Free Monoids over Semigroups in a Monoidal Category: Construction and Applications

The adjunction of a unit to an algebraic structure with a given binary associative 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 coalgebraic structure induces a “dual" algebraic one; here the compatibility of adjoining (co)units and dualization is examined. The extension of this process to starred algebraic structures and to monoid actions is discussed as well. Particular emphasis is given to examples from many areas of mathematics.