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.

翻译：我们提出了一种子结构语境中单子与分配律的范畴论理论。在分配律的研究中，变量语境的结构规则（的缺失）所起的作用已被认识到；我们的理论利用Tronin的言语范畴$\mathbf W$，以统一且与呈现无关的方式形式化了这些子结构情形。我们引入了$\mathbf W$-操作子单子（通过$\mathbf W$中的结构规则定义的单子）和$\mathbf W$-交换单子（在$\mathbf W$中的结构规则下不变的单子）这两类。我们给出了集合范畴$\mathbf{Set}$上单子分配律$ST\to TS$的一个典范构造；当$S$是$\mathbf W$-操作子的且$T$是$\mathbf W$-交换的（在温和条件下）时，这一构造适用。这解释了许多已知和新的分配律。即使当$S$不满足$\mathbf W$-操作子性质时，我们也可以精炼$S$并强制其具备$\mathbf W$-操作子性；这捕捉了Varacca和Winskel的索引估值构造。