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Relation Lifting, With An Application To The Many-Valued Cover Modality, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jirí Velebil
Relation Lifting, With An Application To The Many-Valued Cover Modality, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jirí Velebil
Engineering Faculty Articles and Research
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 …
Completeness For The Coalgebraic Cover Modality, Clemens Kupke, Alexander Kurz, Yde Venema
Completeness For The Coalgebraic Cover Modality, Clemens Kupke, Alexander Kurz, Yde Venema
Engineering Faculty Articles and Research
We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is defined in terms of a categorically defined relation lifting operation.
As the main contributions of our paper we introduce a derivation system, and prove that it provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, …