Categories as algebra. II

A theory of the semidirect product of categories and the derived category of a category morphism is presented. In order to include division (<) in this theory, the traditional setting of these constructions is expanded to include relational arrows. In this expanded setting, a relational morphism phi : M --> N of categories determines an optimal decomposition M < D(W) greater than or equal to N where greater than or equal to4 denotes semidirect product and D(phi) is the derived category of phi. The theory of the semidirect product of varieties of categories, V (*) W, is developed. Associated with each variety V of categories is the collection V-D of relational morphisms whose derived category belongs to V. The semidirect product of varieties and the composition of classes of the form V-D are shown to stand in the relationship (V (*) W)(D) = VDWD. The associativity of the semidirect product of varieties follows from this result. Finally, it is demonstrated that all the results in the article concerning varieties of categories have pseudovariety and monoidal versions. This allows us to furnish a straightforward proof that g(V (*) W) = gV (*) gW for both varieties and pseudovarieties of monoids.