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  1. I have a few questions related to the super-algebra page, some of which I’ve also asked at MathOverflow. I think there’s some value on asking them here too (in particular because I’m hoping Urs might see this, who I think might have thought about this before, or, if not, might find this interesting nonetheless).

    The “abstract idea” section of the super-algebra page reads:

    Superalgebra is universal in the following sense. The crucial super-grading rule (the “Koszul sign rule”, Grassmann 1844, §37, §55)

    ab=(1) deg(a)deg(b)ba a \otimes b = (-1)^{deg(a) deg(b)} b \otimes a

    in the symmetric monoidal category of \mathbb{Z}-graded vector spaces is induced from the subcategory which is the abelian 2-group of metric graded lines. This in turn is the free abelian 2-group (groupal symmetric monoidal category) on a single generator. (This point of view is amplified in the first part of (Kapranov 13), whose second part is about super 2-algebra, more details in Kapranov 15). Generally then super-grading and hence super-algebra arises from the 2-truncation (3-coskeleton) of the free abelian ∞-group on a single generator, which is the sphere spectrum 𝕊\mathbb{S}.

    and then:

    This suggests (as indicated in (Kapranov 13, Kapranov 15)) that in full generality higher supergeometry is to be thought of as 𝕊\mathbb{S}-graded geometry, hence dually as higher algebra with ∞-group of units augmented over the sphere spectrum.

    A similar point of view is put forward in the spectral super-scheme page:

    Definition. Spectral/E E_\infty super-geometry is simply the E-∞ geometry over even periodic ring spectra.

    I’ve been exploring these ideas a bit the last few days, and I think I found a possible notion of 𝔼 \mathbb{E}_{\infty}-superalgebras and 𝔼 \mathbb{E}_{\infty}-supergeometry that differs from the ones suggested by the above quoted pages.

    It starts with the following definition, taken from Bunke–Nikolaus, Section 2: given a monoidal category 𝒞\mathcal{C} and a ring RR, we define a 𝒞\mathcal{C}-graded RR-algebra to be a lax monoidal functor M:(𝒞, 𝒞,1 𝒞)(Mod R, R,R)M\colon(\mathcal{C},\otimes_{\mathcal{C}},\mathbf{1}_{\mathcal{C}})\to(\mathsf{Mod}_R,\otimes_R,R). If moreover 𝒞\mathcal{C} is braided, then a 𝒞\mathcal{C}-graded RR-algebra MM is 𝒞\mathcal{C}-graded-commutative if RR is a braided lax monoidal functor. More generally, one could allow 𝒞\mathcal{C} to be a symmetric monoidal \infty-category; this is relevant when working with spectra, but for rings only the 11-truncation is relevant.

    These recover monoid-graded algebras as the A discA_{\mathsf{disc}}-graded ones, but there’s also a number of other interesting notions arising from this definition. In particular, following Kapranov’s idea, one can consider rings graded by kk-truncations of the sphere spectrum. For k=0k=0, this recovers \mathbb{Z}-graded algebras, but the situation is considerably more interesting for k=1k=1:

    A τ 1𝕊\tau_{\leq1}\mathbb{S}-graded ring consists of a pair (R ,{σ k} k)(R_\bullet,\{\sigma_k\}_{k\in\mathbb{Z}}) with

    • R R_\bullet a \mathbb{Z}-graded ring;
    • {σ k:R kR k} k\{\sigma_k\colon R_k\to R_k\}_{k\in\mathbb{Z}} a family of order 22 automorphisms, one for each R kR_k;

    which is moreover τ 1𝕊\tau_{\leq1}\mathbb{S}-graded commutative if we have

    ab={ba if deg(a)deg(b) is even, σ deg(a)+deg(b)(ab) if deg(a)deg(b) is oddab=\begin{cases}ba &\text{if deg(a)deg(b) is even,}\\\sigma_{\deg(a)+\deg(b)}(ab) &\text{if deg(a)deg(b) is odd}\end{cases}

    for each a,bR a,b\in R_\bullet.

    By choosing σ k(a)=a\sigma_k(a)=-a for all aR ka\in R_k and all kk\in\mathbb{Z}, these recover \mathbb{Z}-graded-commutative algebras as a special case, and in this definition the Koszul rule comes from π 1(𝕊) 2\pi_1(\mathbb{S})\cong\mathbb{Z}_2, while the \mathbb{Z}-grading (vs. 2\mathbb{Z}_2-grading issue that Kapranov mentions) is explained by π 0(𝕊)\pi_0(\mathbb{S})\cong\mathbb{Z}!

    “Higher supersymmetry” now corresponds to higher and higher kk-truncations of 𝕊\mathbb{S} as suggested by Kapranov; though at each nn-categorical level one can only really see “nn-supersymmetry” (e.g. for n=1n=1 this corresponds to \infty-functors 𝕊N (Mod R)\mathbb{S}\to\mathrm{N}_{\bullet}(\mathsf{Mod}_R) being the same as 11-functors Ho(𝕊)Mod R\mathsf{Ho}(\mathbb{S})\to\mathsf{Mod}_R, and in this case we have Ho(𝕊)Ho(τ k𝕊)\mathsf{Ho}(\mathbb{S})\cong\mathsf{Ho}(\tau_{\leq k}\mathbb{S}) for all k1k\geq1, so only τ 1𝕊\tau_{\leq1}\mathbb{S} appears here).

    This suggests a general definition of “(kk-super)-𝔼 n\mathbb{E}_{n}-RR-algebras” as the 𝔼 n\mathbb{E}_{n}-monoids in Fun(τ k𝕊,Mod R)\mathsf{Fun}(\tau_{\leq k}\mathbb{S},\mathsf{Mod}_R), where for n=n=\infty we recover lax symmetric monoidal functors τ k𝕊Mod R\tau_{\leq k}\mathbb{S}\to\mathsf{Mod}_R. Finally, one possible notion of 𝔼 \mathbb{E}_{\infty}-supergeometry would then be the variant of SAG obtained by replacing 𝔼 \mathbb{E}_{\infty}-rings as the basic building blocks with these “(kk-super)-𝔼 \mathbb{E}_{\infty}-rings”.

    (P.S. There are also other very interesting “universal gradings” leading to variants a bit similar to these. In particular, recalling the characterisation of the sphere spectrum as the free 𝔼 \mathbb{E}_{\infty}-group on a point, we may think of considering instead the free 𝔼 n\mathbb{E}_{n}-group on a point, Ω nS n\Omega^n S^n. While τ k𝕊\tau_{\leq k}\mathbb{S}-gradings give a \mathbb{Z}-grading together with actions induced by the first kk stable homotopy groups of spheres, τ kΩ nS n\tau_{\leq k}\Omega^{n}S^{n}-gradings give a \mathbb{Z}-grading again, though this time the induced actions correspond to the unstable homotopy groups of spheres π n+1(S n)\pi_{n+1}(S^n), π n+2(S n)\pi_{n+2}(S^n), \ldots, π n+k(S n)\pi_{n+k}(S^n)! For ordinary rings, this means that gradings by the free braided 22-group are given by a \mathbb{Z}-grading together with a \mathbb{Z}-indexed family of \mathbb{Z}-actions, corresponding to π 0(Ω 2S 2)π 2(S 2)\pi_0(\Omega^2 S^2)\cong\pi_2(S^2)\cong\mathbb{Z} for the gradings and π 1(Ω 2S 2)π 3(S 2))\pi_1(\Omega^2S^2)\cong\pi_3(S^2)\cong\mathbb{Z}) for the actions!)


    Has this construction combining this specific notion of grading with truncations of the sphere spectrum been considered before? (Have you perhaps already thought about this, Urs?)

    Also, are there other nice classes of τ 1𝕊\tau_{\leq1}\mathbb{S}-graded commutative rings besides ordinary rings and \mathbb{Z}-graded-commutative ones?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2021
    • (edited Sep 10th 2021)

    Thanks for writing in.

    Regarding the second but last question: My last thinking about this issue is still that recorded at spectral super-scheme, which you have seen. I have not considered the specific construction you describe, which certainly looks like a natural implementation of Kapranov’s suggestion!

    But does it not – as you point out above – still suffer from the shortcoming of yielding (higher refinements of) only \mathbb{Z}-graded commutativity instead of /2\mathbb{Z}/2-graded commutativity?

    It’s only in the latter generality that most of the interesting effects associated with supersymmetry appear (e.g. the super-Poincaré-algebras are only /2\mathbb{Z}/2-graded).

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 10th 2021

    Is it possible to exploit

    But ordinary /2\mathbb{Z}/2-graded supercommutative superalgebra is equivalently \mathbb{Z}-graded supercommutative superalgebra over the free even periodic \mathbb{Z}-graded supercommutative superalgebra (spectral super-scheme)?

    What would higher refinements of \mathbb{Z}-graded commutativity over the free even periodic \mathbb{Z}-graded supercommutative superalgebra look like?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2021

    So exploiting this fact is what led to the proposal at “spectral super-scheme”: There it is argued that these higher refinements of /2\mathbb{Z}/2-graded commutative superalgebras are E E_\infty-algebras over even periodic ring spectra!

    On the one hand, this connects the idea of higher super-algebra to established practice and examples of algebraic topology. (Notably Charles Rezk had already run into this perspective purely for alg-top reasons.) On the other hand, it still lacks more connection to topics in supersymmetry as appearing in mathematical physics.

  2. Hi Urs, thanks for the reply!

    I think it does have this shortcoming in that τ 1𝕊\tau_{\leq1}\mathbb{S}-graded-commutative rings are not exactly 2\mathbb{Z}_2-graded-commutative rings. However, every 2\mathbb{Z}_2-graded-commutative ring can be naturally regarded as a τ 1𝕊\tau_{\leq1}\mathbb{S}-graded ring by putting everything into degree 00 and 11 and by choosing the σ k\sigma_k automorphisms to be given by aaa\mapsto-a.

    I have yet to work out how things go on the level of morphisms, though. (I hope to do this tomorrow, and then give an update here; I’m not very hopeful with the Koszul rule for morphisms f(ab)=(1) deg(f)deg(a)f(a)f(b)f(a b)=(-1)^{\deg(f)\deg(a)}f(a)f(b), however :/)

    My current intuition on this construction is that it regards 00-supercommutativity as ordinary commutativity, i.e. ab=baa b=b a, together with an additional \mathbb{Z}-gradation (with no extra conditions). Then 2\mathbb{Z}_2-graded commutativity, i.e. ab=(1) deg(a)deg(b)baa b=(-1)^{\deg(a)\deg(b)}b a, is enforced as 11-supercommutativity via π 1(𝕊)\pi_1(\mathbb{S}). The higher refinements are all torsion―we never see anything like \mathbb{Z} again. So e.g. a 22-supercommutative ring in nn-groupoids for n2n\geq2 (we need to consider homotopical versions of rings; the ordinary ones don’t see the higher torsion) might feel like something “having two distinct minus signs” (or more precisely: two order 22 automorphisms), each one satisfying a Koszul rule. Similarly, 33-supercommutativity involves additional automorphisms σ k:A kA k\sigma_k\colon A_k \to A_k, one for each kk\in\mathbb{Z}, satisfying σ k 24=id A k\sigma^{24}_k=\mathrm{id}_{A_k}, and so on.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2021
    • (edited Sep 10th 2021)

    However, every 2\mathbb{Z}_2-graded-commutative ring can be naturally regarded as a τ 1𝕊\tau_{\leq1}\mathbb{S}-graded ring by putting everything into degree 00 and 11 and by choosing the σ k\sigma_k automorphisms to be given by aaa\mapsto-a.

    Absolutey, it’s the formalization of this phenomenon, via that Proposition, which yields the definition proposed at spectral super-scheme!

    I am not saying this cannot be integrated into the perspective you adopted, just saying that it would need to be integrated to be a satisfactory reply to “What is higher superalgebra?” (The same remark applies to Kapronov’s proposal! and in fact it applies to a sizeable community of “NQ-manifold” theorists out there.) In fact, it seems pretty clear for how to integrate this into you scheme: You can enhance it to speak of higher graded algebras over higher graded base rings, and then you just have to require the higher graded base ring to be 2-periodic. I think.

    What I found fascinating about the proposal at spectral super-scheme is that the identification of super-grading with foundations of algebraic topology ranges even deeper than in Kapranov’s proposal: It’s really the notion of E E_\infty-algebra as such that already captures “higher \mathbb{Z}-graded super-algebra” and then the choice of even periodic base ring spectra (which are known to play a pivotal role in chromatic theory etc.) makes it genuinely super-algebraic in the sense of physics.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 10th 2021

    In case it’s of interest to this discussion, I started a stub at 2-periodic sphere spectrum.

  3. I have good news and bad news: the bad news is that the Koszul rule f(ab)=(1) deg(f)deg(a)f(a)f(b)f(ab)=(-1)^{\deg(f)\deg(a)}f(a)f(b) for morphisms of superalgebras f:ABf\colon A\to B isn’t enforced by the notion of a “morphism of τ 1𝕊\tau_{\leq1}\mathbb{S}-graded rings”, defined as a monoidal natural transformation f :R S f_\bullet\colon R_\bullet\to S_\bullet. In detail, the latter is as follows:

    The good news is that with this definition, we do have a fully faithful embedding of categories Gr 2RingsGr τ 1𝕊Rings\mathsf{Gr}_{\mathbb{Z}_2}\mathsf{Rings}\hookrightarrow\mathsf{Gr}_{\tau_{\leq1}\mathbb{S}}\mathsf{Rings}!

    So, how important is it to have the Koszul rule for morphisms f(ab)=(1) deg(f)deg(a)f(a)f(b)f(a b)=(-1)^{\deg(f)\deg(a)}f(a)f(b), instead of just f(ab)=f(a)f(b)f(a b)=f(a)f(b)?

    (There’s another interesting point here: I haven’t straightened this out, but I think if we go one level higher, considering

    • 22-rings, modelled as ring categories, which are equivalent to 11-truncated ring spectra, and
    • k=2k=2, i.e. τ 2𝕊\tau_{\leq2}\mathbb{S};

    then “22-supercommutative 22-rings” are lax pseudomonoidal pseudofunctors τ 2𝕊2Ab\tau_{\leq 2}\mathbb{S}\to\mathsf{2Ab}, and the morphisms between those will contain 22-cells verifying lax monoidal pseudonaturality, which will be indexed by π 2(𝕊) 2\pi_2(\mathbb{S})\cong\mathbb{Z}_2, needing to satisfy a different kind of Koszul rule.)


    Absolutey, it’s the formalization of this phenomenon, via that Proposition, which yields the definition proposed at spectral super-scheme!

    This is something I’ve been thinking about too: commutative rings embed fully faithfully into 𝔼 \mathbb{E}_{\infty}-rings, and for this reason we view higher algebra as a faithful extension of ring theory. With the same phenomenon happening for 2\mathbb{Z}_{2}-graded rings and τ 1𝕊\tau_{\leq1}\mathbb{S}-graded rings, there again being a fully faithful embedding of categories, are there reasons for not “just” considering them as a generalised form of ordinary supersymmetry (completely forgetting for the moment about any homotopy groups higher than π 1\pi_1)?


    You can enhance it to speak of higher graded algebras over higher graded base rings, and then you just have to require the higher graded base ring to be 2-periodic. I think.

    I’m not sure how to extend the definition I mentioned of a “grading by a symmetric monoidal \infty-category” to a “grading by nonconnective spectra” (I’ve been saying “τ 1𝕊\tau_{\leq1}\mathbb{S}-graded rings”, but its really “τ 1(QS 0)\tau_{\leq1}(Q S^0)-graded rings”), however it seems to me that replacing 𝕊\mathbb{S} by the 22-periodified sphere spectrum might lead to a grading even worse than a \mathbb{Z}-grading, since π 0(𝕊[β])\pi_0(\mathbb{S}[\beta]) is completely crazy. Is this correct?


    What I found fascinating about the proposal at spectral super-scheme is that the identification of super-grading with foundations of algebraic topology ranges even deeper than in Kapranov’s proposal: It’s really the notion of E E_\infty-algebra as such that already captures “higher \mathbb{Z}-graded super-algebra” and then the choice of even periodic base ring spectra (which are known to play a pivotal role in chromatic theory etc.) makes it genuinely super-algebraic in the sense of physics.

    There is another point related to this that I’ve been wanting to raise: in spectral super-scheme, you wrote:

    But more is true: the E E_\infty-analog of the integers \mathbb{Z} is the sphere spectrum 𝕊\mathbb{S}, and every E-infinity ring EE is canonically 𝕊\mathbb{S}-graded (Sagave-Schlichtkrull 11, theorem 1.7-18).

    There’s a number of inequivalent definitions of 𝕊\mathbb{S}-graded, though:

    • 𝕊\mathbb{S}-graded as in Sagave–Schlichtkrull;
    • 𝕊\mathbb{S}-graded meaning a lax symmetric monoidal functor QS 0SpQS^0\to\mathsf{Sp};
    • 𝕊\mathbb{S}-graded in the sense that’s useful for Proj\mathrm{Proj}’s in SAG. Rok Gregoric has done (and is doing) work on this, and gives some examples of these here.

    One last point (sorry for the really long message!): FWIU, exterior algebras play the role of polynomial algebras for superalgebras. Rok proposed what I think is the correct definition of a “spectral exterior algebra” here, and it agrees with your definition here up to a delooping, coming from Sym R (Σ(M))Σ ( R (M))\mathrm{Sym}^\bullet_R(\Sigma (M)) \simeq \Sigma^{\bullet}(\bigwedge_R^\bullet(M)).

    A potentially interesting variant definition (which I believe is not the correct analogue of “spectral (higher) exterior algebras”, but seems interesting nonetheless) would be the free τ k𝕊\tau_{\leq k}\mathbb{S}-graded commutative RR-algebra on an RR-module MM, defined as the image of Δ M:τ k𝕊ModSp R\Delta_{M}\colon\tau_{\leq k}\mathbb{S}\to\mathsf{ModSp}_R under the free 𝔼 \mathbb{E}_{\infty}-monoid functor for the Day convolution monoidal structure on Fun(τ k𝕊,ModSp R)\mathsf{Fun}(\tau_{\leq k}\mathbb{S},\mathsf{ModSp}_R).

    In the 11-categorical world, we consider instead the free commutative monoid on Δ M:τ 1Mod R\Delta_{M}\colon\tau_{\leq 1}\to\mathsf{Mod}_R for the Day convolution monoidal structure on Fun(τ 1𝕊,Mod R)\mathsf{Fun}(\tau_{\leq 1}\mathbb{S},\mathsf{Mod}_R), i.e. the free symmetric lax monoidal functor τ 1𝕊Mod R\tau_{\leq 1}\mathbb{S}\to\mathsf{Mod}_R. This looks very interesting: is it the exterior algebra on MM again? If not, is it relevant to superalgebra and supersymmetry?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2021

    Hi, am just back from an offline family weekend.

    I should say that I don’t have the leisure right now to follow you more deeply into this discussion (due to other projects active right now) but I am still interested. The topic is potentially of utmost relevance and may be just waiting for a breakthrough insight in order to take off.

    That said, some quick comments:

    On the MO discussion: Thanks for the pointer! Have added the link to the bottom of the entry spectral super-scheme.

    Regarding incorporating /2\mathbb{Z}/2-grading into your perspective: Don’t you get a symmetric monoidal \infty-category of modules over any given τ n𝕊\tau_n \mathbb{S}-graded algebra in your sense? If so, just repeat the process and next consider such graded algebras in the infinity category of modules over a 2-periodic one.

    Regarding relaxing/generalizing the rules of super-algebra: Part of the interest in the topic here is exactly to understand if such generalizations are natural. So seeing these appear may be a feature instead of a bug. Depends. To sort this out one should try to find good examples and applications.

  4. Hi, Urs! I hope you had a great time with your family :)

    I should say that I don’t have the leisure right now to follow you more deeply into this discussion (due to other projects active right now) but I am still interested. The topic is potentially of utmost relevance and may be just waiting for a breakthrough insight in order to take off.

    No problem! Feel free to drop another message here in the future and restart the discussion when you’re less time-pressed, if you’d like to :)

    Regarding incorporating /2\mathbb{Z}/2-grading into your perspective: Don’t you get a symmetric monoidal \infty-category of modules over any given τ n𝕊\tau_n \mathbb{S}-graded algebra in your sense? If so, just repeat the process and next consider such graded algebras in the infinity category of modules over a 2-periodic one.

    I think we do, but in this case the unpleasant grading will still be here: in the 1-categorical case already a lax symmetric monoidal functor τ 1𝕊Mod /2\tau_{\leq1}\mathbb{S}\to\mathsf{Mod}_{\mathbb{Z}/2} will still carry a \mathbb{Z}-grading, again because of Obj(τ 1𝕊)=\mathrm{Obj}(\tau_{\leq1}\mathbb{S})=\mathbb{Z}. The problem really is the π 0\pi_0 :/

    To get a /2\mathbb{Z}/2-grading under the Bunke–Nikolaus definition, we would need to somehow universally deform 𝕊\mathbb{S} into a spectrum 𝕊˜\tilde{\mathbb{S}} with π 0(𝕊˜)/2\pi_{0}(\tilde{\mathbb{S}})\cong\mathbb{Z}/2 (in particular 2-periodification doesn’t work in this setting, as we instead get a π 0(𝕊[β]) kπ 2k(𝕊)\pi_0(\mathbb{S}[\beta])\cong\bigoplus_{k\in\mathbb{Z}}\pi_{2k}(\mathbb{S})-grading).

    Regarding relaxing/generalizing the rules of super-algebra: Part of the interest in the topic here is exactly to understand if such generalizations are natural. So seeing these appear may be a feature instead of a bug. Depends. To sort this out one should try to find good examples and applications.

    I agree completely! I’ve been trying to find some interesting examples of these things during the past few days, but have yet to succeed (though admittedly I haven’t thought much about them yet. One thing I’m planning to do soon-ish is to compute what the free τ 1\tau_{\leq1}-ring on a module looks like, I think that has a good chance of being neither an ordinary ring nor a graded-commutative one)

  5. Hi Urs,

    A small update unrelated to gradings: I think I found an object 𝕊/2\mathbb{S}/2 which one may call the “mod 2 sphere spectrum”. It satisfies an analogous universal property to /2\mathbb{Z}/2 and has π 0(𝕊/2)/2\pi_0(\mathbb{S}/2)\cong\mathbb{Z}/2. See here for the UP, here for Maxime Ranzi’s proof that it exists and also a description (it is ΩQℝℙ \Omega Q\mathbb{RP}^\infty), and lastly here for a table with its first homotopy groups, together with a comparison with those of 𝕊\mathbb{S}.

    (I have no idea whether this is the right object to consider in Kapranov’s context, but I thought you might like to know about it nonetheless! =)

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTime4 days ago
    • (edited 4 days ago)

    Thanks, looks interesting. BTW, one might denote this Σ + 1B(/2)\Sigma^{\infinity-1}_+ B (\mathbb{Z}/2).

    This construction loses the /24\mathbb{Z}/24 in degree 3, which was a major motivation for Kapranov. That 24 should still be present in the 2-periodic sphere spectrum that David C. highlighted in comment #7 above.

    Neither of the two spectra has E E_\infinity ring structure, though. I am not sure what either is doing for us in regards to superalgebra, but that’s mostly because I haven’t tried to think about it.

  6. For what it’s worth I think Σ + 1B/2\Sigma^{\infty-1}_+\mathbf{B}\mathbb{Z}/2 (thanks for the suggestion!) isn’t the right thing to look at. OTOH there’s an argument for using precisely 𝕊\mathbb{S}: we can only put a (1) deg(a)deg(b)(-1)^{\deg(a) \deg(b)} in the Koszul rule for \mathbb{Z}-graded RR-modules leading to supercommutativity because abelian groups have inverses, carrying an action AA\mathbb{Z}\otimes A\to A of the integers given by akaa\mapsto k a. Moreover, the integers are universal with respect to this property: they are the monoid corepresenting the functor CMonSets\mathsf{CMon}\to\mathsf{Sets} sending a monoid to its set of invertible elements.

    Similarly, QS 0Q S^0 is the corepresenting object for the invertible objects functor Pic:Mon 𝔼 (𝒮)𝒮\mathsf{Pic}\colon\mathsf{Mon}_{\mathbb{E}_\infty}(\mathcal{S})\to\mathcal{S}, and indeed this gives spectra an action 𝕊EE\mathbb{S}\otimes E\to E of the sphere spectrum. So one possibly interesting thing we might want to look at is what kind of symmetric monoidal structures the \infty-category of \mathbb{Z}-graded spectra might have.

    In the classical case of \mathbb{Z}-graded abelian groups (/\mathbb{Z}-graded *\mathbb{Z}*-modules) there are only two symmetric monoidal structures: the trivial one and the Koszul one, corresponding to 1,11,-1\in\mathbb{Z}. But maybe there are more for \mathbb{Z}-graded spectra (/\mathbb{Z}-graded *𝕊\mathbb{S}*-modules)!

    (I’ve asked about this later point here, though I haven’t made any progress on it since :/)

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTime3 days ago

    I certainly think the sphere spectrum 𝕊\mathbb{S} is the right thing to use. Allow me to recall, as you seem to be revolving around this:

    1. Fascinatingly, every E E_\infty-ring is automatically 𝕊\mathbb{S}-graded and thereby already is \mathbb{Z}-graded in a higher homotopical sense;

    2. the only aspect missing for this homotopified \mathbb{Z}-grading to be a genuine super-grading is that it be 2-periodified;

    3. this is provided exactly by restriction to those E E_\infty-rings which are algebras over 2-periodic E E_\infty-spectra.

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTime3 days ago
    • (edited 3 days ago)

    Should examples listed at periodic ring spectrum all be E E_{\infty}? In which case I should remove ’2-periodic sphere spectrum’. Or can an E 2E_2-ring spectrum count?

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTime3 days ago

    Oh, at “periodic ring spectrum” we don’t need to require E E_\infty. But the story at spectral super-scheme seems to require E E_\infty-structure.

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTime3 days ago
    • (edited 3 days ago)

    Vague speculation: I wonder if there’s a multi-initial collection of 2-periodic E E_{\infty}-rings.

  7. Re #14, does that tell us something interesting, in terms of point 3, that the 2-periodic spectra are complex oriented (even cohomology theory)?

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTime2 days ago

    Yes, this is what I am thinking of when I say (#4) that:

    this connects the idea of higher super-algebra to established practice and examples of algebraic topology.

    Because the ingredients of spectral super-schemes according to the proposed definition there are not outlandish, but are exactly what one cares about in chromatic homotopy theory.

    Conversely, this is the reason why Rezk 09, Sec 2 already ran into this kind of higher super-algebra in investigations of Morava E-theories!

    All this makes me think that this AlgTop/super-connection is the right one, and that it now just takes someone to pick this up and run with it.

  8. 1. Fascinatingly, every E E_\infty-ring is automatically 𝕊\mathbb{S}-graded and thereby already is \mathbb{Z}-graded in a higher homotopical sense;

    This is something I’ve been meaning to understand lately. FWIU there are at least three natural definitions of “𝕊\mathbb{S}-graded” in the literature, one of them being the one given by Sagave–Schlichtkrull, mentioned in spectral super-scheme. Is it correct to say that a “Sagave–Schlischtkrull 𝕊\mathbb{S}-grading” on an 𝔼 \mathbb{E}_{\infty}-space AA is the same thing as a morphism of 𝔼 \mathbb{E}_\infty-spaces AQS 0A\to Q S^0 (where one departs from the classical identification {gradings on a monoid A}{monoid morphisms A(,+,0)}\{\text{gradings on a monoid }\,A\}\cong\{\text{monoid morphisms }\,A\to(\mathbb{Z},+,0)\})?

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTime1 day ago

    Yes, that’s what their Thm. 1.7 on p.3 is saying (the notation is a little intransparent, and I keep having to sift through their definitions to remind me about what it all means, but luckily they clarify this in the text right beneath).

    Now Q()Q(-) is old notation for Ω Σ ()\Omega^\infty \Sigma^\infty (-) – the underlying space in degree 0 of the suspension spectrum – so that QS 0Ω Σ S 0Ω 𝕊 Q S^0 \;\simeq\; \Omega^\infty \Sigma^\infty S^0 \;\simeq\; \Omega^\infty \mathbb{S}^\infty is the underlying space of the sphere spectrum, which we might just as well denote by 𝕊\mathbb{S} itself, since the sphere spectrum is connective.

    In conclusion, the conceptual content of that Thm. 1.7 should be that: just as a \mathbb{Z}-grading on a monoid AA is equivalently a monoid homomorphism AA \xrightarrow{\;} \mathbb{Z} to \mathbb{Z} with its additive abelian monoid structure, so every E E_\infty-monoid 𝒜\mathcal{A} already comes equipped with an E E_\infty-monoid homomorphism 𝒜𝕊\mathcal{A} \xrightarrow{\;} \mathbb{S} to the additive E E_\infty-monoid underlying the sphere spectrum.

    So I think this is direct the E E_\infty-analog of the classical identification of \mathbb{Z}-gradings (plus the fascinating claim that for E E_\infty-monoids this exists canonically). Why do you say it “departs” from that?

  9. Thank you very much, Urs! This cleared up most of my misunderstandings :)

    There’s a single small point about which I’m still a bit confused: I see that to every symmetric spectrum EE one can associate an 𝕊\mathbb{S}-graded 𝔼 \mathbb{E}_\infty-space Ω 𝒥(E)\Omega^\mathcal{J}(E) (which is indeed very fascinating!). Why are general E E_\infty-monoids canonically (and non-trivially) 𝕊\mathbb{S}-graded, though?

    (From what I understand, Theorem 1.7 says that the model category of commutative 𝒥\mathcal{J}-spaces is Quillen equivalent to that of 𝔼 \mathbb{E}_\infty-spaces over B𝒥Ω 𝕊\mathbf{B}\mathcal{J}\simeq\Omega^\infty\mathbb{S}, but does it also give a way to go from II-spaces (/𝔼 \mathbb{E}_\infty-spaces) to 𝒥\mathcal{J}-spaces?)

    Why do you say it “departs” from that?

    I was thinking about the other notions of gradings for E E_\infty-spaces: they all “depart” from more or less equivalent characterisations of \mathbb{Z}-gradings of monoids (such as lax monoidal functors discA\mathbb{Z}_\mathsf{disc}\to A or morphisms of monoids AA\to\mathbb{Z}) which however differ in the \infty-setting.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTime14 hours ago
    • (edited 14 hours ago)

    Why are general E E_\infty-monoids canonically (and non-trivially) 𝕊\mathbb{S}-graded, though?

    I assume that by asking “Why?” you do not mean “How does Sagave-Schlichtkrull’s proof work?”, but you do mean to ask for a conceptual explanation.

    I would turn this around: The observation that they are is a peek into the fundamental nature of reality, and is a hint for a deep relation between higher- and super- geometry/algebra, a hint that remains to be fully followed up on.

    At the coarse level of homotopy groups, this is an old observation in algebraic topology, embodied in classical phenomena such as the super Lie algebra structure of Whitehead products.

    My gut feeling here is that ultimately algebraic topology/homotopy theory is part of the explanation of “Why super-mathematics?” Namely: Super-algebra (in an enhanced form) seems to be the algebra/geometry automatically appearing in homotopy-theoretic foundations.

    but does it also give a way to go from \mathcal{I}-spaces to 𝒥\mathcal{J}-spaces?

    Good question, I am wondering about this each time I pick up their article. I don’t know. (Maybe this is explained somewhere in their writing, but I haven’t spent more time with it, I have to admit.)

    The picture I have is:

    While each E E_\infty-spectrum carries a canonical 𝕊\mathbb{S}-grading in that it maps to an E E_\infty-monoid in 𝒥\mathcal{J}-spaces (via their right adjoint functor Ω 𝒥\Omega^{\mathcal{J}} in their equation (4.4)), the plain morphisms of E E_\infty-monoids need not respect this 𝕊\mathbb{S}-grading. Instead, the E E_\infty-morphisms that respect the canonical 𝕊\mathbb{S}-grading are those in the category of E E_\infty-monoids in 𝒥\mathcal{J}-spaces.

  10. I assume that by asking “Why?” you do not mean “How does Sagave-Schlichtkrull’s proof work?”, but you do mean to ask for a conceptual explanation.

    Thanks, Urs! I had something else in mind: Sagave–Schlichtkrull prove that any symmetric spectrum EE has an underlying 𝕊\mathbb{S}-graded 𝔼 \mathbb{E}_\infty-space Ω 𝒥(E)\Omega^\mathcal{J}(E), and by the equivalence between connective spectra and grouplike 𝔼 \mathbb{E}_\infty-spaces, we see that so does any grouplike 𝔼 \mathbb{E}_\infty-space. I’m a bit confused though about whether this applies also to not necessarily grouplike 𝔼 \mathbb{E}_\infty-spaces: do we have an 𝕊\mathbb{S}-graded 𝔼 \mathbb{E}_\infty-space Ω 𝒥(X)\Omega^\mathcal{J}(X) for any such not necessarily grouplike 𝔼 \mathbb{E}_\infty-space XX?

    (To me this seems a bit unlikely―I’d expect them to be graded over 𝔽=def n=0 BΣ n\mathbb{F}\overset{\mathrm{def}}{=}\coprod_{n=0}^\infty\mathbf{B}\Sigma_{n}, though not over Ω 𝕊\Omega^\infty\mathbb{S}. OTOH, there’s also Ω 𝒥(X grp)\Omega^\mathcal{J}(X^{\mathrm{grp}}), but that seems way too lossy…)

    Regardless, thank you very much for your explanation, Urs! In particular I didn’t know about the super Lie algebra structure of Whitehead products you mentioned―this seems very nice, and I’ll really enjoy learning about it! :)

    • CommentRowNumber25.
    • CommentAuthorMarc Hoyois
    • CommentTime11 hours ago

    The result of Sagave-Schlichtkrull seems slightly misleading to me. A more natural statement would be a grading by the Picard \infty-groupoid of invertible spectra. For any presentably symmetric monoidal \infty-category CC, there is by the universal property of Day convolution a symmetric monoidal adjunction between Fun(Pic(C),Spc)Fun(Pic(C),Spc) and CC. The right adjoint sends an (E E_\infty-)object cc to a Pic(C)\Pic(C)-graded (E E_\infty-)space, whose unit component is Map(1,c)Map(1,c). Now given a particular invertible object LPic(C)L\in Pic(C), there is a unique E E_\infty-map Ω 𝕊Pic(C)\Omega^\infty\mathbb{S}\to Pic(C) sending 11 to LL by the universal property of Ω 𝕊\Omega^\infty\mathbb{S}, so any Pic(C)Pic(C)-graded space gives rise to an Ω 𝕊\Omega^\infty\mathbb{S}-graded space.

    If I understand correctly, the Sagave-Schlichtkrull Ω 𝕊\Omega^\infty\mathbb{S}-graded space is obtained via this construction applied to L=Σ𝕊L=\Sigma\mathbb{S}. But going from the Pic(Sp)Pic(Sp)-grading to the Ω 𝕊\Omega^\infty\mathbb{S}-grading seems to forget a lot of information. For example at the level of homotopy groups the E E_\infty-map Ω 𝕊Pic(Sp)\Omega^\infty\mathbb{S}\to Pic(Sp) is a map π n𝕊π n1𝕊\pi_n\mathbb{S}\to \pi_{n-1}\mathbb{S} (for n2n\geq 2).

    On the other hand the map Ω 𝕊Pic(Sp)\Omega^\infty\mathbb{S}\to Pic(Sp) is an equivalence on 11-truncations, so for graded objects in 1-categories there is no difference.

    Sagave–Schlichtkrull prove that any symmetric spectrum EE has an underlying 𝕊\mathbb{S}-graded 𝔼 \mathbb{E}_\infty-space Ω 𝒥(E)\Omega^\mathcal{J}(E)

    EE should be an E E_\infty-ring for this, otherwise Ω 𝒥(E)\Omega^\mathcal{J}(E) is only an Ω 𝕊\Omega^\infty\mathbb{S}-graded space.

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