In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is described as some non-explicit functor which associates a stack $M_G $ to any compact Lie group $G $. It is claimed that this functor can be extended from Lie groups to "extensions of Lie groups by $B\mathbb G_m $", but neither it is explained what kind of objects such extensions really are nor what is really this higher equivariant homotopy. Naturally, it is also claimed that there exists "n-equivariant homotopy theory" for spectra associated to formal groups of higher height.

While I can take some guesses at parts of this statement (e.g. a $B\mathbb G_m $ extension can be considered either as a topological group or as a category in manifolds ), most of it remains a mystery to me. Some googling didn't help me to find any references. Thus the question: what is this higher equivariant theory and where can I read about it?

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    $\begingroup$ I don't quite understand why so say that Lurie claims that $M$ can be extended to "extensions of Lie groups by $BG_m$". I see nowhere that he says this. What he does say is that a suitable extension of $G$, classified by a level $\ell\in H^4(BG,Z)$, is sent by $M$ to a $G_m$-torsor $L$ over $M_G$. Note that $G_m$ is here an object in derived algebraic geometry, not an object related to Lie groups. ... $\endgroup$ Dec 2 '15 at 15:05
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    $\begingroup$ Lurie doesn't spell out concretely what is supposed to be classified by the level $\ell$. But an $\ell$ determines a fiber sequnce $K(Z,3)\to B\widetilde{G}\to BG\to K(Z,4)$, which in turn can be thought of as giving a central extension $K(Z,2)\to \widetilde{G}\to G$. I think that is all to the story that Jacob is telling here, and that's what I tried to extrapolate in my answer. ... $\endgroup$ Dec 2 '15 at 15:11
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    $\begingroup$ I agree that it is reasonable to assign a more concrete geometric meaning to such extensions (I would try to describe it in terms of central extensions of $G$ by $\mathbb{B}U(1)$, as groups in the smooth infinity-topos.) However, assigning such geometric meaning seems to me to be largely orthogonal to what Jacob does in section 5, which is mainly about homotopy theoretic/infinity categorical manipulations. $\endgroup$ Dec 2 '15 at 15:14
  • $\begingroup$ Five years on, I'm really curious whether this story has been fleshed out anywhere. $\endgroup$
    – pupshaw
    Oct 15 '20 at 18:24
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    $\begingroup$ @pupshaw I don't think it has, but perhaps it will be in Lurie's Elliptic IV. Some preparatory work for this appears in Elliptic III. My guess is that what I said in my answer is going to be basically correct as far as it goes, but that Jacob is going to formulate the ultimate results somewhat differently, in particular making use of what is now understood about "ambidexterity", and which will include a proper notion of "equivariant stable homotopy theory for higher groups" (where "higher groups" means "$\pi$-finite spaces"). $\endgroup$ Oct 17 '20 at 18:45

I would also like to know the answer to your question. Since no one has given an answer yet, I'll speculate recklessly and irresponsibly on how this might work (by riffing off of the final paragraph of section 5.1 of the survey).

Before figuring out what "higher equivariant homotopy theory" is, we should first know what "equivariant homotopy theory" is. In this case, we want "global equivariant homotopy theory", aka the homotopy theory of smooth stacks, as introduced by Gepner-Henriques.

They show that the homotopy theory of smooth stacks is modelled by the homotopy theory $PSh(Orb)$ of presheaves of infinity-groupoids on $Orb$. Here $Orb$ is a topologically enriched category (i.e., infinity category) whose objects are compact Lie groups $G$, and whose morphism spaces are given by $$Orb(G,H) := BFun(G,H),$$ a classifying space of the category of smooth functors (so path components of $Orb(G,H)$ correspond to conjugacy classes of homomorphisms $\phi\colon G\to H$, and the path component containing $\phi$ is the classifying space of the centralizer of $\phi$)

Remark: there is an evident map $B\colon Orb(G,H)\to Map(BG,BH)$. When $H$ is an extension of an abelian group by a torus, it is a weak equivalence; but it is not generally a weak equivalence otherwise.

Jacob's construction apparently associates to each derived oriented elliptic curve $E\to Spec(A)$ a functor $M\colon Orb\to Sch_A$ taking values in derived schemes over $A$. The curve $E=M(U(1))$. Section 5.1 suggests extending this construction by incorporating a "level".

So let's define $\widetilde{Orb}$ to have objects $(G,\ell)$, where $G$ is a compact Lie group and $\ell\colon BG \to K(\Lambda, 4)$ a map, where $\Lambda$ is a free abelian group. The morphism space $\widetilde{Orb}((G,\ell), (G',\ell'))$ is the homotopy pullback of $$Orb(G,G') \to Map(BG,BG') \to Map(BG, K(\Lambda',4)) \leftarrow Map(K(\Lambda,4), K(\Lambda',4)).$$ I.e., a "map" $(G,\ell)\to (G',\ell')$ is a homomorphism $\phi\colon G\to G'$ together with $f\colon K(\Lambda,4)\to K(\Lambda',4)$ and a homotopy $f\circ \ell \sim \ell'\circ B\phi$.

Now we can define 2-equivariant homotopy theory to be $PSh(\widetilde{Orb})$

Ideally, at this point you would identify $\widetilde{Orb}$ with something more geometric, e.g., some suitable category of Lie 2-groups. Maybe you would even identify $PSh(\widetilde{Orb})$ with the homotopy theory of mumble mumble smooth 2-stacks mumble (I have no actual idea here). However, we don't need to do this if we just want to play with homotopy theory: we just need $\widetilde{Orb}$.

Note that $Orb$ can be identified as the full subcategory of $\widetilde{Orb}$ consisting of $(G,0\colon BG\to K(0,4))$. I think Jacob is claiming that the functor $M\colon Orb\to Sch_A$ can be extended to a functor from $\widetilde{Orb}$. This wouldn't be an arbitrary extension, but would satisfy some compatiblities, perhaps including:

  • Compatibility with base change: given a homomorphism $\phi\colon H\to G$, require that $M(H,\ell\circ B\phi)\to M(G,\ell)$ be the evident base change along $M(H,0)\to M(G,0)$.

  • Given two levels $\ell\colon G\to K(\Lambda,4)$ and $\ell'\colon G\to K(\Lambda',4)$, $M(G,\ell\oplus \ell')$ should be the pullback of $M(G,\ell)\to M(G,0)\leftarrow M(G,\ell')$. This would imply that that the values of $M$ are determined by those on objects on $Orb$, together with $(G,\ell)$ with $\ell\colon BG\to K(Z,4)$.

  • The previous implies that $M(e,Be\to K(Z,4))$ has the structure of an abelian group object in $Sch_A$. We would require that this object be $\mathbb{G}_m$.

  • Require every $M(G,BG\to K(Z,4))$ be a $\mathbb{G}_m$-torsor.

Then Jacob's theorem is possibly something like: an extension of $M$ satisfying a list of compatibilities is entirely determined by specifying a value at the cup-product level $\ell\colon BU(1)\times BU(1)\to K(Z,4)$, which itself should satisfy some properties/structure ("symmetric biextension of $M(U(1))=E$ by $\mathbb{G}_m$").

One might guess that n-equivariant homotopy theory is built from pairs $(G,\ell)$, where $\ell \colon BG\to Z$ is a map to some suitable type of $(n+2)$-truncated space $Z$. How this ought to be specified probably depends on what you want to accomplish with it. I don't have any idea about that.

  • $\begingroup$ I like your approach. Two notes: $Be \to K(Z, 4)$ should be not $\mathbb G_m$ but rather $B\mathbb G_m$ with a suitable rigid structure (over complex numbers it should be $\mathbb C(x)^* / \mathbb C^*$). Also I think maps $BG \to K(Z, 4)$ may be not rigid enough, we need to actually get a group extension by $B\mathbb G_m$. $\endgroup$ Dec 1 '15 at 0:08

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