Free Monoids over Semigroups in a Monoidal Category Construction and Applications

The adjunction of a unit to an algebraic structure with a given binary asso-ciative operation is discussed by interpreting such structures as semigroups and monoids respectively in a monoidal category. This approach then allows for results on the adjunction of counits to coalgebraic structures with a binary co-associative co-operation as well. Special attention is paid to situations where a given coalge-braic structure induces a “dual ” algebraic one; here the compatibility of adjoining (co)units and dualization is examined. The extension of this process to starred al-gebraic structures and to monoid actions is discussed as well. Particular emphasis is given to examples from many areas of mathematics