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An equational approach to enriched distributivity. (English) Zbl 1524.18009

Summary: The familiar adjunction between ordered sets and completely distributive lattices can be extended to generalised metric spaces, that is, categories enriched over a quantale (a lattice of “truth values”), via an appropriate distributive law between the “down-set” monad and the “up-set” monad on the category of quantale-enriched categories. If the underlying lattice of the quantale is completely distributive, and if powers distribute over non-empty joins in the quantale, then this distributive law can be concretely formulated in terms of operations, equations and choice functions, similar to the familiar distributive law of lattices.

MSC:

18B35 Preorders, orders, domains and lattices (viewed as categories)
18C05 Equational categories
06D10 Complete distributivity
18D20 Enriched categories (over closed or monoidal categories)
06F07 Quantales
08A65 Infinitary algebras
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