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Limits in categories of Vietoris coalgebras

Published online by Cambridge University Press:  10 August 2018

DIRK HOFMANN
Affiliation:
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal Email: dirk@ua.pt, a28224@ua.pt
RENATO NEVES
Affiliation:
INESC TEC (HASLab) & Universidade do Minho, Braga, Portugal Email: nevrenato@di.uminho.pt
PEDRO NORA
Affiliation:
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal Email: dirk@ua.pt, a28224@ua.pt

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.

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Paper
Copyright
Copyright © Cambridge University Press 2018 

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References

Adámek, J. (2005). Introduction to coalgebra. Theory and Applications of Categories 14 (8) 157199.Google Scholar
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.Google Scholar
Alur, R. (2015). Principles of Cyber-Physical Systems, MIT Press.Google Scholar
Adámek, J., Milius, S. and Moss, L.S. (2018). Initial Algebras and Terminal Coalgebras. In preparation.Google Scholar
Barr, M. (1993). Terminal coalgebras in well-founded set theory. Theoretical Computer Science 114 (2) 299315.CrossRefGoogle Scholar
Barbosa, L.S. (2003). Towards a calculus of state-based software components. Journal of Universal Computer Science 9 (8) 891909.Google Scholar
Bezhanishvili, G., Bezhanishvili, N. and Harding, J. (2012). Modal compact Hausdorff spaces. Journal of Logic and Computation 25 (1) 135.CrossRefGoogle Scholar
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.Google Scholar
Bezhanishvili, N., Fontaine, G. and Venema, Y. (2010). Vietoris bisimulations. Journal of Logic and Computation 20 (5) 10171040.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Bonsangue, M.M., Kurz, A. and Rewitzky, I.M. (2007). Coalgebraic representations of distributive lattices with operators. Topology and its Applications 154 (4) 778791.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Chen, L.-T. (2014). On a purely categorical framework for coalgebraic modal logic. PhD thesis, University of Birmingham.Google Scholar
Cignoli, R., Lafalce, S. and Petrovich, A. (1991). Remarks on Priestley duality for distributive lattices. Order 8 (3) 299315.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Doberkat, E.E. (2009). Stochastic coalgebraic logic, Monographs in Theoretical Computer Science, An EATCS Series, Springer.CrossRefGoogle Scholar
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.Google Scholar
Engelking, R. (1989). General Topology, Sigma Series in Pure Mathematics, 2nd ed. Heldermann Verlag, Berlin.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory – Selected Topics in Point-Set Topology, New Mathematical Monographs, vol. 22, Cambridge University Press.CrossRefGoogle Scholar
Gumm, P.H. and Schröder, T. (2001). Products of coalgebras. Algebra Universalis 46 (1) 163185.CrossRefGoogle Scholar
Hausdorff, F. (1914). Grundzüge der Mengenlehre, Veit & Comp., Leipzig.Google Scholar
Hasuo, I. and Jacobs, B. (2011). Traces for coalgebraic components. Mathematical Structures in Computer Science 21 (2) 267320.CrossRefGoogle Scholar
Hofmann, D. (1999). Natürliche Dualitäten und das verallgemeinert Stone-Weierstraß Theorem. PhD thesis, University of Bremen.Google Scholar
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.Google Scholar
Hughes, J. (2001). A study of categories of algebras and coalgebras. PhD thesis, Carnegie Mellon University.Google Scholar
Jacobs, B. (2016). Introduction to Coalgebra: Towards Mathematics of States and Observations, Cambridge University Press.Google Scholar
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.Google Scholar
Kelley, J. (1975). General Topology, Nostrand, Van. Reprinted by Springer-Verlag, Graduate Texts in Mathematics, 27.Google Scholar
Kock, A. (1972). Strong functors and monoidal monads. Archiv der Mathematik 23 (1) 113120.CrossRefGoogle Scholar
Kupke, C., Kurz, A. and Venema, Y. (2004). Stone coalgebras. Theoretical Computer Science 327 (1–2) 109134.CrossRefGoogle Scholar
Kurz, A. (2001). Logics for coalgebras and applications to computer science, BoD–Books on Demand.Google Scholar
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.CrossRefGoogle Scholar
MacLane, S. (1971). Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York.Google Scholar
Manes, E.G. (2002). Taut monads and T0-spaces. Theoretical Computer Science 275 (1–2) 79109.CrossRefGoogle Scholar
Michael, E. (1951). Topologies on spaces of subsets. Transactions of the American Mathematical Society 71 (1) 152182.CrossRefGoogle Scholar
Möbus, A. (1983). Alexandrov compactification of relational algebras. Archiv der Mathematik 40 (6) 526537.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Neves, R. and Barbosa, L.S. (2017). Languages and models for hybrid automata: A coalgebraic perspective. Theoretical Computer Science. DOI: 10.1016/j.tcs.2017.09.038CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Panangaden, P. (2009). Labelled Markov Processes, Imperial College Press.CrossRefGoogle Scholar
Petrovich, A. (1996). Distributive lattices with an operator. Studia Logica 56 (1–2) 205224. Special issue on Priestley duality.CrossRefGoogle Scholar
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.Google Scholar
Power, J. and Watanabe, H. (1998). An axiomatics for categories of coalgebras. Electronic Notes in Theoretical Computer Science 11 158175.CrossRefGoogle Scholar
Rutten, J.J.M.M. (2000). Universal coalgebra: A theory of systems. Theoretical Computer Science 249 (1) 380.CrossRefGoogle Scholar
Schalk, A. (1993). Algebras for Generalized Power Constructions. PhD thesis, Technische Hochschule Darmstadt.Google Scholar
Simmons, H. (1982). A couple of triples. Topology and its Applications 13 (2) 201223.CrossRefGoogle Scholar
Stauner, T. (2001). Systematic development of hybrid systems. PhD thesis, Technische Uuniversität München.Google Scholar
Tabuada, P. (2009). Verification and Control of Hybrid Systems – A Symbolic Approach, Springer.CrossRefGoogle Scholar
Tholen, W. (2009). Ordered topological structures. Topology and its Applications 156 (12) 21482157.CrossRefGoogle Scholar
Vietoris, L. (1922). Bereiche zweiter Ordnung. Monatshefte für Mathematik und Physik 32 (1) 258280.CrossRefGoogle Scholar
Viglizzo, I.D. (2005). Coalgebras on measurable spaces. PhD thesis, Department of Mathematics, Indiana University.Google Scholar
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.Google Scholar
Waterhouse, W.C. (1972). An empty inverse limit. Proceedings of the American Mathematical Society 36 (2) 618.Google Scholar
Zenor, P. (1970). On the completeness of the space of compact subsets. Proceedings of the American Mathematical Society 26 (1) 190192.CrossRefGoogle Scholar
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