Applied Categorical Structures

, Volume 23, Issue 6, pp 829–858 | Cite as

Representable \((\mathbb {T}, V)\)-categories

  • D. Chikhladze
  • M. M. ClementinoEmail author
  • D. Hofmann


Working in the framework of \((\mathbb {T},\textbf {V})\)-categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad \(\mathbb {T}\), we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for \((\mathbb {T},\textbf {V})\)-categories.


Monad Kock-Zöberlein monad Multicategory Topological space \((\mathbb {T},\protect \textbf {V})\)-category 

Mathematics Subject Classifications (2010)

18C20 18D15 18A05 18B30 18B35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barr, M.: Relational algebras, in: Lecture Notes in Math 137, pp 39–55. Springer, Berlin (1970)Google Scholar
  2. 2.
    Betti, R., Carboni, A., Street, R., Walters, R.: Variation through enrichment. J. Pure Appl. Algebra 29, 109–127Google Scholar
  3. 3.
    Borceux, F.: Handbook of Categorical Algebra 2: Categories and Structures. Cambridge University Press, Cambridge (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chikhladze, D.: Lax formal theory of monads, monoidal approach to bicategorical structures and generalized operads (2014)Google Scholar
  5. 5.
    Clementino, M.M., Hofmann, D.: Lawvere completeness in topology. Appl. Categ. Structures 17, 175–210 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Clementino, M.M., Hofmann, D.: Relative injectivity as cocompleteness for a class of distributors. Theory Appl. Categ. 21(12), 210–230 (2008)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Clementino, M.M., Tholen, W.: Metric, topology and multicategory – a common approach. J. Pure Appl. Algebra 179, 13–47 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    EscardĂł, M.H., Flagg, R.: Semantic domains, injective spaces and monads. In: Mathematical foundations of programming semantics. Proceedings of the 15th conference, Tulane Univ., New Orleans, LA, April 28 - May 1, 1999. Amsterdam: Elsevier, Electronic Notes in Theoretical Computer Science 20, electronic paper No.15 (1999)Google Scholar
  9. 9.
    Flagg, R.C.: Algebraic theories of compact pospaces. Topology Appl. 77(3), 277–290 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: A compendium of continuous lattices. Springer-Verlag, Berlin (1980)zbMATHCrossRefGoogle Scholar
  11. 11.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J., Mislove, M.W., Scott, D.: Continuous lattices and domains, volume 93 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  12. 12.
    Hermida, C.: Representable multicategories. Adv. Math. 151(2), 164–225 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Hermida, C.: From coherent structures to universal properties. J. Pure Appl. Algebra 165(1), 7–61 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hochster, M.: Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142, 43–60 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Hofmann, D.: Topological theories and closed objects. Adv. Math. 215(2), 789–824 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Hofmann, D.: Injective spaces via adjunction. J. Pure Appl. Algebra 215(3), 283–302 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Hofmann, D.: Duality for distributive spaces. Theory Appl. Categ. 28(3), 66–122 (2013)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Hofmann, D., Seal, G., Tholen, W. (eds.): Monoidal Topology. A Categorical Approach to Order, Metric and Topology, Encyclopedia Math. Appl. 153, Cambridge Univ. Press (2014)Google Scholar
  19. 19.
    Jung, A.: Stably compact spaces and the probabilistic powerspace construction. In: Desharnais, J., Panangaden, P. (eds.) Domain-theoretic methods in probabilistic processes, Vol. 87, p 2004Google Scholar
  20. 20.
    Kelly, G.M.: Basic concepts of enriched category theory, London Math. Soc. Lecture Notes 64. Cambridge University Press, Cambridge (1982)Google Scholar
  21. 21.
    Kelly, G.M., Lack, S.: On property-like structures. Theory Appl. Categ. 3, 213–250 (1997)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Kock, A.: Monads for which structures are adjoint to units. J. Pure Appl. Algebra 104(1), 41–59 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lawson, J.: Stably compact spaces. Math. Structures Comput. Sci. 21, 125–169 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Lawvere, F.W.: Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43, 135–166 (1973)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nachbin, L.: Topologia e Ordem. Univ. of Chicago Press, 1950. In Portuguese, English translation: Topology and Order, Van Nostrand, Princeton (1965)Google Scholar
  26. 26.
    Priestley, H.A.: Representation of distributive lattices by means of ordered stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)Google Scholar
  27. 27.
    Rosebrugh, R.D., Wood, R.J.: Distributive laws and factorization. J. Pure Appl. Algebra 175(1–3), 327–353 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Simmons, H.: A couple of triples. Topology Appl. 13, 201–223 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Stone, M.H.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40(1), 37–111 (1936)MathSciNetGoogle Scholar
  30. 30.
    Stone, M.H.: Topological representations of distributive lattices and Brouwerian logics. Časopis pro pěstování matematiky a fysiky 67(1), 1–25 (1938)Google Scholar
  31. 31.
    Street, R.: Formal theory of monads. J. Pure Appl. Algebra 2, 149–168 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Tholen, W.: Ordered topological structures. Topology Appl. 156, 2148–2157 (2009)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • D. Chikhladze
    • 1
  • M. M. Clementino
    • 1
    Email author
  • D. Hofmann
    • 2
  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2.CIDMA – Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal

Personalised recommendations