4 research outputs found
A Fibrational Approach to Automata Theory
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
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words