Generalized Eilenberg Theorem I: Local Varieties of Languages

We investigate the duality between algebraic and coalgebraic recognition of languages to derive a generalization of the local version of Eilenberg's theorem. This theorem states that the lattice of all boolean algebras of regular languages over an alphabet {\Sigma} closed under derivatives is isomorphic to the lattice of all pseudovarieties of {\Sigma}-generated monoids. By applying our method to different categories, we obtain three related results: one, due to Gehrke, Grigorieff and Pin, weakens boolean algebras to distributive lattices, one weakens them to join-semilattices, and the last one considers vector spaces over the binary field.