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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2021
    • (edited Oct 21st 2021)

    added pointer to

    (via user varkor, here)

    diff, v10, current

    • CommentRowNumber2.
    • CommentAuthornonemenon
    • CommentTimeSep 25th 2023
    • (edited Sep 25th 2023)

    From what I can gather, sheaves on (some) Vect (with some Grothendieck topology I don’t recall) constitute a standard model for generalized smooth spaces. I still recall my alarm when I first learned that this produces a synthetic differential geometry equivalent to that obtained by taking sheaves on the category of cartesian spaces and all smooth maps between them, which I believe is called Cart when equipped with (some Grothendieck topology I don’t recall). Resolving my surprise concerning this identification was formative in developing my intuition for the various flavors of completion 2-functors such as Karoubi envelopes and Cauchy completion in the enriched context (and Yoneda etc).

    Forgive me if a flaw in my understanding has resulted in a silly question, but I am quite curious what would be obtained by extending this recipe along the analogy/abstraction [R : R-Bimod] :: [V : V-Prof]. In short: what are “spaces locally modelled on enriched categories” in the sense that tangent spaces are (enriched) categories, tensor fields are diagrams of profunctors indexed by the shape ∫ of the base, etc.

    Does anyone know if this has been addressed in the literature? I would love to read more! Thank you

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2023

    sheaves on (some) Vect\mathbf{Vect} (with some Grothendieck topology I don’t recall) constitute a standard model for generalized smooth spaces.

    Do you have a pointer to what you have in mind here?

    the category of cartesian spaces and all smooth maps between them, which I believe is called Cart\mathbf{Cart}

    Yes. I had turned attention to this category as a site for smooth spaces in Def. 1.2.197 of dcct (p. 144) and the idea stuck (previous authors had insisted on using categories of open subsets of Euclidean spaces, instead).

    • CommentRowNumber4.
    • CommentAuthornonemenon
    • CommentTimeSep 25th 2023

    Hmm. After struggling to track down where I got the idea about Vect serving as a nice site for smooth spaces, I wonder if I got confused reading an older version of CartSp or smooth set. Maybe my alarm was actually good sensibility all along, haha.

    Thank you for the reference. I will keep chewing on the relationship between Vect and Prof; right now I’m toying with slices Prof /1Prof_{/1} as analogs of Vect /k[1]AffVect_{/k[1]} \simeq Aff.

    • CommentRowNumber5.
    • CommentAuthoranuyts
    • CommentTimeFeb 16th 2024

    Prof = Kl(PSh)

    diff, v11, current