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nLab > Latest Changes: condensed E-infinity ring

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    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2022

    If possible, please add some indication on where this is headed; Some construction or further development where it’s important that the E ∞E_\infty-rings are condensed.

    A reference would be great, for instance, where one might see condensed E ∞E_\infty-rings in action.

    Without any such indication, it feels a little hollow to churn out entries on “condensed Xs” as X varies.

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeMay 28th 2022

    I’m not sure what the motivation for the original author was, but Peter Scholze writes on page 44 in his Lectures on condensed mathematics about his definition of a type of condensed ring called pre-analytic rings:

    This definition is in some ways preliminary. In particular, one should allow the pre-analytic ring to be “derived”, i.e. AA may be a simplicial ring and A[S]A[S] a simplicial module (as usual, simplicial objects are considered here as objects of the appropriate ∞\infty-category, obtained by inverting homotopy equivalences). Only with this extra flexibility a good general theory can be developed. In that generality, one loses the property that the relevant “derived categories” are actually the derived category of their heart, as usual in the context of modules over simplicial rings; the rest of the discussion below immediately extends.

    However, simplicial abelian groups model connective spectra in ∞Grpd\infty Grpd if I remember correctly, rather than general spectra, so I think the simplicial rings considered here by Scholze only model connective A ∞A_\infty-rings. Similarly with the simplicial modules.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2022

    I’m not sure what the motivation for the original author was

    What’s your motivation with the entry (condensed E-infinity ring)? Do you have an example in mind? An application? Some comment on the definition? Any of that would be good to add to the entry! Even just an outlook would be good. Maybe one wants to announce that all of XYZ theory is eventually better done with condensed E ∞E_\infty-rings? I have some vague ideas about that, but I’d be interested in hearing what you think is the case.

    However, simplicial abelian groups model connective spectra

    Connectivity is the least worry here, one can pass to the stable Dold-Kan correspondence to go unbounded. But no matter if connective or not, simplicial abelian groups capture only the HℤH \mathbb{Z}-module spectra, and this are equivalently just the chain complexes known from homological algebra.

    (The reason why Lurie’s “Derived Algebraic Geometry” program was eventually renamed to “Spectral Algebraic Geometry” was, I gather, to better highlight that spectra are so much richer than dg- and simplicial abelian structures.)

  5. added ideas section

    Anonymous

    diff, v2, current

  6. fixing definition according to this comment

    Anonymous

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2022

    Same discussion applies as we had in the other thread here.

    I have added a first sentence of the form we had agreed there, and I have removed this line:

    While Peter Scholze’s presentations of condensed mathematics only talks about H-ℤ\mathbb{Z}-module spectra in derived geometry, theoretically one should be able to extend it to general spectra in spectral geometry.

    Because it seems that Peter’s lecture never mentions HℤH\mathbb{Z}-module spectra! In but a footnote it vaguely mentions simplicial abelian groups. From there I had pointed out to you that these may be thought of as presenting HℤH\mathbb{Z}-module spectra.

    diff, v4, current

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