Completely Scott closed set and its applications

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-02-19 DOI:10.1016/j.topol.2025.109283

Licong Sun, Bin Pang

引用次数: 0

Abstract

In this paper, we propose a concept of completely Scott closed sets and use it to study links between convex spaces and continuous lattices. Firstly, we take three equivalent approaches to construct a convex space from a continuous lattice. Secondly, we construct an adjunction between the category of convex spaces and the opposite category of continuous lattices via completely Scott closed sets. This adjunction exactly induces the concept of sober convex spaces which gives rise to a categorical duality between them and algebraic lattices. Finally, we prove that completely Scott closed sets form a monad over the category of convex spaces and obtain an isomorphism between the category of sober convex spaces and the Eilenberg–Moore category of this monad.

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完全斯科特闭集及其应用

本文提出了完全Scott闭集的概念，并用它来研究凸空间与连续格之间的联系。首先，我们用三种等价的方法从连续晶格构造凸空间。其次，通过完全Scott闭集构造凸空间的类别与连续格的相对类别之间的连接。这一附加性精确地导出了清醒凸空间的概念，并由此产生了它们与代数格之间的直言对偶性。最后，我们证明了完全Scott闭集在凸空间范畴上形成了一个单子，并得到了清醒凸空间范畴与这个单子的Eilenberg-Moore范畴之间的同构。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.