论文标题
Barr-of-ofact类别和软捆式表示
Barr-Exact Categories and Soft Sheaf Representations
论文作者
论文摘要
长期以来，人们一直知道，通用代数A的捆绑表示的关键要素是由通勤一致的分布晶格组成。 在本文中，我们通过用Barr-extactory代替代数类别来扩展该理论的范围，从而包含许多“非代码”示例。我们的方法基于K-sheaf的概念：直观地，在开放子集上定义了系套，而K-Sheaves则在紧凑的子集上定义。在整个过程中，我们都考虑在完整的晶格上而不是空间上的滑轮。这使我们能够获得无点的捆捆表示，从而将空间替换为框架。 这些结果用于构建紧凑型空间类别的双重构造，并恢复Banaschewski和Vermeulen的无点式造型Gelfand Rings（Quaest。Math。，2011）。
It has long been known that a key ingredient for a sheaf representation of a universal algebra A consists in a distributive lattice of commuting congruences on A. The sheaf representations of universal algebras (over stably compact spaces) that arise in this manner have been recently characterised by Gehrke and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of the notion of softness. In this paper, we extend the scope of this theory by replacing varieties of algebras with Barr-exact categories, thus encompassing a number of "non-algebraic" examples. Our approach is based on the notion of K-sheaf: intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain point-free versions of sheaf representations whereby spaces are replaced with frames. These results are used to construct sheaf representations for the dual of the category of compact ordered spaces, and to recover Banaschewski and Vermeulen's point-free sheaf representation of commutative Gelfand rings (Quaest. Math., 2011).
