A Fibrational Approach to Automata Theory

TL;DR

This paper introduces a fibrational framework linking automata theory, language varieties, and monoids in predual categories, providing new insights and proofs for classical theorems like Eilenberg's variety theorem.

Contribution

It develops a fibrational approach to automata theory that unifies various concepts and offers new proofs for fundamental results in the field.

Findings

Established isomorphisms between opfibrations and language varieties

Connected local and global varieties of languages with monoid pseudovarieties

Provided a new proof of Eilenberg's variety theorem within a unified framework

Abstract

For predual categories C and D we establish isomorphisms between opfibrations representing local varieties of languages in C, local pseudovarieties of D-monoids, and finitely generated profinite D-monoids. The global sections of these opfibrations are shown to correspond to varieties of languages in C, pseudovarieties of D-monoids, and profinite equational theories of D-monoids, respectively. As an application, we obtain a new proof of Eilenberg's variety theorem along with several related results, covering varieties of languages and their coalgebraic modifications, Straubing's C-varieties, fully invariant local varieties, etc., within a single framework.