Relation lifting, with an application to the many-valued cover modality

We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the "powerset monad&q...

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Published inLogical methods in computer science Vol. 9, Issue 4
Main Authors Bilkova, Marta, Kurz, Alexander, Petrisan, Daniela, Velebil, Jiri
Format Journal Article
LanguageEnglish
Published Logical Methods in Computer Science e.V 25.10.2013
Subjects
computer science - logic in computer science
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Summary:We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the "powerset monad" on categories, one is the preservation by T of "exactness" of certain squares. Both characterisations are generalisations of the "classical" results known for set functors: the first characterisation generalises the existence of a distributive law over the genuine powerset monad, the second generalises preservation of weak pullbacks. The results presented in this paper enable us to compute predicate liftings of endofunctors of, for example, generalised (ultra)metric spaces. We illustrate this by studying the coalgebraic cover modality in this setting.
ISSN:1860-5974
1860-5974
DOI:10.2168/LMCS-9(4:8)2013