# Two modes of recognition: algebra, coalgebra, and languages

## TL;DR

This paper connects algebraic and coalgebraic approaches in categorical language theory, defining regular maps that generalize classical automata languages and establishing a Kleene-like theorem for their characterization.

## Contribution

It introduces a unified framework linking algebraic and coalgebraic language theories for monads, generalizing regular languages and automata behaviors.

## Key findings

- Regular maps form a subcategory of the Kleisli category.
- A Kleene-like theorem characterizes regular morphisms.
- Framework applies to various automata types like non-deterministic, tree, fuzzy, and weighted automata.

## Abstract

The aim of the paper is to build a connection between two approaches towards categorical language theory: the coalgebraic and algebraic language theory for monads. For a pair of monads modelling the branching and the linear type we defined regular maps that generalize regular languages known in classical non-deterministic automata theory. These maps are behaviours of certain automata (i.e. they possess a coalgebraic nature), yet they arise from Eilenberg-Moore algebras and their homomorphisms (by exploiting duality between the category of Eilenberg-Moore algebras and saturated coalgebras). Given some additional assumptions, we show that regular maps form a certain subcategory of the Kleisli category for the monad which is the composition of the branching and linear type. Moreover, we state a Kleene-like theorem characterising the regular morphisms category in terms of the smallest subcategory closed under certain operations. Additionally, whenever the branching type monad is taken to be the powerset monad, we show that regular maps are described as maps recognized by certain functors whose codomains are categories with all finite hom-sets. We instantiate our framework on classical non-deterministic automata, tree automata, fuzzy automata and weighted automata.