Skip to main content
×
×
Home

Limits in categories of Vietoris coalgebras

  • DIRK HOFMANN (a1), RENATO NEVES (a2) and PEDRO NORA (a1)
Abstract

Motivated by the need to reason about hybrid systems, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one – intuitively, the topological analogue of a Kripke polynomial functor. Among other results, we prove that every Vietoris polynomial functor admits a final coalgebra if it respects certain conditions concerning separation axioms and compactness. When the functor is restricted to some of the categories induced by these conditions, the resulting categories of coalgebras are even complete.

As a practical application, we use these developments in the specification and analysis of non-deterministic hybrid systems, in particular to obtain suitable notions of stability and behaviour.

Copyright
References
Hide All
Adámek, J. (2005). Introduction to coalgebra. Theory and Applications of Categories 14 (8) 157199.
Adámek, J., Herrlich, H. and Strecker, G.E. (1990). Abstract and Concrete Categories: The Joy of Cats, Pure and Applied Mathematics, John Wiley & Sons Inc., New York. Republished in: Reprints in Theory and Applications of Categories, No. 17 (2006) pp. 1507.
Alur, R. (2015). Principles of Cyber-Physical Systems, MIT Press.
Adámek, J., Milius, S. and Moss, L.S. (2018). Initial Algebras and Terminal Coalgebras. In preparation.
Barr, M. (1993). Terminal coalgebras in well-founded set theory. Theoretical Computer Science 114 (2) 299315.
Barbosa, L.S. (2003). Towards a calculus of state-based software components. Journal of Universal Computer Science 9 (8) 891909.
Bezhanishvili, G., Bezhanishvili, N. and Harding, J. (2012). Modal compact Hausdorff spaces. Journal of Logic and Computation 25 (1) 135.
Baldan, P., Bonchi, F., Kerstan, H. and König, B. (2014). Behavioral metrics via functor lifting. In: Raman, V. and Suresh, S.P. (eds.) Proceedings of the 34th International Conference on Foundation of Software Technology and Theoretical Computer Science, December 15–17, 2014, New Delhi, India, LIPIcs, vol. 29, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 403–415.
Bezhanishvili, N., Fontaine, G. and Venema, Y. (2010). Vietoris bisimulations. Journal of Logic and Computation 20 (5) 10171040.
Balan, A. and Kurz, A. (2011). Finitary functors: From set to preord and poset. In: Corradini, A., Klin, B. and Cîrstea, C. (eds.) Algebra and Coalgebra in Computer Science – Proceedings of the 4th International Conference, Winchester, UK, August 30–September 2, 2011, Lecture Notes in Computer Science, vol. 6859, Springer, 8599.
Bonsangue, M.M., Kurz, A. and Rewitzky, I.M. (2007). Coalgebraic representations of distributive lattices with operators. Topology and its Applications 154 (4) 778791.
Balan, A., Kurz, A. and Velebil, J. (2013). Positive fragments of coalgebraic logics. In: Heckel, R. and Milius, S. (eds.) Algebra and Coalgebra in Computer Science – Proceedings of the 5th International Conference, Warsaw, Poland, September 3–6, 2013, vol. 8089, Lecture Notes in Computer Science, Springer, 51–65.
Bourbaki, N. (1942). Éléments de mathématique. 3. Pt. 1: Les structures fondamentales de l'analyse. Livre 3: Topologie générale. Chap. 3: Groupes topologiques. Chap. 4: Nombres réels, Hermann & Cie, Paris.
Bonsangue, M.M., Rutten, J.J.M.M. and Silva, A. (2009). An algebra for kripke polynomial coalgebras. In: Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, 11–14 August 2009, IEEE Computer Society, Los Angeles, CA, USA, 49–58.
Chen, L.-T. (2014). On a purely categorical framework for coalgebraic modal logic. PhD thesis, University of Birmingham.
Cignoli, R., Lafalce, S. and Petrovich, A. (1991). Remarks on Priestley duality for distributive lattices. Order 8 (3) 299315.
Clementino, M.M. and Tholen, W. (1997). A characterization of the Vietoris topology. In: Proceedings of the 12th Summer Conference on General Topology and its Applications, North Bay, ON, vol. 22, 71–95.
Dahlqvist, F., Danos, V. and Garnier, I. (2016). Giry and the machine. In: Proceedings of the 32nd Conference on the Mathematical Foundations of Programming Semantics, Electronic Notes in Theoretical Computer Science, vol. 325, pp. 85–110.
Doberkat, E.E. (2009). Stochastic coalgebraic logic, Monographs in Theoretical Computer Science, An EATCS Series, Springer.
Duda, R. (1972). One result on inverse limits and hyperspaces. In: Novák, J. (ed.) General Topology and its Relations to Modern Analysis and Algebra. Proceedings of the Third Prague Topological Symposium, 1971, Academia Publishing House of the Czechoslovak Academy of Sciences, Praha, 99102.
Engelking, R. (1989). General Topology, Sigma Series in Pure Mathematics, 2nd ed. Heldermann Verlag, Berlin.
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W. and Scott, D.S. (1980). A Compendium of Continuous Lattices, Springer-Verlag, Berlin.
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W. and Scott, D.S. (2003). Continuous lattices and domains, Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge University Press, Cambridge.
Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory – Selected Topics in Point-Set Topology, New Mathematical Monographs, vol. 22, Cambridge University Press.
Gumm, P.H. and Schröder, T. (2001). Products of coalgebras. Algebra Universalis 46 (1) 163185.
Hausdorff, F. (1914). Grundzüge der Mengenlehre, Veit & Comp., Leipzig.
Hasuo, I. and Jacobs, B. (2011). Traces for coalgebraic components. Mathematical Structures in Computer Science 21 (2) 267320.
Hofmann, D. (1999). Natürliche Dualitäten und das verallgemeinert Stone-Weierstraß Theorem. PhD thesis, University of Bremen.
Hofmann, D., Neves, R. and Nora, P. (2018). Generating the algebraic theory of C(X): The case of partially ordered compact spaces. Theory and Applications of Categories 33 (12) 276295.
Hughes, J. (2001). A study of categories of algebras and coalgebras. PhD thesis, Carnegie Mellon University.
Jacobs, B. (2016). Introduction to Coalgebra: Towards Mathematics of States and Observations, Cambridge University Press.
Jung, A. (2004). Stably compact spaces and the probabilistic powerspace construction. In: Desharnais, J. and Panangaden, P. (eds.) Domain-Theoretic Methods in Probabilistic Processes, ENTCS, vol. 87, Elsevier, 520.
Kelley, J. (1975). General Topology, Nostrand, Van. Reprinted by Springer-Verlag, Graduate Texts in Mathematics, 27.
Kock, A. (1972). Strong functors and monoidal monads. Archiv der Mathematik 23 (1) 113120.
Kupke, C., Kurz, A. and Venema, Y. (2004). Stone coalgebras. Theoretical Computer Science 327 (1–2) 109134.
Kurz, A. (2001). Logics for coalgebras and applications to computer science, BoD–Books on Demand.
Linton, F.E.J. (1969). Coequalizers in categories of algebras. In: Seminar on Triples and Categorical Homology Theory, Springer, Berlin, 7590. Republished in: Reprints in Theory and Applications of Categories.
MacLane, S. (1971). Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York.
Manes, E.G. (2002). Taut monads and T0-spaces. Theoretical Computer Science 275 (1–2) 79109.
Michael, E. (1951). Topologies on spaces of subsets. Transactions of the American Mathematical Society 71 (1) 152182.
Möbus, A. (1983). Alexandrov compactification of relational algebras. Archiv der Mathematik 40 (6) 526537.
Nachbin, L. (1965). Topology and Order. Translated from the Portuguese by Bechtolsheim, Lulu. Van Nostrand Mathematical Studies, No. 4. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London.
Neves, R. and Barbosa, L.S. (2016). Hybrid automata as coalgebras. In: Sampaio, A. and Wang, F. (eds.) Theoretical Aspects of Computing – 13th International Colloquium, Taipei, Taiwan, ROC, October 24–31, 2016, Lecture Notes in Computer Science, vol. 9965, 385402.
Neves, R. and Barbosa, L.S. (2017). Languages and models for hybrid automata: A coalgebraic perspective. Theoretical Computer Science. DOI: 10.1016/j.tcs.2017.09.038
Neves, R., Barbosa, L.S., Hofmann, D. and Martins, M.A. (2016). Continuity as a computational effect. Journal of Logical and Algebraic Methods in Programming 85 (5) 10571085.
Panangaden, P. (2009). Labelled Markov Processes, Imperial College Press.
Petrovich, A. (1996). Distributive lattices with an operator. Studia Logica 56 (1–2) 205224. Special issue on Priestley duality.
Pompeiu, D. (1905). Sur la continuité des fonctions de variables complexes. Annales de la Faculté des Sciences de l'Université de Toulouse pour les Sciences Mathématiques et les Sciences Physiques. 2ième Série 7 (3) 265315.
Power, J. and Watanabe, H. (1998). An axiomatics for categories of coalgebras. Electronic Notes in Theoretical Computer Science 11 158175.
Rutten, J.J.M.M. (2000). Universal coalgebra: A theory of systems. Theoretical Computer Science 249 (1) 380.
Schalk, A. (1993). Algebras for Generalized Power Constructions. PhD thesis, Technische Hochschule Darmstadt.
Simmons, H. (1982). A couple of triples. Topology and its Applications 13 (2) 201223.
Stauner, T. (2001). Systematic development of hybrid systems. PhD thesis, Technische Uuniversität München.
Tabuada, P. (2009). Verification and Control of Hybrid Systems – A Symbolic Approach, Springer.
Tholen, W. (2009). Ordered topological structures. Topology and its Applications 156 (12) 21482157.
Vietoris, L. (1922). Bereiche zweiter Ordnung. Monatshefte für Mathematik und Physik 32 (1) 258280.
Viglizzo, I.D. (2005). Coalgebras on measurable spaces. PhD thesis, Department of Mathematics, Indiana University.
Venema, Y. and Vosmaer, J. (2014). Modal logic and the vietoris functor. In: Bezhanishvili, G. (ed.) Leo Esakia on Duality in Modal and Intuitionistic Logics, Springer, Dordrecht, Netherlands, 119153.
Waterhouse, W.C. (1972). An empty inverse limit. Proceedings of the American Mathematical Society 36 (2) 618.
Zenor, P. (1970). On the completeness of the space of compact subsets. Proceedings of the American Mathematical Society 26 (1) 190192.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed