Monads and Distributive Laws in Substructural Contexts (Extended Version)

TL;DR

This paper develops a categorical framework for monads and distributive laws within substructural contexts, formalizing the influence of structural rules using verbal categories and introducing new classes of monads.

Contribution

It introduces $ extbf W$-operadic and $ extbf W$-commutative monads, providing a canonical construction of distributive laws applicable to various known and new cases.

Findings

Defined $ extbf W$-operadic and $ extbf W$-commutative monads.

Constructed a canonical distributive law $ST o TS$ on $ extbf{Set}$.

Unified treatment of known and new distributive laws in substructural contexts.

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.