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nLab > Latest Changes: condensed spectrum

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  1. the (infinity,1)-category of condensed infinity-groupoids has all finite (infinity,1)-limits, so it should have spectrum objects.

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorjbian
    • CommentTimeMay 29th 2022
    • (edited May 29th 2022)
    There are two things one can consider:
    1. sheaves of spectra on the pro-etale site. (condensed X is sheaves of X on proetale site, for X = Sp)
    2. stabilization of the category of condensed anima. (the approach currently in this entry)

    It’s actually not clear that the two are equivalent. One potential issue is that condensed anima don’t form an infinity topos due to subtle set-theoretic reasons, so the usual identification of sheaves of spectra on a topos & its stabilization doesn’t immediately apply here.

    A more severe issue is that one would certainly like to equip condensed spectra with a smash product - e.g. in order to study condensed ring spectra and modules. In approach 1 this is the obvious point-wise symmetric monoidal structure. In approach 2, one has to actually construct a smash product. This isn’t straightforward - you can (Goodwillie-)differentiate the cartesian monoidal structure on condensed anima to get “multiplication maps” on the stabilization, but this is a priori just a stable infinity operad, and not a symmetric monoidal structure - there’s something non-trivial to prove here.

    So in short, approach 1 is a simpler definition and comes for free with a smash product. Approach 2 (what’s in this article) is potentially equivalent, but there’s some non-trivial things to prove even if it does turn out to be equivalent.

    Let me also mention that I think it’s very worthwhile spending time thinking about stabilizations of these categories of condensed objects. For example when you localize to rational or v_n periodic spaces these stabilzation adjunctions give you algebraic models for spaces - it’s certainly interesting to ask to what degree these hold in the condensed setting. In a slightly orthogonal direction thinking about stabilization of categories of augmented condensed ring spectra allows one to set up deformation theory in this setting. So the spirit of the approach taken in these articles is certainly worth exploring, but it’s also probably good to have simple & correct definitions of the basic objects in these articles :)

  3. added ideas section

    Anonymous

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2022

    I have expanded out

    H-ℤ\mathbb{Z}-module spectra in derived geometry

    to

    HℤH\mathbb{Z}-module spectra in derived geometry (via the stable Dold-Kan correspondence)

    Let’s see, maybe a better lead-in to this entry could be as follows:

    By a condensed spectrum one should generally mean a spectrum in the context of condensed mathematics, hence a spectrum object internal to condensed ∞\infty-groupoids.

    A special case of this general notion has essentially been considered in [[concrete reference goes here]]: The condensed simplicial abelian groups discussed there may be understood, under the stable Dold-Kan correspondence, to represent (the condensed version of) the particular case of connective HℤH\mathbb{Z}-module spectra. These are used to/needed to/desireable for…

    diff, v3, current

  5. modified lead to Urs Schreiber suggestion

    Anonymous

    diff, v4, current

  6. fixing definition according to this comment

    Anonymous

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2022

    Thanks, that’s looking better.

    I have adjusted the second paragraph a tad more, pointing within Scholze’s lecture specifically to footnote 12, and trying to indicate that this footnote is far from discussing the definition of a condensed spectrum.

    In the same vein, I have added to the References-section the following disclaimer:

    A discussion of condensed spectra in the literature seems not to be available yet. For general background on condensed mathematics see: …

    diff, v6, current

  8. Added “hypercomplete”

    diff, v7, current

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