In this talk we introduce our recent work on semi-galois categories. As the name suggests, semi-galois categories are extension of galois categories: While galois categories are dual to profinite groups, semi-galois categories are dual to profinite monoids. Originally, we introduced this class of categories in order to axiomatize a certain branch in formal language theory of computer science, known as Eilenberg theory, which concerns a systematic classification of regular languages, finite monoids, and deterministic finite automata; but recently observed that semi-galois categories are inherently related to class field theory as well. In this talk, (1) starting from a background of semi-galois categories and motivation in formal language theory, (2) we introduce the axiom, basic general properties and examples of semi-galois categories, and then (3) discuss an arithmetic (or F_1) analogue of a classical theorem known as Christol's theorem (an automata-theoretic characterization of when a formal power series over finite field is algebraic over the polynomial ring F_q[t]) and its natural relation to semi-galois categories.