The Block Product of Categories and Tilson's Division

Abstract

This paper introduces the notion of block product of two categories. It is the first step in the working out of a useful definition of the block product of two C-varieties which is needed for the theory of decompositions of monoids by means of wreath or block products. The construction of the block product of two categories is a particular case of a more general notion: the strong semidirect product of an action. In this first part of a two parts work we define the strong semidirect product functor and give some properties of this functor with respect to Tilson's division. In the second part we construct a left adjoint of the strong semidirect product functor and we define the kernel of a relational morphism of categories. With these notions we define the block product of two C-varieties and prove a kernel theorem in the category settings which extends a work of J. Rhodes and B. Tilson.