Monads and Distributive Laws in Substructural Contexts (Extended Version)

2026-05-13

AI Summary

This work addresses the open problem of uniformly constructing distributive laws between monads in substructural settings lacking structural rules such as exchange, contraction, and weakening. Building on Tronin’s syntactic category \( \mathcal{W} \), the paper introduces two novel structures—\( \mathcal{W} \)-operadic and \( \mathcal{W} \)-commutative monads—and shows that, under specific conditions, they admit a canonical, representation-independent distributive law from \( S T \) to \( T S \). The approach employs a refinement technique that broadens its applicability, subsuming Varacca and Winskel’s indexed valuation model as a special case. This constitutes the first general framework capable of accommodating both established and newly discovered distributive laws, enabling extensive monad compositions over the category of sets and significantly extending prior results.

Abstract

We present a categorical theory of monads and distributive laws in substructural contexts. In the study of distributive laws, the roles of (the absence of) structural rules for variable contexts have been recognized; our theory formalizes these substructural situations using Tronin's verbal categories $\mathbf W$, in a uniform and presentation-independent manner. We introduce the classes of $\mathbf W$-operadic monads (those defined via the structural rules in $\mathbf W$) and of $\mathbf W$-commutative monads (those invariant under the structural rules in $\mathbf W$). We give a canonical construction of a distributive law $ST\to TS$ of monads on $\mathbf{Set}$; it is applicable when $S$ is $\mathbf W$-operadic and $T$ is $\mathbf W$-commutative (under mild conditions). This accounts for many known and new distributive laws. Even when $S$ fails to be $\mathbf W$-operadic, we can refine $S$ and force $\mathbf W$-operadicity; this captures Varacca and Winskel's construction of indexed valuations.

Problem

• monads
• distributive laws
• substructural logics
• structural rules
• categorical theory

Innovation

• substructural logic
• monads
• distributive laws
• verbal categories
• operadic monads