Applied Categorical Structures ( IF 0.6 ) Pub Date : 2021-03-03 , DOI: 10.1007/s10485-021-09634-0 Themba Dube , Dorca Nyamusi Stephen
A map that is well known in C(X), dating back to the work of L. Gillman, M. Henriksen and M. Jerison in 1954, is here used to construct a morphism of quantales, whose right adjoint we show to be the map that sends a subset A of \(\beta X\) to the ideal
$$\begin{aligned} {\varvec{O}}^A =\{f\in C(X)\mid A\subseteq {{\,\mathrm{int}\,}}_{\beta X}{{\,\mathrm{cl}\,}}_{\beta X}Z(f)\} \end{aligned}$$of C(X) if and only if X is a P-space. We show that when viewed this way, this map characterizes R.G. Woods’ WN-maps in terms of commutativity of a certain diagram in the category of quantales. All this is achieved most economically by working with locales instead of topological spaces. The Lindelöf reflection allows us to present “countable” analogues of the results just mentioned and we highlight the similarities and disparities.
中文翻译：
将理想映射到子区域
一张以C（X）闻名的地图，其历史可追溯到L. Gillman，M。Henriksen和M. Jerison于1954年的工作，此处用于构造量子词素的态素，我们将其右边的伴奏显示为映射发送一个子集甲的\（\测试X \）理想
$$ \ begin {aligned} {\ varvec {O}} ^ A = \ {f \ in C（X）\ mid A \ subseteq {{\，\ mathrm {int} \，}} _ {\ beta X} {{\，\ mathrm {cl} \，}} _ {\ beta X} Z（f）\} \ end {aligned} $$的Ç（X）当且仅当X是P -space。我们表明，以这种方式查看时，该图通过定量图类别中的某些图的可交换性来表征RG Woods的WN-图。通过与语言环境（而不是拓扑空间）合作，可以最经济地实现所有这些目标。Lindelöf的反思使我们能够提出刚才提到的结果的“可数”类似物，并且我们强调了相似性和差异性。
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