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Basic Concepts of Quantale-Enriched Topologies

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Abstract

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.

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Notes

  1. Here we have used the fact that the forgetful functor \(\textsf {Mod}_r(\mathfrak {Q})\xrightarrow {\,\,\,} \textsf {Sup}\) has a left adjoint functor (see [24, p. 174]).

  2. Since directed subsets are nonempty, note that the bottom element is not necessarily a zero element of \(\diamond \).

  3. For the definition of the Kleisli category associated with a monad (resp. algebraic theory) we refer to [25].

  4. We recall that a quantale homomorphism is not necessarily unital.

  5. Here \(\beta \) refers to the lax distributive law of the ultrafilter monad over the \(\mathfrak {Q}\)-powerset monad (cf. [20, Prop. 3.1]).

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Acknowledgements

We thank the referee for constructive comments and suggestions leading to an essential improvement of our paper. As one example among others we mention the referee’s encouragement to give a more categorical framework of enriched quasi-magmas being now Sect. 3.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, 48080, Bilbao, Spain

    Javier Gutiérrez García

  2. Fakultät für Mathematik und Naturwissenschaften, Bergische Universität, 42097, Wuppertal, Germany

    Ulrich Höhle

  3. Wydział Matematyki i Informatyki, Uniwersytet im. Adama Mickiewicza, 61-614, Poznan, Poland

    Tomasz Kubiak

Authors
  1. Javier Gutiérrez García
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  2. Ulrich Höhle
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Correspondence to Javier Gutiérrez García.

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Communicated by Jiří Rosický

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JGG acknowledges support from the Basque Government (Grant IT974-16)

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Gutiérrez García, J., Höhle, U. & Kubiak, T. Basic Concepts of Quantale-Enriched Topologies. Appl Categor Struct 29, 983–1003 (2021). https://doi.org/10.1007/s10485-021-09639-9

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