Monads and Distributive Laws in Substructural Contexts (Extended Version)

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.

38 pages, LICS 2026