Semi-galois Categories I: The Classical Eilenberg Variety Theory

Recently, Eilenberg's variety theorem was reformulated in the light of Stone's duality theorem. On one level, this reformulation led to a unification of several existing Eilenberg-type theorems and further generalizations of these theorems. On another level, this reformulation is also a natural continuation of a research line on profinite monoids that has been developed since the late 1980s. The current paper concerns the latter in particular. In this relation, this paper introduces and studies the class of semi-galois categories, i.e. an extension of galois categories; and develops a particularly fundamental theory concerning semi-galois categories: That is, (I) a duality theorem between profinite monoids and semi-galois categories; (II) a coherent duality-based reformulation of two classical Eilenberg-type variety theorems due to Straubing [30] and Chaubard et al. [10]; and (III) a Galois-type classification of closed subgroups of profinite monoids in terms of finite discrete cofibrations over semi-galois categories.