8 Categories and Groupoids

Abstract

In this chapter, we consider a generalization of monoid algebras that will be used in the next chapter to study inverse monoid algebras, namely the algebra of a small category. Further examples include incidence algebras of posets (cf. Appendix C) and path algebras of quivers. We show that the Clifford-Munn-Ponizovskiĭ theory applies equally well to categories. The parametrization of the simple modules for the algebra of a finite category given here could also be obtained from a result of Webb [Web07], reducing to the monoid case, and the Clifford-Munn-Ponizovskiĭ theory, but we give a direct proof. Since category algebras are contracted semigroup algebras, these results also follow from the original results of Munn and Ponizovskiĭ (cf. [CP61, Chapter 5]). A basic reference on category theory is Mac Lane [Mac98]. Category algebras were considered at least as far back as Mitchell [Mit72].