The seminal paper [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)] by B. Tilson showed that, whether explicitly or implicitly, categories play a role in the study of semidirect products of monoids and of monoid varieties. The main new tool was the derived category of a relational morphism of monoids (a notion with precursors elsewhere in mathematics, as it turns out) and the main new theorem, the derived category theorem, which in its varietal form states that a monoid \(M\) belongs to the semidirect product \({\mathcal U}* {\mathcal V}\) of two monoid varieties if and only if there is a relational morphism from \(M\) to a monoid in \({\mathcal V}\) whose derived category belongs to the category variety \(g{\mathcal U}\) generated by \({\mathcal U}\).
Since then, several papers have been devoted to extending these ideas to their “natural” setting of category varieties, always with the intent of applying the results to decompositions of monoids. One problem has been to define the requisite notions correctly. The main purpose of the paper under review is to provide correct definitions for the semidirect product of categories, Cayley graph of categories and derived category (and its two-sided analogue) of a relational morphism of categories. A parallel purpose is then to use these definitions to prove a new derived category theorem and to derive the important equation \(g{\mathcal U}* g{\mathcal V}= g({\mathcal U}* {\mathcal V})\) for monoid varieties. At least two incorrect proofs of this equation appear in the literature, one due to the reviewer.
Several further applications are given to monoid varieties and pseudovarieties.
{Reviewer’s remark: The author and Tilson have collaborated on a sequel to Tilson’s paper [Categories as algebra. II, preprint], in which the topic is addressed from an even more general viewpoint and alternative proofs are given for many of the results in the paper under review}.