Lax Extensions of Coalgebra Functors

Abstract

We discuss the use of relation lifting in the theory of setbased coalgebra. On the one hand we prove that the neighborhood functor does not extend to a relation lifting of which the associated notion of bisimilarity coincides with behavorial equivalence. On the other hand we argue that relation liftings may be of use for many other functors that do not preserve weak pullbacks, such as the monotone neighborhood functor. We prove that for any relation lifting L that is a lax extension extending the coalgebra functor T and preserving diagonal relations, L-bisimilarity captures behavioral equivalence. We also show that if T is finitary, it admits such an extension iff there is a separating set of finitary monotone predicate liftings for T .

Enhanced coalgebraic bisimulation

“…This result is based on a definition of relational lifting for weak pullback preserving functors. We leave as future work the study of either a more broadly applicable notion of bisimulation, as in (Gorín and Schröder, 2013), or a more general definition of relation lifting, which applies to arbitrary functors on Set, as in (Marti and Venema, 2012). While in our work relators must preserve binary composition, in (Levy, 2011) a framework has been developed which only laxly preserves composition.…”

Simulations and Bisimulations for Coalgebraic Modal Logics

“…In [6] it is shown that so-called lax extensions of T preserving diagonals induce notions of bisimulation that are sound and complete for behavioural equivalence, and that a finitary functor has such an extension iff it admits a separating set of finitary monotone predicate liftings. Our result, while otherwise working with similar assumptions, does not suppose finitaryness of the functor.…”

The Logic of Exact Covers: Completeness and Uniform Interpolation

“…On the other hand, this simplicity comes at a price: the loss of monotonicity. However, logics based on the cover modality are very unlikely to be applicable to general neighbourhood frames [25], the simplest non-monotonic modal logic. Despite its conceptual simplicity, the logic of exact covers does have independent applications, as witnessed by the interpolation theorems of the previous section.…”

“…This was later generalised by Marti and Venema [25] for functors that have a lax extension preserving diagonals, but Theorem 2 of op.cit. acknowledges that this must necessarily fail for (not necessarily monotone) neighbourhood frames.…”