Distributive laws for Lawvere theories

TL;DR

This paper develops a comprehensive framework for combining algebraic structures via distributive laws for Lawvere theories, extending existing theories and providing explicit descriptions of composite theories.

Contribution

It introduces four new approaches to distributive laws for Lawvere theories, connecting them with monads and establishing comparison functors to existing frameworks.

Findings

Four approaches to distributive laws for Lawvere theories are proposed.

Comparison functors show correspondence between Lawvere theories and finitary monads.

Framework facilitates generalisation and explicit construction of composite theories.

Abstract

Distributive laws give a way of combining two algebraic structures expressed as monads; in this paper we propose a theory of distributive laws for combining algebraic structures expressed as Lawvere theories. We propose four approaches, involving profunctors, monoidal profunctors, an extension of the free finite-product category 2-monad from Cat to Prof, and factorisation systems respectively. We exhibit comparison functors between CAT and each of these new frameworks to show that the distributive laws between the Lawvere theories correspond in a suitable way to distributive laws between their associated finitary monads. The different but equivalent formulations then provide, between them, a framework conducive to generalisation, but also an explicit description of the composite theories arising from distributive laws.