论文标题
定量代数和公制单元的分类
Quantitative Algebras and a Classification of Metric Monads
论文作者
论文摘要
定量代数为作用于操作的度量空间上的$σ$ - 代数。 Mardare，Panangaden和Plotkin引入了1个基本品种，作为定量方程式提出的定量代数类别。我们证明，对于超级空间的类别$ \ mathsf {umet} $，这些品种这些品种在$ \ mathsf {umet} $上进行了族裔对应于强烈的单个单子。对于公制空间的类别$ \ mathsf {metsf {metsf {mathsf {mathsf {mathsf {mathsf {mathsf {mathsf {mathsf {mathsf {mathsf {mathsf {mathsf {mathsf} $ compaces的类别中，前提是强烈的内部函数在构图下关闭。 对于无数的红衣主教$λ$，$λ$ - 元的定量代数与强烈$λ$可访问的单位之间存在类似的两者。此外，我们提出了Mardare等人介绍的$λ$基本品种之间的徒点，并在$ \ mathsf {metsf {met} $上介绍了$λ$可供应的单元。最后，提出了与$ \ mathsf {metsf {met} $上的一般丰富的$λ$可访问的单子，以呈现给广义品种的生物通信。
Quantitative algebras are $Σ$-algebras acting on metric spaces, where operations are nonexpanding. Mardare, Panangaden and Plotkin introduced 1-basic varieties as categories of quantitative algebras presented by quantitative equations. We prove that for the category $\mathsf{UMet}$ of ultrametric spaces such varieties bijectively correspond to strongly finitary monads on $\mathsf{UMet}$. The same holds for the category $\mathsf{Met}$ of metric spaces, provided that strongly finitary endofunctors are closed under composition. For uncountable cardinals $λ$ there is an analogous bijection between varieties of $λ$-ary quantitative algebras and monads that are strongly $λ$-accessible. Moreover, we present a bijective correspondence between $λ$-basic varieties as introduced by Mardare et al and enriched, surjections-preserving $λ$-accesible monads on $\mathsf{Met}$. Finally, for general enriched $λ$-accessible monads on $\mathsf{Met}$ a bijective correspondence to generalized varieties is presented.
