Authors: Marta Bilkova ; Alexander Kurz ; Daniela Petrisan ; Jiri Velebil

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.

• 03B45 - Modal logic (including the logic of norms)
• 03G30 - Categorical logic, topoi
• 06F07 - Quantales
• 18C15 - Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
• 18D20 - Enriched categories (over closed or monoidal categories)