Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-31T10:50:24.788Z Has data issue: false hasContentIssue false
  • English
  • Français

The formal ball model for -categories

Published online by Cambridge University Press:  02 December 2010

MATEUSZ KOSTANEK  and
PAWEŁ WASZKIEWICZ
MATEUSZ KOSTANEK
Affiliation:
Theoretical Computer Science, Jagiellonian University, ul. S. Łojasiewicza 6, 30-348 Kraków, Poland Email: pqw@tcs.uj.edu.pl.
PAWEŁ WASZKIEWICZ
Affiliation:
Theoretical Computer Science, Jagiellonian University, ul. S. Łojasiewicza 6, 30-348 Kraków, Poland Email: pqw@tcs.uj.edu.pl.
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalise the construction of the formal ball model for metric spaces due to A. Edalat and R. Heckmann in order to obtain computational models for separated -categories. We fully describe -categories that are

  1. (a) Yoneda complete

  2. (b) continuous Yoneda complete

via their formal ball models. Our results yield solutions to two open problems in the theory of quasi-metric spaces by showing that:

  1. (a) a quasi-metric space X is Yoneda complete if and only if its formal ball model is a dcpo, and

  2. (b) a quasi-metric space X is continuous and Yoneda complete if and only if its formal ball model BX is a domain that admits a simple characterisation of approximation.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Issue 1 , February 2011 , pp. 41 - 64
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramsky, S. and Jung, A. (1994) Domain Theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science 3, Oxford University Press 1–168.Google Scholar
Albert, M. H. and Kelly, G. M. (1998) The closure of a class of colimits. Journal of Pure and Applied Algebra 51 1–17.CrossRefGoogle Scholar
Ali-Akbari, M., Honari, B., Pourmahdian, M. and Rezaii, M. M. (2009) The space of formal balls and models of quasi-metric spaces. Mathematical Structures in Computer Science 19 337–355.CrossRefGoogle Scholar
America, P. and Rutten, J. J. M. M. (1989) Solving Reflexive Domain Equations in a Category of Complete Metric Spaces. Journal of Computer and System Sciences 39 (3)343–375.CrossRefGoogle Scholar
Bonsangue, M. M., van Breugel, F. and Rutten, J. J. M. M. (1998) Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding. Theoretical Computer Science 193 (1–2)1–51.CrossRefGoogle Scholar
Clementino, M. M. and Hofmann, D. (2003) Topological Features of Lax Algebras. Applied Categorical Structures 11 267–286.CrossRefGoogle Scholar
Clementino, M. M. and Hofmann, D. (2008) Relative injectivity as cocompleteness for a class of distributors. Theory and Applications of Categories 21 210–230.Google Scholar
Clementino, M. M. and Tholen, W. (2003) Metric, Topology and Multicategory – a Common Approach. Journal of Pure and Applied Algebra 179 13–47.Google Scholar
Clementino, M. M., Hofmann, D. and Tholen, W. (2004) One Setting for All: Metric, Topology, Uniformity, Approach Structure. Applied Categorical Structures 12 (2)127–154.CrossRefGoogle Scholar
Clementino, M. M. and Hofmann, D. (2009) Lawvere completeness in Topology. Applied Categorical Structures 17 175–210 (available at arXiv:math.CT/0704.3976).CrossRefGoogle Scholar
Edalat, A. (1995a) Domain theory and integration. Theoretical Computer Science 151 163–193.CrossRefGoogle Scholar
Edalat, A. (1995b) Dynamical systems, measures and fractals via domain theory. Information and Computation 120 (1)32–48.Google Scholar
Edalat, A. and Heckmann, R. (1998) A computational model for metric spaces. Theoretical Computer Science 193 (1–2)53–73.Google Scholar
Edalat, A., Krznarić, M. and Lieutier, A. (2003) Domain-theoretic solution of differential equations (scalar fields). In Proceedings of MFPS XIX. Electronic Notes in Theoretical Computer Science 83.Google Scholar
Edalat, A. and Lieutier, A. (2004) Domain theory and differential calculus (functions of one variable). Mathematical Structures in Computer Science 14 (6)771–802.CrossRefGoogle Scholar
Flagg, R. C. (1997) Quantales and Continuity Spaces. Algebra Universalis 37 257–276.CrossRefGoogle Scholar
Flagg, R. C. and Kopperman, R. (1995) Fixed Points and Reflexive Domain Equations in Categories of Continuity Spaces. Electronic Notes in Theoretical Computer Science 1.Google Scholar
Flagg, R. C. and Kopperman, R. (1997) Continuity Spaces: Reconciling Domains and Metric Spaces. Theoretical Computer Science 177 (1)111–138.Google Scholar
Flagg, R. C., Sünderhauf, P. and Wagner, K. R. (1996) A Logical Approach to Quantitative Domain Theory. Topology Atlas Preprint no. 23, available on-line at: http://at.yorku.ca/e/a/p/p/23.htm.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous lattices and domains. Encyclopedia of mathematics and its applications 93.Google Scholar
Hofmann, D. (2007) Topological Theories and Closed Objects. Advances in Mathematics 215 789–824.Google Scholar
Kelly, G. M. (1982) Basic concepts of enriched category theory. London Mathematical Society Lecture Note Series 64, Cambridge University Press. (Also: Reprints in Theory and Applications of Categories 10 (2005).)Google Scholar
Kelly, G. M. and Lack, S. (2000) On the Monadicity of Categories with Chosen Colimits. Theory and Applications of Categories 7 148–170.Google Scholar
Kelly, G. M. and Schmitt, V. (2005) Notes on Enriched Categories with Colimits of Some Class. Theory and Applications of Categories 14 399–423.Google Scholar
Kock, A. (1995) Monads for which Structures are Adjoint to Units. Journal of Pure and Applied Algebra 104 41–59.Google Scholar
Kostanek, M. and Waszkiewicz, P. (2010) The limit–colimit coincidence theorem for -categories. Mathematical Structures in Computer Science 20 267–284.Google Scholar
Künzi, H-P. A. and Schellekens, M. P. (2002) On the Yoneda completion of a quasi-metric space. Theoretical Computer Science 278 (1–2)159–194.CrossRefGoogle Scholar
Lai, H. and Zhang, D. (2007) Complete and directed complete Omega-categories. Theoretical Computer Science 388 (1–3)1–25.CrossRefGoogle Scholar
Lawson, J. D. (1997) Spaces of maximal points. Mathematical Structures in Computer Science 7 543–555.CrossRefGoogle Scholar
Lawson, J. D. (1998) Computation on Metric Spaces via Domain Theory. Topology and Its Applications 85 247–263.Google Scholar
Lawson, J. D. (2008) Metric Spaces and FS-domains. Theoretical Computer Science 405 73–74.Google Scholar
Lawvere, F. W. (1973) Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43 135–166. (Also reprinted in Theory and Applications of Categories 11–37 (2002).)CrossRefGoogle Scholar
Romaguera, S. and Valero, O. (2009) A quantitative computational model for complete metric spaces via formal balls. Mathematical Structures in Computer Science 19 541–563.Google Scholar
Romaguera, S. and Valero, O. (2010) New results on formal balls for quasi-metric spaces – preprint.Google Scholar
Rutten, J. J. M. M. (1996) Elements of Generalized Ultrametric Domain Theory. Theoretical Computer Science 170 349–381.CrossRefGoogle Scholar
Rutten, J. J. M. M. (1998) Weighted colimits and formal balls in generalized metric spaces. Topology and its Applications 89 179–202.Google Scholar
Schmitt, V. (2006) Flatness, Preorders and General Metric Spaces. To appear in Georgian Mathematical Journal (available at arXiv:math.CT/0602463).Google Scholar
Vickers, S. (2005) Localic Completion of Generalized Metric Spaces I. Theory and Applications of Categories 14 328–356.Google Scholar
Wagner, K. R. (1994) Solving Recursive Domain Equations with Enriched Categories, Ph.D. Thesis, Carnegie Mellon University.Google Scholar
Waszkiewicz, P. (2009) On domain theory over Girard quantales. Fundamenta Informaticae 92 1–24.CrossRefGoogle Scholar
Zhang, Q.-Y. and Fan, L. (2005) Continuity in Quantitative Domains. Fuzzy Sets and Systems 154 (1)118–131.CrossRefGoogle Scholar