We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category \(\mathcal {K}\). In order to do that, we introduce the 2-category of relative monads in a 2-category \(\mathcal {K}\) with relative monad morphisms and relative monad transformations as 1- and 2-cells, respectively. We relate our definition to the 2-category of monads in \(\mathcal {K}\) defined by Street. Using this perspective, we prove two Beck-type theorems regarding relative distributive laws. We also describe what does it mean to have Eilenberg–Moore and Kleisli objects in this context and give examples in the 2-category of locally small categories.

我们在相对单子和单子之间引入分配律的概念。我们称其为相对分配律，并将其定义在任何 2 类\(\mathcal {K}\)中。为了做到这一点，我们在 2 类\(\mathcal {K}\)中引入了 2 类相对单子，其中相对单子态射和相对单子变换分别作为 1 和 2 单元。我们将我们的定义与Street 定义的\(\mathcal {K}\)中的 2 类单子联系起来。利用这个视角，我们证明了关于相对分配律的两个贝克型定理。我们还描述了在这种情况下拥有 Eilenberg-Moore 和 Kleisli 对象意味着什么，并在局部小类别的 2 类中给出示例。