Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: condensed spectrum

Bottom of Page

1 to 8 of 8

  1.  

  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

1 to 8 of 8

  1.