Presenting Distributive Laws
Marcello M. Bonsangue (Leiden University), Helle Hvid Hansen (Radboud, University Nijmegen), Alexander Kurz (University of Leicester), Jurriaan Rot, (Leiden University)

TL;DR
This paper explores how to derive distributive laws for monads with equations from those of free monads, aiding in the analysis of context-free languages and algebra-coalgebra interactions.
Contribution
It introduces a method to obtain distributive laws for non-free monads from free monads, facilitating their use in language and semantics analysis.
Findings
Established a construction for distributive laws with equations
Applied the method to show equivalence of context-free language representations
Enhanced understanding of algebra-coalgebra interactions in categorical semantics
Abstract
Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages.
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