Abstract
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.
Chapter PDF
References
Adámek, J., Trnková, V.: Automata and Algebras in Categories. Kluwer Academic Publishers, Norwell (1990)
Baltag, A.: A logic for coalgebraic simulation. Electronic Notes in Theoretical Computer Science 33, 42–60 (2000)
Peter Gumm, H., Schröder, T.: Types and coalgebraic structure. Algebra Universalis 53, 229–252 (2005)
Hansen, H.H., Kupke, C.: A coalgebraic perspective on monotone modal logic. Electronic Notes in Theoretical Computer Science 106, 121–143 (2004)
Hansen, H.H., Kupke, C., Pacuit, E.: Bisimulation for Neighbourhood Structures. In: Mossakowski, T., Montanari, U., Haveraaen, M. (eds.) CALCO 2007. LNCS, vol. 4624, pp. 279–293. Springer, Heidelberg (2007)
Hansen, H.H., Kupke, C., Pacuit, E.: Neighbourhood structures: Bisimilarity and basic model theory. Logical Methods in Computer Science 5(2) (2009)
Hughes, J., Jacobs, B.: Simulations in coalgebra. In: Theor. Comp. Sci. Elsevier (2003)
Kelly, G.M.: Basic concepts of enriched category theory (2005)
Kupke, C., Kurz, A., de Venema, Y.: Completeness for the coalgebraic cover modality (accepted for publication)
Kurz, A., Leal, R.A.: Equational coalgebraic logic. Electronic Notes in Theoretical Computer Science, 333–356 (2009)
Levy, P.B.: Similarity Quotients as Final Coalgebras. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 27–41. Springer, Heidelberg (2011)
Marti, J.: Relation liftings in coalgebraic modal logic. Master’s thesis, ILLC, University of Amsterdam (2011)
Pattinson, D.: Coalgebraic modal logic: Soundness, completeness and decidability of local consequence. Theoretical Computer Science 309(1–3), 177–193 (2003)
Rutten, J.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249(1), 3–80 (2000)
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)
Schröder, L.: Expressivity of coalgebraic modal logic: The limits and beyond. Theoretical Computer Science 390(2-3), 230–247 (2008); Foundations of Software Science and Computational Structures
Schubert, C., Seal, G.J.: Extensions in the theory of lax algebras. Theories and Applications of Categories 21(7), 118–151 (2008)
Thijs, A.: Simulation and Fixpoint Semantics. PhD thesis, University of Groningen (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 IFIP International Federation for Information Processing
About this paper
Cite this paper
Marti, J., Venema, Y. (2012). Lax Extensions of Coalgebra Functors. In: Pattinson, D., Schröder, L. (eds) Coalgebraic Methods in Computer Science. CMCS 2012. Lecture Notes in Computer Science, vol 7399. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32784-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-32784-1_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32783-4
Online ISBN: 978-3-642-32784-1
eBook Packages: Computer ScienceComputer Science (R0)