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nLab > Latest Changes: locally compact locale

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJul 21st 2013

  2. Added to locally compact locale a class of examples of locally compact locales which may constructively fail to be spatial, namely spectra of rings.

    In the process I noticed that our entry on the spectrum of a commutative ring needs some love. For instance, it is missing a constructively sensible construction (as a locale or directly as a topos, as referenced in this recent math.SE question), the formulation of its universal property, and a discussion of its relation to Hakim’s very general spectrum construction and to the relative spectrum construction commonly used in algebraic geometry. When time allows, I’ll have a go at these additions.

    • CommentRowNumber3.
    • CommentAuthorDean
    • CommentTimeFeb 22nd 2025
    I would like to edit this page to include references to two important results as well as, later, including the proofs from these papers:

    (1) A Categorical Proof of the Equivalence of Local Compactness and Exponentiability in Locale Theory (Christopher F. Townsend, 2006)

    http://www.numdam.org/item/?id=CTGDC_2006__47_3_233_0

    Townsend, Christopher F. A categorical proof of the equivalence of local compactness and exponentiability in locale theory. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 47 (2006) no. 3, pp. 233-239. http://www.numdam.org/item/CTGDC_2006__47_3_233_0/

    (2) Function spaces in the category of locales (Martin Hyland, 1981)

    https://wayf.springernature.com/?redirect_uri=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2FBFb0089910&search=New+York+University

    Hyland, J.M.E. (1981). Function spaces in the category of locales. In: Banaschewski, B., Hoffmann, RE. (eds) Continuous Lattices. Lecture Notes in Mathematics, vol 871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089910
    [1] Hyland, J.M.E. Function space in the category of locales. In "Continuous lattices", Lecture Notes in Math. vol. 871 (Springer-Verlag, 1981), 264-281.

    I'm not sure of the correct citation format.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2025
    • (edited Feb 22nd 2025)

    I have added the references, like this:

      * Christopher F. Townsend: *A categorical proof of the equivalence of local compactness and exponentiability in locale theory*, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]], **47** 3 (2006) 233-239 [[numdam:CTGDC_2006__47_3_233_0](http://www.numdam.org/item/CTGDC_2006__47_3_233_0)]

  * [[J. Martin E. Hyland]]: *Function spaces in the category of locales* in: *Continuous Lattices*, Lecture Notes in Mathematics **871**, Springer (1981) [[doi:10.1007/BFb0089910](https://doi.org/10.1007/BFb0089910)]

    diff, v6, current

1 to 4 of 4

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