Abstract

It is well-known that the set of lists over a set X is the free monoid on X. This is in fact true in any monoidal category, for list objects defined as having an initiality property akin to primitive recursion. In certain applications to be discussed during the talk, it is important to extend the notion of monoid to that of T-monoid by further adding a compatible monad algebra structure to monoids. Correspondingly, the notion of list object may be extended to that of T-list object by adding monad algebra structure with respect to which the universal iterator is a homomorphism. We shall see that T-list objects give rise to free T-monoids, and consider practical settings where one can give an explicit construction of them in terms of initial algebras. Finally, I shall concentrate on an application, introducing a notion of near semiring category, and showing how such categories are an appropriate setting to consider operads with compatible algebraic structure. Along the way I will sketch a theory of parametrised initiality for algebras, and point out applications to the theory of abstract syntax. This is joint work with Marcelo Fiore.