Skip to main content
SpringerLink
Log in

Characterization of a category for monoidal topology

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

This paper characterizes one of the categories for monoidal topology of M. M. Clementino, D. Hofmann, G. J. Seal, and W. Tholen in terms of the Sierpinski object of E. G. Manes. In particular, we describe the categories of preordered sets and premetric spaces (in the sense of F. W. Lawvere) in terms of modules over a quantale.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats. Dover Mineola, New York (2009)

  2. Barr, M.: Relational algebras. In: Reports of the Midwest Category Seminar, vol. IV, Lect. Notes Math. 137, pp. 39–55 (1970)

  3. Bělohlávek R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Logic 128, 277–298 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang C.L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24, 182–190 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clementino M.M., Hofmann D.: Topological features of lax algebras. Appl. Categ. Structures 11, 267–286 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Clementino M.M., Hofmann D., Tholen W.: One setting for all: Metric, topology, uniformity, approach structure. Appl. Categ. Struct. 12, 127–154 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clementino M.M., Tholen W.: Metric, topology and multicategory—a common approach. J. Pure Appl. Algebra 179, 13–47 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clementino M.M., Tholen W.: Proper maps for lax algebras and the Kuratowski–Mrówka theorem. Theory Appl. Categ. 27, 327–346 (2013)

    MathSciNet  Google Scholar 

  9. Diers Y.: Categories of algebraic sets. Appl. Categ. Struct. 4, 329–341 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Diers Y.: Affine algebraic sets relative to an algebraic theory. J. Geom. 65, 54–76 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Erné M., Koslowski J., Melton A., Strecker G.E.: A primer on Galois connections. Ann. N. Y. Acad. Sci. 704, 103–125 (1993)

    Article  Google Scholar 

  12. Hofmann, D., Seal, G.J., Tholen, W. (eds.): Monoidal Topology: A Categorical Approach to Order, Metric and Topology. Cambridge University Press (2014)

  13. Hofmann D., Tholen W.: Lax algebra meets topology. Topology Appl. 159, 2434–2452 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Johnstone, P.T.: Stone Spaces. Cambridge University Press (1982)

  15. Kelly, G.M.: Basic Concepts of Enriched Category Theory. Cambridge University Press (1982)

  16. Kruml, D., Paseka, J.: Algebraic and Categorical Aspects of Quantales. In: M. Hazewinkel (ed.) Handbook of Algebra, vol. 5, pp. 323–362. Elsevier (2008)

  17. Lawvere F.W.: Metric spaces, generalized logic and closed categories. Repr. Theory Appl. Categ. 1, 1–37 (2002)

    MathSciNet  Google Scholar 

  18. Lowen R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56, 621–633 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lowen, R.: Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford: Clarendon Press (1997)

  20. MacDonald, J., Sobral, M.: Aspects of Monads. In: M. Pedicchio, W. Tholen (eds.) Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory, pp. 213–268. Cambridge University Press (2004)

  21. Manes E.G.: Compact Hausdorff objects. General Topology Appl. 4, 341–360 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  22. Manes, E.G.: Algebraic Theories. Springer (1976)

  23. Rosenthal, K.I.: Quantales and Their Applications. Addison Wesley Longman (1990)

  24. Seal G.J.: Canonical and op-canonical lax algebras. Theory Appl. Categ. 14, 221–243 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Solovyov, S.: On the category \({\mathcal{Q}}\)-Mod. Algebra Univers. 58, 35–58 (2008)

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2896/2, 616 69, Brno, Czech Republic

    Sergey A. Solovyov

  2. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlarska 2, 611 37, Brno, Czech Republic

    Sergey A. Solovyov

Authors
  1. Sergey A. Solovyov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey A. Solovyov.

Additional information

Presented by T. Kowalski.

Dedicated to Brian Davey on the occasion of his 65th birthday

This research was supported by the ESF Project No. CZ.1.07/2.3.00/20.0051 “Algebraic methods in Quantum Logic” of the Masaryk University in Brno, Czech Republic.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Solovyov, S.A. Characterization of a category for monoidal topology. Algebra Univers. 74, 389–410 (2015). https://doi.org/10.1007/s00012-015-0352-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-015-0352-1

2010 Mathematics Subject Classification

Keywords and phrases

Search

Navigation