, Volume 26, Issue 2, pp 177–196 | Cite as

Modules on Involutive Quantales: Canonical Hilbert Structure, Applications to Sheaf Theory



We explain the precise relationship between two module-theoretic descriptions of sheaves on an involutive quantale, namely the description via so-called Hilbert structures on modules and that via so-called principally generated modules. For a principally generated module satisfying a suitable symmetry condition we observe the existence of a canonical Hilbert structure. We prove that, when working over a modular quantal frame, a module bears a Hilbert structure if and only if it is principally generated and symmetric, in which case its Hilbert structure is necessarily the canonical one. We indicate applications to sheaves on locales, on quantal frames and even on sites.


Quantale Module Principal element Principal symmetry Inner product Sheaf 


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  1. 1.
    Betti, R., Carboni, A.: Notion of topology for bicategories. Cah. Topol. Géom. Differ. 24, 19–22 (1983)MATHMathSciNetGoogle Scholar
  2. 2.
    Freyd, P.J., Scedrov, A.: Categories, Allegories, vol. 39. North-Holland Mathematical Library, Amsterdam (1990)Google Scholar
  3. 3.
    Heymans, H., Stubbe, I.: On principally generated Q-modules in general, and skew local homeomorphisms in particular. Ann. Pure Appl. Logic, in press (2009). doi: 10.1016/j.apal.2009.05.001 MATHGoogle Scholar
  4. 4.
    Kelly, G.M.: Basic Concepts of Enriched Category Theory. Cambridge University Press, Cambridge (1982) (Also available as: Reprints in Theory Appl. Categ. 10, 2005)MATHGoogle Scholar
  5. 5.
    Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic. Springer, New York (1992)MATHGoogle Scholar
  6. 6.
    Mesablishvili, B.: Every small Sl-enriched category is Morita equivalent to an Sl-monoid. Theory Appl. Categ. 13, 169–171 (2004)MATHMathSciNetGoogle Scholar
  7. 7.
    Mulvey, C.J., Pelletier, J.W.: A quantisation of the calculus of relations. Can. Math. Soc. Conf. Proc. 13, 345–360 (1992)MathSciNetGoogle Scholar
  8. 8.
    Paseka, J.: Hilbert Q-modules and nuclear ideals in the category of \(\bigvee\)-semilattices with a duality. Electron. Notes Theor. Comput. Sci. 29, 1–19 (1999)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Paseka, J.: Morita equivalence in the context of Hilbert modules. In: Proceedings of the Ninth Prague Topological Symposium (2001). Topology Atlas, North Bay, pp. 223–251 (2002)Google Scholar
  10. 10.
    Paseka, J.: Interior tensor product of Hilbert modules. Contributions to General Algebra 13 (Velké Karlovice 1999/Dresden 2000). Heyn, Klagenfurt, pp. 253–263 (2003)Google Scholar
  11. 11.
    Resende, P.: Etale groupoids and their quantales. Adv. Math. 208, 147–209 (2007)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Resende, P.: Groupoid Sheaves as Hilbert modules. Preprint available on the arXiv:0807.4848v1 (2008)
  13. 13.
    Resende, P., Rodrigues, E.: Sheaves as modules. Appl. Categ. Structures, on-line. (2009). doi: 10.1007/s10485-008-9131-x Google Scholar
  14. 14.
    Rosenthal, K.I.: The Theory of Quantaloids, Pitman Research Notes in Mathematics Series 348. Longman, Harlow (1996)Google Scholar
  15. 15.
    Stubbe, I.: Categorical structures enriched in a quantaloid: categories, distributors and functors. Theory Appl. Categ. 14, 1–45 (2005a)MATHMathSciNetGoogle Scholar
  16. 16.
    Stubbe, I.: Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid. Appl. Categ. Struct. 13, 235–255 (2005b)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Stubbe, I.: Categorical structures enriched in a quantaloid: tensored and cotensored categories. Theory Appl. Categ. 16, 283–306 (2006)MATHMathSciNetGoogle Scholar
  18. 18.
    Stubbe, I.: \({\cal Q}\)-modules are \({\cal Q}\)-suplattices. Theory Appl. Categ. 19, 50–60 (2007)MATHMathSciNetGoogle Scholar
  19. 19.
    Walters, R.F.C.: Sheaves on sites as cauchy complete categories. J. Pure Appl. Algebra 24, 95–102 (1982)MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of AntwerpAntwerpBelgium

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