The categorical framework for our axioms of quantale-enriched topologies is the theory of modules in the monoidal category \(\textsf {Sup}\) and its free right modules generated by power sets. To express the intersection axiom we introduce the structure of a quasi-magma on a quantale. By selecting appropriate quantales and their corresponding quasi-magmas, we show that some well-established mathematical structures become quantale-enriched topologies. These include, among others, the closed left ideal lattices of non-commutative \(C^*\)algebras, lower regular function frames of approach spaces as well as quantale-valued topological spaces.

中文翻译：

---

富含Quantale的拓扑的基本概念

我们的富集量子拓扑的公理的分类框架是单等分类\（\ textsf {Sup} \）中的模块理论及其由幂集生成的自由权限模块。为了表达相交公理，我们介绍了一个量子岩上的准岩浆的结构。通过选择适当的量子量及其相应的准岩浆，我们证明了一些完善的数学结构成为量子量丰富的拓扑。其中包括非交换\（C ^ * \）代数的闭合左理想晶格，进近空间的较低正则函数框架以及量子值拓扑空间。