Scientific Modelling with Coalgebra-Algebra Homomorphisms

TL;DR

This paper explores the use of coalgebra-algebra homomorphisms in scientific modeling, extending their applicability beyond unique homomorphisms by leveraging monadic structures for more flexible pattern capturing.

Contribution

It introduces dual techniques using (co)monadic structures to generalize coalgebra-algebra homomorphisms for scientific models without requiring universal properties.

Findings

Applicable to a broad range of real-world scientific models

Provides a flexible framework for modeling without universal properties

Extends the theoretical foundation of recursive function definitions

Abstract

Many recursive functions can be defined elegantly as the unique homomorphisms, between two algebras, two coalgebras, or one each, that are induced by some universal property of a distinguished structure. Besides the well-known applications in recursive functional programming, several basic modes of reasoning about scientific models have been demonstrated to admit such an exact meta-theory. Here we explore the potential of coalgebra--algebra homomorphism that are not a priori unique, for capturing more loosely specifying patterns of scientific modelling. We investigate a pair of dual techniques that leverage (co)monadic structure to obtain reasonable genericity even when no universal properties are given. We show the general applicability of the approach by discussing a surprisingly broad collection of instances from real-world modelling practice.