Profinite categories, implicit operations and pseudovarieties of categories Original Research Article

The last decade has seen two methodological advances of particular direct import for the theory of finite monoids and indirect import for that of rational languages. The first has been the use of categories (considered as “algebras over graphs”) as a framework in which to study monoids and their homomorphisms, the second has been the use of implicit operations to study pseudovarieties of monoids. Still more recent work has emphasized the role of profiniteness in finite monoid theory. This paper fuses these three topics by means of a general study of profinite categories, with applications to C-varieties (pseudovarieties of categories) in general, to those C-varieties arising from M-varieties (pseudovarieties of monoids) in particular, to implicit operations on categories and to recognizable languages over graphs.