the Sch"utzenberger product to the level of monoids in an algebraic category $\mathcal{D}$, leading to a uniform view of the corresponding constructions for monoids (Sch"utzenberger), ordered monoids (Pin), idempotent semirings (Kl'ima and Pol'ak) and algebras over a field (Reutenauer). In addition, assuming that $\mathcal{D}$ is part of a Stone-type duality, we derive a characterization of the languages recognized by Sch"utzenberger products.