Abstract :
This paper offers definitions for the semidirect product of categories, Cayley graphs for categories, and the kernel of a relational morphism of categories which will allow us to construct free (profinite) objects for the semidirect product of two (pseudo) varieties of categories analogous to the monoid case. The main point of this paper is to prove g(V*W)=gV*gW. Previous attempts at this have contained errors which, the author feels, are due to an incorrect usage of the wreath product. In this paper, we make no use of wreath products, but use instead representations of the free objects. Analogous results hold for semigroups and semigroupoids.
We then give some applications by computing pseudoidentities for various semidirect products of pseudovarieties. A further application will appear in a forthcoming joint work with Almeida where we compute finite iterated semidirect products such as A*G*J*G*J*cdots, three dots, centered*G*J. We obtain analogous results for the two-sided semidirect product. This paper also recovers the incorrectly proven result of Jones and Pustejovsky, needed to prove that DS is local.
We conclude by showing that our semidirect product can be used to classify split extensions of groupoids, generalizing the classical theory for groups.
