论文标题
关系结构类别的指控性
Exponentiability in categories of relational structures
论文作者
论文摘要
对于关系喇叭理论$ \ MATHBB {t} $，我们为对象和形态的指示性提供了有用的足够条件，$ \ Mathbb {t} \ text { - } \ Mathsf { - Mathsf {modsf {mod} $ of $ \ MATHBB {此类类别的众所周知的例子，这些类别在编程语言语言研究中的最新应用包括预定的集合和（扩展）度量空间的类别。结果，我们为$ \ mathbb {t} \ text { - } \ mathsf {mod} $获得了有用的足够条件，将是笛卡尔封闭的，本地载笛子的关闭，甚至是quasitopos;特别是，我们为笛卡尔的封闭提供了两种不同的解释。我们的结果恢复了niefield和Clementino所显示的某些条件的（充分性） - 霍夫曼（Hofmann）表征了部分有序集的类别的指示性，以及类别$ v \ text { - } \ Mathsf {catsf {cat} $的小型$ v $ - 用于某些通勤Unital Quantales $ v $的小型$ V $ - 类别。
For a relational Horn theory $\mathbb{T}$, we provide useful sufficient conditions for the exponentiability of objects and morphisms in the category $\mathbb{T}\text{-}\mathsf{Mod}$ of $\mathbb{T}$-models; well-known examples of such categories, which have found recent applications in the study of programming language semantics, include the categories of preordered sets and (extended) metric spaces. As a consequence, we obtain useful sufficient conditions for $\mathbb{T}\text{-}\mathsf{Mod}$ to be cartesian closed, locally cartesian closed, and even a quasitopos; in particular, we provide two different explanations for the cartesian closure of the categories of preordered and partially ordered sets. Our results recover (the sufficiency of) certain conditions that have been shown by Niefield and Clementino--Hofmann to characterize exponentiability in the category of partially ordered sets and the category $V\text{-}\mathsf{Cat}$ of small $V$-categories for certain commutative unital quantales $V$.
