The additive vs. multiplicative grading issue is a good point: repeating the story above for “$\tau_{\leq1}(\Omega^\infty\mathbb{S})^\times$-graded algebras”, i think we don’t quite get supercommutative superalgebras, but rather objects $A_\bullet$ consisting of two abelian groups $A_{-1}$ and $A_1$, multiplication maps $A_i\times A_j\to A_{i j}$, a unit $1_A\in A_1$, and a supercommutativity rule of the form

$a b = (-1)^{\begin{pmatrix}deg(a)\\2\end{pmatrix}\begin{pmatrix}deg(b)\\2\end{pmatrix}}b a,$saying that the multiplication maps

$\begin{aligned} A_{1}\times A_{1} &\to A_1,\\ A_{-1}\times A_{1} &\to A_{-1},\\ A_{1}\times A_{-1} &\to A_{-1} \end{aligned}$are commutative, while the map $A_{-1}\times A_{-1}\to A_{1}$ is anticommutative. Have you ever seen a supercommutativity rule of this form appear anywhere?

Speaking of different sign rules, there’s a second such issue I was wondering about: in signs in supergeometry, you discuss the Deligne and Bernstein conventions for super dgas,

$\begin{aligned} \alpha_i\alpha_j &= (-1)^{(n_i n_j+\sigma_i\sigma_j)}\alpha_j\alpha_i,\\ \alpha_i\alpha_j &= (-1)^{(n_i+\sigma_i)\cdot(n_j+\sigma_j)}\alpha_j\alpha_i \end{aligned}$for when we have a grading of the form $\mathbb{Z}\times\mathbb{Z}/2$. This kind of grading appears also when we consider the action of the “multiplicative” tensor product $\otimes\colon Ho(\mathbb{S})\times Ho(\mathbb{S})\to Ho(\mathbb{S})$ on the morphisms of $Ho(\mathbb{S})$, coming from $\mathbb{Z}\times\mathbb{Z}/2\cong\pi_0(\mathbb{S})\times\pi_1(\mathbb{S})$, but there instead we have a rule of the form

$((n_i,\sigma_i),(n_j,\sigma_j))\mapsto n_j\sigma_i+n_i\sigma_j$(in turn coming from the equality $sgn(\sigma\otimes\tau)=m sgn(\sigma)+n sgn(\tau)$ for products of permutations). Have you maybe wondered before about whether one gets anything interesting from using a sign rule of the form

$\alpha_i\alpha_j = (-1)^{(n_j\sigma_i+n_i\sigma_j)}\alpha_j\alpha_i$instead of the Deligne or Bernstein ones? (Edit: I just noticed I had swapped $i$ and $j$ above; the correct sign rule isn’t $(-1)^{(n_i\sigma_i+n_j\sigma_j)}$, but rather $(-1)^{(n_j\sigma_i+n_i\sigma_j)}$. This happens to be the “Bernstein rule minus the Deligne one”: $(-1)^{(n_j\sigma_i+n_i\sigma_j)}=(-1)^{(n_i+\sigma_i)\cdot(n_j+\sigma_j)-(n_i n_j+\sigma_i\sigma_j)}$.)

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.

Thanks for the pointer, Urs! I saw this a long time ago, found it *very* nice, and made a note to learn it better, but ended up forgetting… I really should go back to this! :)

]]>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.

]]>Ah, right. I completely failed to noticed that the grading would then be “multiplicative”. Thanks, Urs!

]]>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 $\{ \pm 1 \} \subset \mathbb{Z}$.

]]>Still on the $\mathbb{Z}$ vs. $\mathbb{Z}/2$ issue in Kapranov’s proposal: what about replacing the sphere spectrum with its ∞-group of units $(\Omega^\infty\mathbb{S})^\times$?

In the classical case we have $(\mathbb{Z},\times,1)^\times=\{-1,1\}\cong\mathbb{Z}/2$, giving

$\begin{aligned} \pi_0((\Omega^\infty\mathbb{S})^\times) &\cong \pi_0(\Omega^\infty\mathbb{S})^\times\\ &\cong \pi_0(\mathbb{S})^\times\\ &\cong \mathbb{Z}^\times\\ &\cong \mathbb{Z}/2. \end{aligned}$Meanwhile, $\pi_n((\Omega^\infty\mathbb{S})^\times))\cong\pi_n(\mathbb{S})$ for $n\geq1$, keeping in particular $\pi_1(\mathbb{S})\cong\mathbb{Z}/2$, $\pi_2(\mathbb{S})\cong\mathbb{Z}/2$, and $\pi_3(\mathbb{S})\cong\mathbb{Z}/24$ (differently from $\Sigma^{\infty-1}\mathbb{RP}^\infty$).

]]>Hi Dmitri,

Sure! I’ll add this (and some other things) there in a bit.

]]>Re #30: Would you considering transfering your understanding to the nLab article then? It could certainly benefit from having more details.

]]>Hi Marc,

I think I finally understood it! Thank you very much! :)

]]>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.

]]>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.

Thanks! This is really nice!

The universal property is HA 4.8.1.10 (4) (in the special case of Cor. 4.8.1.12): If AA is symmetric monoidal, the Day convolution on PSh(A)PSh(A) is the universal cocomplete symmetric monoidal category under A where the tensor product preserves colimits in each variable.

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?)

]]>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.

]]>Hi Marc,

Thank you very much for your explanation! I’ve been quite confused about this lately.

by the universal property of Day convolution a symmetric monoidal adjunction between $Fun(Pic(C),Spc)$ and $C$.

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?)

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?

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

Thanks!

]]>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.

]]>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 $E$ has an underlying $\mathbb{S}$-graded $\mathbb{E}_\infty$-space $\Omega^\mathcal{J}(E)$, and by the equivalence between connective spectra and *grouplike* $\mathbb{E}_\infty$-spaces, we see that so does any grouplike $\mathbb{E}_\infty$-space. I’m a bit confused though about whether this applies also to not necessarily grouplike $\mathbb{E}_\infty$-spaces: do we have an $\mathbb{S}$-graded $\mathbb{E}_\infty$-space $\Omega^\mathcal{J}(X)$ for any such not necessarily grouplike $\mathbb{E}_\infty$-space $X$?

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

]]>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.

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 $E$ one can associate an $\mathbb{S}$-graded $\mathbb{E}_\infty$-space $\Omega^\mathcal{J}(E)$ (which is indeed *very* fascinating!). Why are general $E_\infty$-monoids canonically (and non-trivially) $\mathbb{S}$-graded, though?

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

Why do you say it “departs” from that?

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

]]>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?

]]>1. Fascinatingly, every $E_\infty$-ring is automatically $\mathbb{S}$-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 “$\mathbb{S}$-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 $\mathbb{S}$-grading” on an $\mathbb{E}_{\infty}$-space $A$ is the same thing as a morphism of $\mathbb{E}_\infty$-spaces $A\to Q S^0$ (where one departs from the classical identification $\{\text{gradings on a monoid }\,A\}\cong\{\text{monoid morphisms }\,A\to(\mathbb{Z},+,0)\}$)?

]]>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.

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

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

]]>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.

]]>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?

]]>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.

]]>