Lax Extensions of Coalgebra Functors
We discuss the use of relation lifting in the theory of set-based 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.
Keywordscoalgebra relation lifting predicate lifting bisimilarity
- 6.Hansen, H.H., Kupke, C., Pacuit, E.: Neighbourhood structures: Bisimilarity and basic model theory. Logical Methods in Computer Science 5(2) (2009)Google Scholar
- 7.Hughes, J., Jacobs, B.: Simulations in coalgebra. In: Theor. Comp. Sci. Elsevier (2003)Google Scholar
- 8.Kelly, G.M.: Basic concepts of enriched category theory (2005)Google Scholar
- 9.Kupke, C., Kurz, A., de Venema, Y.: Completeness for the coalgebraic cover modality (accepted for publication)Google Scholar
- 10.Kurz, A., Leal, R.A.: Equational coalgebraic logic. Electronic Notes in Theoretical Computer Science, 333–356 (2009)Google Scholar
- 12.Marti, J.: Relation liftings in coalgebraic modal logic. Master’s thesis, ILLC, University of Amsterdam (2011)Google Scholar
- 15.Santocanale, L., de Venema, Y.: Uniform interpolation for monotone modal logic. In: Beklemishev, L., Goranko, V., Shehtman, V. (eds.) Advances in Modal Logic, vol. 8, pp. 350–370. College Publications (2010)Google Scholar
- 18.Thijs, A.: Simulation and Fixpoint Semantics. PhD thesis, University of Groningen (1996)Google Scholar