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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 $\mathbb{Z}/2$-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 $\mathbb{Z}/2$-graded).
Is it possible to exploit
But ordinary $\mathbb{Z}/2$-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?
So exploiting this fact is what led to the proposal at “spectral super-scheme”: There it is argued that these higher refinements of $\mathbb{Z}/2$-graded commutative superalgebras are $E_\infty$-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.
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However, every $\mathbb{Z}_2$-graded-commutative ring can be naturally regarded as a $\tau_{\leq1}\mathbb{S}$-graded ring by putting everything into degree $0$ and $1$ and by choosing the $\sigma_k$ automorphisms to be given by $a\mapsto-a$.
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 $E_\infty$-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.
In case it’s of interest to this discussion, I started a stub at 2-periodic sphere spectrum.
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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 $\mathbb{Z}/2$-grading into your perspective: Don’t you get a symmetric monoidal $\infty$-category of modules over any given $\tau_n \mathbb{S}$-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.
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Thanks, looks interesting. BTW, one might denote this $\Sigma^{\infinity-1}_+ B (\mathbb{Z}/2)$.
This construction loses the $\mathbb{Z}/24$ 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 $E_\infinity$ 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.
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I certainly think the sphere spectrum $\mathbb{S}$ is the right thing to use. Allow me to recall, as you seem to be revolving around this:
1. Fascinatingly, every $E_\infty$-ring is automatically $\mathbb{S}$-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 $E_\infty$-rings which are algebras over 2-periodic $E_\infty$-spectra.
Should examples listed at periodic ring spectrum all be $E_{\infty}$? In which case I should remove ’2-periodic sphere spectrum’. Or can an $E_2$-ring spectrum count?
Oh, at “periodic ring spectrum” we don’t need to require $E_\infty$. But the story at spectral super-scheme seems to require $E_\infty$-structure.
Vague speculation: I wonder if there’s a multi-initial collection of 2-periodic $E_{\infty}$-rings.
Re #14, does that tell us something interesting, in terms of point 3, that the 2-periodic spectra are complex oriented (even cohomology theory)?
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.
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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 $Q(-)$ is old notation for $\Omega^\infty \Sigma^\infty (-)$ – the underlying space in degree 0 of the suspension spectrum – so that $Q S^0 \;\simeq\; \Omega^\infty \Sigma^\infty S^0 \;\simeq\; \Omega^\infty \mathbb{S}^\infty$ is the underlying space of the sphere spectrum, which we might just as well denote by $\mathbb{S}$ 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 $A$ is equivalently a monoid homomorphism $A \xrightarrow{\;} \mathbb{Z}$ to $\mathbb{Z}$ with its additive abelian monoid structure, so every $E_\infty$-monoid $\mathcal{A}$ already comes equipped with an $E_\infty$-monoid homomorphism $\mathcal{A} \xrightarrow{\;} \mathbb{S}$ to the additive $E_\infty$-monoid underlying the sphere spectrum.
So I think this is direct the $E_\infty$-analog of the classical identification of $\mathbb{Z}$-gradings (plus the fascinating claim that for $E_\infty$-monoids this exists canonically). Why do you say it “departs” from that?
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Why are general $E_\infty$-monoids canonically (and non-trivially) $\mathbb{S}$-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 $\mathcal{J}$-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 $E_\infty$-spectrum carries a canonical $\mathbb{S}$-grading in that it maps to an $E_\infty$-monoid in $\mathcal{J}$-spaces (via their right adjoint functor $\Omega^{\mathcal{J}}$ in their equation (4.4)), the plain morphisms of $E_\infty$-monoids need not respect this $\mathbb{S}$-grading. Instead, the $E_\infty$-morphisms that respect the canonical $\mathbb{S}$-grading are those in the category of $E_\infty$-monoids in $\mathcal{J}$-spaces.
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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 $C$, there is by the universal property of Day convolution a symmetric monoidal adjunction between $Fun(Pic(C),Spc)$ and $C$. The right adjoint sends an ($E_\infty$-)object $c$ to a $\Pic(C)$-graded ($E_\infty$-)space, whose unit component is $Map(1,c)$. Now given a particular invertible object $L\in Pic(C)$, there is a unique $E_\infty$-map $\Omega^\infty\mathbb{S}\to Pic(C)$ sending $1$ to $L$ by the universal property of $\Omega^\infty\mathbb{S}$, so any $Pic(C)$-graded space gives rise to an $\Omega^\infty\mathbb{S}$-graded space.
If I understand correctly, the Sagave-Schlichtkrull $\Omega^\infty\mathbb{S}$-graded space is obtained via this construction applied to $L=\Sigma\mathbb{S}$. But going from the $Pic(Sp)$-grading to the $\Omega^\infty\mathbb{S}$-grading seems to forget a lot of information. For example at the level of homotopy groups the $E_\infty$-map $\Omega^\infty\mathbb{S}\to Pic(Sp)$ is a map $\pi_n\mathbb{S}\to \pi_{n-1}\mathbb{S}$ (for $n\geq 2$).
On the other hand the map $\Omega^\infty\mathbb{S}\to Pic(Sp)$ is an equivalence on $1$-truncations, so for graded objects in 1-categories there is no difference.
Sagave–Schlichtkrull prove that any symmetric spectrum $E$ has an underlying $\mathbb{S}$-graded $\mathbb{E}_\infty$-space $\Omega^\mathcal{J}(E)$
$E$ should be an $E_\infty$-ring for this, otherwise $\Omega^\mathcal{J}(E)$ is only an $\Omega^\infty\mathbb{S}$-graded space.
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Could you please elaborate on this a bit? I’m unsure about how to construct this adjunction (which universal property of Day convolution are you using?)
The universal property is HA 4.8.1.10 (4) (in the special case of Cor. 4.8.1.12): If $A$ is symmetric monoidal, the Day convolution on $PSh(A)$ is the universal cocomplete symmetric monoidal category under A where the tensor product preserves colimits in each variable.
Also, (if I understand it correcty) the construction you describe builds $\Omega^\mathcal{J}(E)$ as a lax symmetric monoidal functor $\Omega^\infty\mathbb{S}\to Spc$, though Sagave–Schlichtkrull seem to view an $\Omega^\infty\mathbb{S}$-grading as instead a morphism of $\mathbb{E}_\infty$-spaces $\Omega^\mathcal{J}(E)\to\Omega^\infty\mathbb{S}$, “going in the opposite direction”. Do you know how these two definitions compare with each other?
They are equivalent by a symmetric monoidal version of straightening/unstraightening. Namely, if $K$ is an $E_\infty$-space, the equivalence $Spc/K = Fun(K,Spc)$ is symmetric monoidal. I don’t know a direct reference for this but I guess it can be deduced from HA 4.8.1.12 which characterizes the RHS.
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My apologies for asking about this again, but how should we apply the universal property of Day convolution to get the symmetric monoidal adjunction $Fun(Pic(C),Spc)\rightleftarrows C$? (I’ve been trying to figure this out for a while, but I’m still unsure about how to proceed :/… Are we using that $C$ is presentable for there to be a left adjoint to the Yoneda embedding?)
The universal property gives a unique symmetric monoidal colimit-preserving functor $Fun(Pic(C),Spc)\to C$ extending the inclusion $Pic(C)\subset C$. It has a right adjoint, which is the restricted Yoneda embedding. There is no need for C to be presentable, it just needs colimits and a compatible symmetric monoidal structure.
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Re #30: Would you considering transfering your understanding to the nLab article then? It could certainly benefit from having more details.
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Hi Théo
Nominally that’s true. But since we mean to be grading by the additive group of integers, or by its additive quotient by 2, it is unclear (to me, at least) what is gained by pointing to the multiplicative subgroup $\{ \pm 1 \} \subset \mathbb{Z}$.
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I don’t want to discourage you from exploring, just saying where I am not immediately seeing where it’s headed.
Maybe there is something to this idea of regarding $GL(1,\mathbb{S})$ as the $E_\infty$-space encoding super-algebra, not sure, one would have to play with it.
I still think what is needed next is a better supply of plausible examples of objects that should qualify as spectral super-algebras.
I also still think that $E_\infty$-algebras over even periodic ring spectra is the class to look at, but back when I thought about this a little I failed to identify examples that would more directly relate to supersymmetry in physics.
As you may know, the idea was to see if the tower of universal central extensions emanating from the ordinary superpoint (here) might have an interesting spectral enhancement if instead one starts with some incarnation of a spectral superpoint.
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