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arXiv · 2605.13533

Monads and Distributive Laws in Substructural Contexts (Extended Version)

Soichiro Fujii·Yun Chen Tsai·Yoàv Montacute·Ichiro Hasuo
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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.

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Soichiro Fujii, Yun Chen Tsai, Yoàv Montacute, Ichiro Hasuo. 2026-05-13. Monads and Distributive Laws in Substructural Contexts (Extended Version). https://arxiv.org/abs/2605.13533

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