度量紧凑豪斯多夫空间的巴尔协约性

Marco Abbadini, Dirk Hofmann
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引用次数: 0

摘要

紧凑公域空间是一类重要的公域空间,但它们定义的范畴缺乏许多重要性质,如完备性和共完备性。在最近对 "公域理论 "和斯通类型的研究中,出现了一个更一般的概念,即(分离的)公紧凑豪斯多夫空间(metric compact Hausdorffspace),作为纳奇宾的紧凑有序空间的公对应。这些空间保持了紧凑度量空间的许多重要特征,而且,值得注意的是,由此产生的范畴表现得更好。此外,我们还可以利用纳奇宾紧凑有序空间理论的灵感来解决度量结构的问题。在本文中,我们将继续这一研究方向:在分离度量紧凑 Hausdorff 空间范畴中,我们将正则单态表征为嵌入,将外貌表征为投射态。此外,我们还证明了从对象 $X$ 出来的外形变可以通过其内核度量在 $X$ 上进行内部编码,而内核度量被表征为 $X$ 上度量下面的连续度量;这就为表示商对象提供了一种方便的方法。最后,作为主要结果,我们证明了其对偶范畴具有代数色彩:它是巴尔-精确的。虽然我们证明了它不可能是一个无穷代数的变项,但它是否是一个无穷变项仍是未知数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Barr-coexactness for metric compact Hausdorff spaces
Compact metric spaces form an important class of metric spaces, but the category that they define lacks many important properties such as completeness and cocompleteness. In recent studies of "metric domain theory" and Stone-type dualities, the more general notion of a (separated) metric compact Hausdorff space emerged as a metric counterpart of Nachbin's compact ordered spaces. Roughly speaking, a metric compact Hausdorff space is a metric space equipped with a \emph{compatible} compact Hausdorff topology (which does not need to be the induced topology). These spaces maintain many important features of compact metric spaces, and, notably, the resulting category is much better behaved. Moreover, one can use inspiration from the theory of Nachbin's compact ordered spaces to solve problems for metric structures. In this paper we continue this line of research: in the category of separated metric compact Hausdorff spaces we characterise the regular monomorphisms as the embeddings and the epimorphisms as the surjective morphisms. Moreover, we show that epimorphisms out of an object $X$ can be encoded internally on $X$ by their kernel metrics, which are characterised as the continuous metrics below the metric on $X$; this gives a convenient way to represent quotient objects. Finally, as the main result, we prove that its dual category has an algebraic flavour: it is Barr-exact. While we show that it cannot be a variety of finitary algebras, it remains open whether it is an infinitary variety.
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