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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2017
    • (edited Jun 12th 2018)

    I have created an entry spectral symmetric algebra with some basics, and with pointers to Strickland-Turner’s Hopf ring spectra and Charles Rezk’s power operations.

    In particular I have added amplification that even the case that comes out fairly trivial in ordinary algebra, namely Sym RRSym_R R is interesting here in stable homotopy theory, and similarly Sym R(Σ nR)Sym_R (\Sigma^n R).

    I am wondering about the following:

    In view of the discussion at spectral super scheme, then for RR an even periodic ring spectrum, the superpoint over RR has to be

    R 0|1=Spec(Sym RΣR)Spec(R(nBΣ(n) n) +). R^{0 \vert 1} \;=\; Spec(Sym_R \Sigma R) \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\mathbb{R}^n} \right)_+ \right) \,.

    This of course is just the base change/extension of scalars under Spec of the “absolute superpoint”

    𝕊 0|1Spec(Sym 𝕊(Σ𝕊)) \mathbb{S}^{0\vert 1} \simeq Spec(Sym_{\mathbb{S}} (\Sigma \mathbb{S}))

    (which might deserve this notation even though the sphere spectrum is of course not even periodic).

    This looks like a plausible answer to the quest that David C. and myself were on in another thread, to find a plausible candidate in spectral geometry of the ordinary superpoint 0|1\mathbb{R}^{0 \vert 1}, regarded as the base of the brane bouquet.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 16th 2017

    Unfortunately, rather than follow you on this, I have a stack of philosophy of medicine essays to mark.

    But, were one to look to start bouquet building, is there a natural choice of RR? In the ordinary case, you chose \mathbb{R} over \mathbb{C}. Does that suggest KOKO sooner than KUKU (at K-theory spectrum)? Is there an even-periodic version of MO, as MP is to MU? Do you just sum over even suspensions?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 16th 2017

    Or did I mean the MRMR of MR cohomology theory, and its possible periodic version?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2017

    Unfortunately, rather than follow you on this

    Luckily I am progressing on geological time scales with this, so you will have an easy time catching up later.

    is there a natural choice of RR

    It seems the only natural choice is R=𝕊R = \mathbb{S}. That of course is not 2-periodic. So the story might be this: down in “absolute geometry”, we start with Spec(Sym 𝕊Σ𝕊)Spec(Sym_{\mathbb{S}}\Sigma \mathbb{S}). Trying to understand this will make us want make us look at it locally, first by passing to the cover Spec(MU)Spec(𝕊)Spec(MU) \to Spec(\mathbb{S}) and then further localize to Morava E-theory Spec(E)Spec(MU)Spec(E) \to Spec(MU) or to some other complex oriented cohomology theory. The “restriction” of Spec(Sym 𝕊Σ𝕊)Spec(Sym_{\mathbb{S} \Sigma \mathbb{S}}) to this Spec(E)Spec(E) is the spectral superscheme Spec(Sym EΣE)Spec(Sym_E \Sigma E), our actual specral superpoint.

    Something like this.

    But all this is vain speculation. To make progress we need to find some actual spectral analog of a branch in the bouquet. I bet once we see this, the rest will fall into place.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2017

    The thing I understand now (whence the edits in spectral symmetric algebra) and which I did not appreciate before, is that Sym EΣESym_E \Sigma E is not a boring analog of Sym [1]Sym_{\mathbb{R}} \mathbb{R}[1], but an interesting one.

    You may remember that I had suggested Spec(Sym EΣE)Spec(Sym_E \Sigma E) in some earlier discussion here as the correct spectral superpoint, but then I said that it looks like this just yields the theory as over \mathbb{R}, with the EE coefficients just running along.

    What I didn’t realize before is that there is this coefficient of the Thom spaces of the universal bundles over the classifying spaces of the symmetric groups involved (here) coming from the fact that the permutaion action (that makes the graded-symmetric algebra SymSym) is much more interesting on smash powers of a module spectrum then on tensor powers of a vector space.

    It is easy to see some grand speculations growing out of this: If we take our ground ring EE to be MU, then

    MU 0|1=Spec(Sym MUΣMU)=Spec(MU(nBΣ(n) τ n) +) MU^{0 \vert 1} = Spec(Sym_{MU} \Sigma MU) = Spec( MU \wedge (\underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\tau_n} )_+ )

    Now since BΣ nEmbed({1,,n}, )/Σ(n)B \Sigma_n \simeq Embed(\{1,\cdots, n\}, \mathbb{R}^\infty)/\Sigma(n) and using that π 0(MUX +)\pi_0(MU \wedge X_+) is cobordism classes of closed even dimensional manifolds (with stable almost complex structure) in XX, we see that

    π 0(MU(nBΣ(n) τ n) +) \pi_0 \left( MU \wedge (\underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\tau_n} )_+\right)

    is something like a Fock space of brane worldvolumes.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 16th 2017

    To make progress we need to find some actual spectral analog of a branch in the bouquet.

    Is that largely a matter of choosing the right spectral (group scheme?) cohomology?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2017
    • (edited Mar 16th 2017)

    To make progress we need to find some actual spectral analog of a branch in the bouquet.

    Is that largely a matter of choosing the right spectral (group scheme?) cohomology?

    It’s about computing a “maximal invariant higher central extension” of one of these spectral group schemes and checking wether that proceeds at all in analogy with the rational bouquet story.

    So if we start with the spectral superpoint R 0|1=Spec(Sym EΣE)R^{0 \vert 1} = Spec( Sym_E \Sigma E), first of all we need to regard it somehow a spectral group scheme, hence regard Sym EΣESym_E \Sigma E as a Hopf ring spectrum. That ought to work as in Strickland-Turner, with the coproduct being on shifted generators θθ 1+θ 2\theta \mapsto \theta_1 + \theta_2.

    Second then we need to determine the maximal central extension of this group scheme, and then see what that has to do with the super-translation super Lie algebra 1|1\mathbb{R}^{1 \vert 1} (which is the maximal central extension of 0|1\mathbb{R}^{0 \vert 1}).

    Or maybe even better than such explicit computations would be a general argument that if we start with a spectral super group scheme which somehow corresponds to some super L L_\infty-algebra, that then its maximal higher invariant central extensions as super group schemes corresponds somehow to the maximal higher central extension of that super L L_\infty-algebra. Hence an argument that the construction of “super equivariant Whitehead towers” (or whatever they are to be called) is somehow preserved by rationalization (or by something like this).

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 16th 2017

    Won’t spectral exterior algebras need to feature?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2017
    • (edited Mar 17th 2017)

    Sure, that for the superpoint is an example: Sym R(ΣR)Sym_R (\Sigma R). This is the spectral analog of the exterior algebra on one generator, which is Sym k(k[1])Sym_k (k[1]). More generally, the spectral analog of the Grassmann algebra on nn generators Sym k(k n[1])=Sym k(k n[1])=Sym k((k[1]) n)Sym_k (k^n[1]) = Sym_k (k^{\oplus^n}[1]) = Sym_k \left((k[1])^{\oplus n}\right) is Sym R((ΣR) n)Sym_R \left((\Sigma R)^{\oplus^n}\right), where the direct sum of spectra is their wedge sum =\oplus = \vee.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 17th 2017

    For some reason I’d never seen a free graded-commutative algebra over kk written like that.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 20th 2017

    Does a periodic version of 𝕊\mathbb{S} ever appear?

    𝕊P= nΣ 2n𝕊 \mathbb{S} P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} \mathbb{S}
    • CommentRowNumber12.
    • CommentAuthorCharles Rezk
    • CommentTimeMar 20th 2017

    I don’t think there’s any way to give it an E E_\infty-ring structure.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 20th 2017
    • (edited Mar 20th 2017)

    According to Adeel in another thread (here), there is discussion of spectral symmetric algebras in Jacob Lurie’s opus. That’s what I’d expect, but all I found so far is a brief remark below prop. 2.20 in “Spectral Schemes” (here).

    It’s natural to expect that the Hopf cosemiring spectrum structure on symmetric algebras that Strickland-Tuner 97 find on Sym 𝕊𝕊Sym_{\mathbb{S}} \mathbb{S} lifts to the corresponding E E_\infty-structure and generalizes to other spectral symmetric algebras, reflecting the additive and multiplicativ structure that one expects to see on any kind of affine line.

    Is this discussed anywhere?

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 20th 2017

    The spectral affine line over the sphere spectrum is denoted I\mathbf{I} in Adeel’s Brave new motivic homotopy theory I (3.2.1).

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2017

    Thanks for the pointer. I have included this and some discussion at spectral symmetric algebra.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 21st 2017

    Presumably we’re in linear HoTT territory, the !-modality instantiating the bosonic Fock space construction. I wonder if there is a possible modal treatment of the fermionic Fock space.

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 22nd 2017
    • (edited Mar 22nd 2017)
    Sym R(Σ nR) R(nS n/Σ(n)) + R(n(BΣ(n)) τ n), \begin{aligned} Sym_R(\Sigma^n R) & \simeq R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} S^n/\Sigma(n) \right)_+ \\ & \simeq R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} (B \Sigma(n))^{\tau_n} \right) \end{aligned} \,,

    There’s an unbound nn on the left, but it would appear not in the expressions on the right.

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 22nd 2017
    • (edited Mar 22nd 2017)
    R 0|1 Spec(Sym R(ΣR)) RSym 𝕊(Σ𝕊) \begin{aligned} R^{0 \vert 1} &\coloneqq Spec \left( Sym_R (\Sigma R) \right) \\ & \simeq R \wedge Sym_{\mathbb{S}}(\Sigma \mathbb{S}) \end{aligned}

    There should be SpecSpec continuing on the right? So also at spectral super-scheme.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2017
    • (edited Mar 30th 2017)

    There’s an unbound nn on the left, but it would appear not in the expressions on the right.

    Thanks, fixed now. It needs to be like so

    Sym R(Σ nR) R(kS k/Σ(k)) + R(k(BΣ(k)) nτ k), \begin{aligned} Sym_R(\Sigma^n R) & \simeq R \wedge \left( \underset{k \in \mathbb{N}}{\coprod} S^k/\Sigma(k) \right)_+ \\ & \simeq R \wedge \left( \underset{k \in \mathbb{N}}{\coprod} (B \Sigma(k))^{n \tau_k} \right) \end{aligned} \,,

    e.g. slide 4 of Charles’ pdf.

    There should be SpecSpec continuing on the right? So also at spectral super-scheme.

    Yes, thanks. Also fixed now.

    • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 30th 2017

    There must be now an nn missing from the upper right term. Should that be S nk/Σ(k)S^{n k}/\Sigma(k)?

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2017

    Yes. Good that you are paying closer attention than I am! I am fixing it in the entry.

    • CommentRowNumber22.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 12th 2018

    Since it came up again on g+, what is the difficulty in doing what Urs asks for in #7? What does it mean to find the cohomology of R 0|1=Spec(Sym EΣE)R^{0 \vert 1} = Spec( Sym_E \Sigma E) as a spectral group scheme? Is such a cohomology representable? If so, what plays the role of B 1\mathbf{B}^1 \mathbb{R} as it appears here?

  1. Added an example on free E_\infty-algebras, which are constructed using the spectral symmetric algebra construction. Added the universal property as well.

    Anonymous

    diff, v12, current

  2. Universal property for free algebra on many generators

    Anonymous

    diff, v13, current

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