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nLab > Latest Changes: Gabriel–Ulmer duality

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  1. Fixed a dead link to the Lack-Power paper

    Anonymous

    diff, v14, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 9th 2019

    I added the recent

    which considers this and related dualities in the enriched setting.

    diff, v15, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 9th 2019

    If we may think of ordinary Gabriel-Ulmer duality as operating between essentially algebraic theories and their categories of models, how should we think of the enriched version between 𝒱−Lex\mathcal{V}-\mathbf{Lex}, the 2-category of finitely complete 𝒱\mathcal{V}-categories (𝒱\mathcal{V}-categories with finite weighted limits), finite limit preserving 𝒱\mathcal{V}-functors, and 𝒱\mathcal{V}-natural transformations, and 𝒱−Lfp\mathcal{V}-\mathbf{Lfp}, the 2-category of locally finitely presentable 𝒱\mathcal{V}-categories, right adjoint 𝒱\mathcal{V}-functors that preserve filtered colimits, and 𝒱\mathcal{V}-natural transformations?

    Can I think of a finitely complete 𝒱\mathcal{V}-category as a kind of theory?

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 9th 2019
    • (edited Jul 9th 2019)

    So in the case where 𝒱\mathcal{V} is the reals or the real interval, i.e., something along the lines of a Lawvere metric space, there appears to some connection to continuous logic

    • Simon Cho, Categorical semantics of metric spaces and continuous logic, (arXiv:1901.09077)

    with a “continuous subobject classifier”.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJul 9th 2019

    Re: #3: yes? (-:

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 9th 2019

    Admittedly it may not be a very “syntactic” kind of theory…

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2019

    Gave an instance of the enriched version – meet semilattices and algebraic lattices.

    diff, v19, current

    • CommentRowNumber8.
    • CommentAuthorvarkor
    • CommentTimeMar 14th 2023

    Use en-dash in page name.

    diff, v25, current

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 24th 2023

    I thought I’d alerted the nForum to an earlier edit

    For an extension of Gabriel–Ulmer duality to a duality between Cauchy-complete clans and locally finitely presentable categories equipped with a well-behaved kind of weak factorization system see

    (Hence the start of clan.)

    diff, v28, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2024

    added missing publication data to several of the references, and added pointer to:

    diff, v31, current

  11. The linked file is an absolute mess and is not ready to be referenced for any reasonable purposes.

    Ivan

    diff, v32, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2024

    have touched the formatting of the Idea-section, for better clarity

    diff, v33, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2024

    re #11 (almost missed this, as our messages overlapped): Thanks for saying, I admit that I took only the briefest look at it, should have checked more closely.

1 to 13 of 13

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