Injective Hulls of Quantale-Enriched Multicategories

A Publisher Correction to this article was published on 23 July 2021

This article has been updated


In this communication we generalize some recent results of Rump to categories enriched in a commutative quantale V. Using these results, we show that every quantale-enriched multicategory admits an injective hull. Finally, we expose a connection between the Isbell adjunction and the construction of injective hulls for topological spaces made by Banaschewski in 1973.

This is a preview of subscription content, access via your institution.

Change history


  1. 1.

    In particular \(a((-), y) = a(u_X, y)\).


  1. 1.

    Banaschewski, B.: The filter space of a lattice: Its role in general topology. In: Proc. Univ. Houston, Lattice Theory Conference, Houston, pp. 147–155 (1973)

  2. 2.

    Barr, M.: Relational algebras. In: MacLane, S. (ed.) Reports of the Midwest Category Seminar IV, pages 39–55. Springer Berlin Heidelberg, (1970). Authors: H. Applegate, M. Barr, B. Day, E. Dubuc, Phreilambud, A. Pultr, R. Street, M. Tierney, S. Swierczkowski

  3. 3.

    Bénabou, J.: Les distributeurs. Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, rapport 33, (1973)

  4. 4.

    Betti, R., Carboni, A., Street, R., Walters, R.: Variation through enrichment. J. Pure Appl. Algebra 29(2), 109–127 (1983)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chikhladze, D., Clementino, M.M., Hofmann, D.: Representable \((T, V)\)-categories. Appl. Categ. Struct. 23(6), 829–858 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Clementino, M.M., Hofmann, D.: Lawvere completeness in topology. Appl. Categ. Struct. 17(2), 175–210 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Clementino, M.M., Hofmann, D.: Relative injectivity as cocompleteness for a class of distributors. Theory Appl. Categ. 21(12), 210–230 (2009)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Clementino, M.M., Tholen, W.: Metric, topology and multicategory a common approach. J. Pure Appl. Algebra 179(1), 13–47 (2003)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Day, B.: Construction of biclosed categories. Bull. Aust. Math. Soc. 5(1), 139–140 (1971)

    Article  Google Scholar 

  10. 10.

    Day, B., Street, R.: Kan extensions along promonoidal functors. Theory Appl. Categ. 1,(1995)

  11. 11.

    Erné, M.: The ABC of order and topology. In: Herrlich, H., Porst, H.-E. (eds.) Category Theory at Work, volume 18 of Research and Exposition in Mathematics, pp. 57–83. Heldermann Verlag, Berlin (1991). With Cartoons by Marcel Erné

  12. 12.

    Escardó, M.H.: Injective spaces via the filter monad. In: Proceedings of the \(12^{{\rm th}}\) Summer Conference on General Topology and its Applications (North Bay, ON, 1997), vol. 22, pp. 97–100 (1997)

  13. 13.

    Escardó, M.H.: Properly injective spaces and function spaces. Topol. Appl. 89(1–2), 75–120 (1998)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Escardó, M.H., Flagg, R.C.: Semantic domains, injective spaces and monads. Electron. Notes Theor. Comput. Sci. 20, 229–244 (1999)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Flagg, R.C.: Quantales and continuity spaces. Algebra Universalis 37(3), 257–276 (1997)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hofmann, D., Seal, G.J., Tholen, W.: editors. Monoidal Topology. A Categorical Approach to Order, Metric, and Topology, volume 153 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, (July 2014). Authors: M. M. Clementino, E. Colebunders, D. Hofmann, R. Lowen, R. Lucyshyn-Wright, G. J. Seal and W. Tholen

  17. 17.

    Hoffmann, R.E.: The injective hull and the \({\cal{C}}{\cal{L}}\)-compactification of a continuous poset. Can. J. Math. 37(5), 810–853 (1985)

    Article  Google Scholar 

  18. 18.

    Hofmann, D.: Injective spaces via adjunction. J. Pure Appl. Algebra 215(3), 283–302 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hofmann, D.: A four for the price of one duality principle for distributive spaces. Order 30(2), 643–655 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hofmann, D.: The enriched Vietoris monad on representable spaces. J. Pure Appl. Algebra 218(12), 2274–2318 (2014)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hofmann, D., Tholen, W.: Lawvere completion and separation via closure. Appl. Categ. Struct. 18(3), 259–287 (2010)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Hofmann, K.H., Mislove, M.W.: A continuous poset whose compactification is not a continuous poset. The square is the injective hull of a discontinuous CL-compact poset. Seminar on Continuity in Semilattices, Technische Universität Darmstadt, memo (1982)

  23. 23.

    Joyal, A., Tierney, M.: An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc., 51(309):vii+71, (1984)

  24. 24.

    Kelly, G.M.: Basic concepts of enriched category theory, volume 64 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1982. Republished in: Reprints in Theory and Applications of Categories. No. 10, pp. 1–136 (2005)

  25. 25.

    Kock, A.: Monads for which structures are adjoint to units. J. Pure Appl. Algebra 104(1), 41–59 (1995)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Lambek, J.: Deductive systems and categories II. standard constructions and closed categories. In: Hilton, P. J. (ed.), Category Theory, Homology Theory and their Applications I, pp. 76–122. Springer, Berlin (1969)

  27. 27.

    Lambek, J., Barr, M., Kennison, J.F., Raphael, R.: Injective hulls of partially ordered monoids. Theory Appl. Categ. 26(13), 338–348 (2012)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Lawvere, F.W., Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano, 43(1), 135–166, : Republished. Reprints in Theory and Applications of Categories, No. 1(2002), 1–37 (Dec. 1973)

  29. 29.

    Leinster, T.: Higher Operads. Cambridge University Press, Higher Categories (2004)

    Book  Google Scholar 

  30. 30.

    MacNeille, H.M.: Partially ordered sets. Trans. Am. Math. Soc. 42(3), 416–416 (1937)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Restall, G.: Relevant and substructural logics. In: Gabbay, D.M., Woods, J. (eds.), Logic and the Modalities in the Twentieth Century, volume 7 of Handbook of the History of Logic, pp. 289–398. North-Holland, (2006)

  32. 32.

    Rump, W.: Quantum B-algebras. Central Eur. J. Math. 11(11), 1881–1899 (2013)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Rump, W.: The completion of a quantum B-algebra. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. LVII-3 (2016), (2016)

  34. 34.

    Rump, W., Yang, Y.C.: Non-commutative logical algebras and algebraic quantales. Ann. Pure Appl. Logic 165(2), 759–785 (2014)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Scott, D.: Continuous lattices. In Lawvere, F.W. (ed.), Toposes, Algebraic Geometry and Logic, volume 274 of Lecture Notes in Mathematics, pp. 97–136. Springer, 1972. Dalhousie University, Halifax, January 16–19 (1971)

  36. 36.

    Stubbe, I.: Categorical structures enriched in a quantaloid: categories, distributors and functors. Theory Appl. Categ. 14(1), 1–45 (2005)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Stubbe, I.: An introduction to quantaloid-enriched categories. Fuzzy Sets Syst. 256, 95–116 (2014)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Wood, R.J.: Abstract pro arrows I. Cahiers de Topologie et Géométrie Différentielle Catégoriques 23(3), 279–290 (1982)

    Google Scholar 

  39. 39.

    Wood, R.J.: Proarrows II. Cahiers de Topologie et Géométrie Différentielle Catégoriques 26(2), 135–168 (1985)

    MathSciNet  MATH  Google Scholar 

Download references


I am grateful to D. Hofmann for valuable discussions about the content of the paper and to I. Stubbe for the valuable suggestions he gave me during his visit to Aveiro. The author would like to thank the anonymous referee who kindly reviewed the earlier version of this manuscript and provided valuable suggestions and comments which lead to an overall improvement of the paper.

The author acknowledges partial financial assistance by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project POCI-01-0145-FEDER-030947, and project UID/MAT/04106/2019 (CIDMA). The author is also supported by FCT grant PD/BD/128187/2016.

Author information



Corresponding author

Correspondence to Eros Martinelli.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original online version of this article was revised: An author contribution section that was inadvertently included in the original article was removed.

Communicated by Maria Manuel Clementino.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Martinelli, E. Injective Hulls of Quantale-Enriched Multicategories. Appl Categor Struct (2021).

Download citation


  • Quantales
  • Quantale-enriched categories
  • Injective objects
  • Injective hulls
  • Quantale-enriched multicategories