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" 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 fittings of endofunctors of for example, generalised fultrajmetric spaces. We illustrate this by studying the coalgebraic cover modality in this setting.
relation lifting cover modality commutative quantale module exact square enriched categories coalgehra modal logic