Complete Elgot Monads and Coalgebraic Resumptions

TL;DR

This paper explores the relationship between complete Elgot monads and coalgebraic resumption monads, providing a categorical characterization that links iterative effects with coalgebraic structures.

Contribution

It characterizes complete Elgot monads via their algebra categories, connecting them with coalgebraic resumption monads using parametrized monads.

Findings

Characterization of the category of coalgebraic resumption monads.

Complete Elgot monads are precisely those with algebras coherently structured as coalgebraic resumption algebras.

Provides a categorical framework linking iterative effects and coalgebraic semantics.

Abstract

Monads are extensively used nowadays to abstractly model a wide range of computational effects such as nondeterminism, statefulness, and exceptions. It turns out that equipping a monad with a (uniform) iteration operator satisfying a set of natural axioms allows for modelling iterative computations just as abstractly. The emerging monads are called complete Elgot monads. It has been shown recently that extending complete Elgot monads with free effects (e.g. operations of sending/receiving messages over channels) canonically leads to generalized coalgebraic resumption monads, previously used as semantic domains for non-wellfounded guarded processes. In this paper, we continue the study of the relationship between abstract complete Elgot monads and those that capture coalgebraic resumptions, by comparing the corresponding categories of (Eilenberg-Moore) algebras. To this end we first provide a characterization of the latter category; even more generally, we formulate this characterization in terms of Uustalu's parametrized monads. This is further used for establishing a characterization of complete Elgot monads as precisely those monads whose algebras are coherently equipped with the structure of algebras of coalgebraic resumption monads.