Strong monads are important for several applications, in particular, in the denotational semantics of effectful languages, where strength is needed to sequence computations that have free variables. Strength is non-trivial: it can be difficult to determine whether a monad has any strength at all, and monads can be strong in multiple ways. We therefore review some of the most important known facts about strength and prove some new ones. In particular, we present a number of equivalent characterizations of strong functor and strong monad, and give some conditions that guarantee existence or uniqueness of strengths. We look at strength from three different perspectives: actions of a monoidal category V, enrichment over V, and powering over V. We are primarily motivated by semantics of effects, but the results are also useful in other contexts.
翻译:强大的寺院对于几种应用非常重要,特别是在有效语言的分解语义中,需要力量来对具有自由变量的计算进行排序。 力量是非三重性的:很难确定寺院是否具有任何力量,而寺院可以以多种方式强大。 因此,我们审查一些已知的关于力量的最重要事实,并证明一些新的事实。 特别是,我们提出了一些对强力真菌和强力寺院的等同描述,并提供了一些保证力量存在或独特性的条件。 我们从三个不同的角度审视力量:一分子类别五的行动,对V的浓缩,对V的动力。 我们主要受效果的语义学驱动,但结果在其他情况下也很有用。