aa r X i v : . [ m a t h . A T ] J u l ON EQUIVARIANT TOPOLOGICAL MODULAR FORMS
DAVID GEPNER AND LENNART MEIER
Abstract.
Following ideas of Lurie, we give in this article a general construction ofequivariant elliptic cohomology without restriction to characteristic zero. Specializingto the universal elliptic curve we obtain in particular equivariant spectra of topologicalmodular forms. We compute the fixed points of these spectra for the circle group andmore generally for tori.
Contents
1. Introduction 12. Orbispaces 53. Abelian group objects in ∞ -categories 114. Elliptic curves, formal completions and orientations 155. Equivariant elliptic cohomology 196. Eilenberg–Moore type statements 237. Elliptic cohomology is symmetric monoidal 258. Elliptic cohomology and equivariant spectra 279. The circle-equivariant elliptic cohomology of a point 28Appendix A. Quotient ∞ -categories 31Appendix B. Comparison of models for the global orbit category 33Appendix C. The ∞ -category of G -spectra 36References 431. Introduction
The aim of this paper is to construct an integral theory of equivariant elliptic cohomologyfor an arbitrary compact Lie group and prove some of its basic properties. In particular, thisapplies to elliptic cohomology based on the universal elliptic curve over the moduli stackof elliptic curve, yielding compatible G -equivariant spectra of topological modular forms TMF for every compact abelian Lie group G . The construction follows the ideas sketchedin [Lur09b] and builds crucially on the theory Lurie detailed in subsequent work. One ofour main results beyond the construction is the computation of the fixed points of TMF with respect to the circle-group T , identifying these with TMF ⊕ ΣTMF .1.1.
Motivation and main results.
The G -equivariant complex K-theory of a pointagrees per definition with the representation ring of G . Together with Bott periodicitythis gives for example KU even T (pt) ∼ = Z [ t ± ] and KU odd T (pt) = 0 . The goal of the presentpaper is to repeat this calculation at the level of equivariant elliptic cohomology.The basic idea of equivariant elliptic cohomology is inspired from the following algebro-geometric interpretation of the computation above: The spectrum Spec Z [ t ± ] coincideswith the multiplicative group G m . Thus, for every T -space X the K-theory KU even T ( X ) defines a quasi-coherent sheaf on G m . In elliptic cohomology, one replaces G m by an ellipticcurve and thus T -equivariant elliptic cohomology takes values in quasi-coherent sheaves onan elliptic curve.There haven been several realizations of this basic idea. The first was given by Gro-jnowski [Gro07] in 1995, taking a point τ in the upper half-plane and the resulting complexelliptic curve C / Z + τ Z as input and producing a sheaf-valued G -equivariant cohomologytheory for every compact connected Lie group G . This was motivated by applications ingeometric representation theory (see [GKV95], and e.g. [AO16] [YZ17] [FRV +
18] for later de-velopments). Another motivation was provided by Miller’s suggestion to use T -equivariantelliptic cohomology to reprove the rigidity of the elliptic genus, which was realized by Rosu[Ros01] and Ando–Basterra [AB02]. We do not want to summarize all work done on equi-variant elliptic cohomology, but want to mention Greenlees’s work on rational T -equivariantcohomology [Gre05], work of Kitchloo [Kit19], Rezk [Rez19a], Spong [Spo19] and Berwick-Evans–Tripathy [BET19] in the complex-analytic setting and the work of Devoto [Dev96]on equivariant elliptic cohomology for finite groups and its applications to moonshine phe-nomena in [Gan09].Many more works could be named, but for us the most relevant work has been done byLurie in [Lur09b] [Lur19]. The starting point is the definition of an elliptic cohomologytheory, consisting of an even-periodic ring spectrum R , an elliptic curve E over π R , and anisomorphism between the formal group associated with R and that associated with E . Lurierefined this to the notion of an oriented spectral elliptic curve , consisting of an even-periodic E ∞ -ring spectrum R , an elliptic curve E over R and an equivalence between the formalgroups Spf R CP ∞ and b E over R . Thus we have moved completely into the land of spectralalgebraic geometry, instead of the hybrid definition of an elliptic cohomology theory. Thisseems to be of key importance in order to obtain integral equivariant elliptic cohomologytheories, without restriction to characteristic zero. Following the outline given in [Lur09b]and extending work of [Lur19] from finite groups to compact Lie groups, we associate toevery oriented spectral elliptic curve E a “globally” equivariant elliptic cohomology theory.To make this precise, we work with orbispaces, an ∞ -category incorporating G -equivarianthomotopy theory for all compact Lie groups G at once, which was introduced in [GH07].As any topological groupoid or stack (such as the stack [ X/G ] associated to a G -space X )determines an orbispace, they are a convenient source category for cohomology theorieswhich are equivariant for all G simultaneously, often called global cohomology theories .We construct from the datum of an oriented elliptic curve E over an E ∞ -ring R , forany orbispace Y , a stack Ell( Y ) in the world of spectral algebraic geometry. For example,in the case of B T = [pt / T ] , the stack Ell( B T ) is precisely the spectral elliptic curve E and Ell( B C n ) agrees with the n -torsion E [ n ] . Moreover, for any compact abelian Lie group G pushforward of the structure sheaf along Ell([
X/G ]) → Ell( B G ) defines a contravariantfunctor from (finite) G -spaces to quasi-coherent sheaves on Ell( B G ) . In particular, we obtaina functor E ll T : ( finite T -spaces ) op → QCoh( E ) . Thus, we obtain a derived realization of the basic idea of T -equivariant elliptic cohomologysketched above. Taking homotopy groups defines indeed functors E ll i T = π − i E ll T to quasi-coherent sheaves on the underlying classical elliptic curve. Finally, to push the T -equivariantelliptic cohomology functor into the realm of ordinary equivariant homotopy theory, wepostcompose E ll T with the global sections functor Γ , to obtain a functor Γ E ll T : ( finite T -spaces ) op → QCoh( E ) Γ → Spectra . N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 3
We show this functor is representable by a genuine T -spectrum, which we also denote R . Inparticular, Γ E ll T (pt) coincides with the T -fixed points R B T . This allows us to formulatethe main computation of this article.
Theorem 1.1.
Restriction and degree-shifting transfer determine an equivalence of spectra R ⊕ Σ R → R B T . Instead of an oriented spectral elliptic curve over an affine spectral scheme, we might alsodirectly work with the universal oriented elliptic curve. The associated equivariant theoryhas as underlying spectrum
TMF , the Goerss–Hopkins–Miller spectrum of topological mod-ular forms. Thus we may speak of a genuine T -spectrum TMF (and likewise of a genuine G -spectrum TMF for all compact abelian Lie groups G ) and easily deduce from our maintheorem the following corollaries: Corollary 1.2.
Restriction and degree-shifting transfers determine an equivalence of spectra
TMF ⊕ ΣTMF → TMF B T . This coincides with the ring spectrum of global sections of the structure sheaf of the universaloriented spectral elliptic curve. In particular, the reduced theory ] TMF( B T ) ≃ fib(TMF B T → TMF) ≃ ΣTMF is an invertible
TMF -module.
Corollary 1.3.
Restriction and degree-shifting transfers determine an equivalence of spectra M S ⊂{ ,...,n } Σ | S | TMF → TMF B T n . Note that this in particular amounts to a computation of the graded ring
TMF ∗ T n (pt) .The previous corollary also implies a dualizability property for equivariant TMF , which isin stark contrast with the situation for K-theory, where π KU G = R ( G ) has infinite rankover π KU = Z if G is a compact Lie group of positive dimension. Corollary 1.4.
For every compact abelian Lie group G , the G -fixed points TMF B G are adualizable TMF -module.
We conjecture actually the corresponding statement to be true for all compact Lie groups.For finite groups, this is a consequence of Lurie’s Tempered Ambidexterity Theorem [Lur19]in his setting of equivariant elliptic cohomology.1.2.
Outline of the paper.
We start with a section about orbispaces. We will constructthe ∞ -category of orbispaces S Orb = P (Orb) as presheaves on the global orbit category Orb , whose objects are the classifying stacks B G for compact Lie groups G . Our treatmentof this is ∞ -categorical and essentially self-contained.Section 3 is devoted to a general framework for constructing global cohomology theoriesusing a preoriented abelian group object in a presentable ∞ -category X . This is stronglyinspired by the sketch provided in [Lur09b]. More precisely, we associate to a preorientedabelian group object a functor S Orb → X .In Section 4, we introduce our main example of such an X , namely the ∞ -category Shv( M ) of sheaves on a given non-connective spectral Deligne–Mumford stack M . Moreover, we give We use the notation R B T for what is more commonly denoted R T , namely the T -fixed point spectrum, asour notation both stresses the importance of the stack B T and avoids confusion with the function spectrum Map(Σ ∞ + T , R ) , which we will also have opportunity to use. DAVID GEPNER AND LENNART MEIER a precise definition of an orientation of an elliptic curve over such an M . While not necessaryfor the definition of equivariant elliptic cohomology, it is crucial for its finer properties.The actual construction of equivariant elliptic cohomology is given in Section 5. From apreoriented elliptic curve, we will both construct a functor Ell : S Orb → Shv( M ) and variantstaking values in quasi-coherent sheaves.In Section 7 we show that the resulting functor E ll T : ( finite T -spaces ) op → QCoh( E ) is symmetric monoidal. This is based on the orientation b E ≃ Spf R B T (if M = Spec R ). Acrucial ingredient is the study of fiber products of the form Spf R BC n × Spf R B T Spf R BC m .This will be done in the equivalent setting of coalgebras and their cospectra in Section 6,employing essentially the convergence of the Eilenberg–Moore spectral sequence.Using the symmetric monoidality and the universal property of G -spectra as invertingrepresentation spheres (see Appendix C), we show in Section 8 that E ll T actually factorsover finite T -spectra. This allows us to use the Wirthmüller isomorphism calculating thedual of Σ ∞ T + to compute the dual of E ll T ( T ) , which is the key ingredient in the proof ofour main theorem in Section 9. Note that while by construction Γ E ll T (pt) coincides withglobal sections of O E and this is an object purely in spectral algebraic geometry, we actuallyuse T -equivariant homotopy theory for its computation.We end with three appendices. Appendix A discusses quotient ∞ -categories, which areused in our treatment of orbispaces. Appendix B compares our treatment of the globalorbit category Orb with the original treatment in [GH07]. Finally, Appendix C gives an ∞ -categorical construction of (genuine) G -spectra for arbitrary compact Lie groups G andcompares it to orthogonal G -spectra, and uses work of Robalo [Rob15] to establish a uni-versal property of the ∞ -category of G -spectra. Conventions.
In general, we will freely use the terminology of ∞ -categories and spectralalgebraic geometry, for which we refer to the book series [Lur09a], [Lur12] and [Lur16]. Inparticular, an ∞ -category will be for us a quasicategory. One difference though is that wewill assume all (non-connective) spectral Deligne–Mumford stacks to be locally noetherian,i.e. they are étale locally of the form Spec A with π A noetherian and π i A a finitely generated π A -module for i ≥ . Moreover, we assume (non-connective) spectral Deligne–Mumfordstacks to be quasi-separated, i.e. the fiber product of any two affines over such a stack isquasi-compact again. If we write M = ( M , O M ) , then M denotes the underlying ∞ -toposof the (non-connective) spectral Deligne–Mumford stack M and O M its structure sheaf. Incontrast, Shv( M ) will denote the sheaves on the big étale site (see Definition 4.3).We use S as notation for the ∞ -category of spaces and Sp for the ∞ -category of spectra.Likewise, we use S G for the ∞ -category of G -spaces and Sp G for the ∞ -category of genuine G -spectra (see Appendix C for details). We use the notation E B H for the H -fixed points ofa G -spectrum E for H ⊂ G . We will use P for S -valued presheaves. When speaking abouttopological spaces, we assume them to be compactly generated and weak Hausdorff.We will generally follow the convention that Map denotes mapping spaces (or mappinggroupoids), while
Map denotes internal mapping objects; depending on the context, thesemight be mapping spectra or mapping (topological) groupoids. This choice of notationemphasizes the fact that we always work ∞ -categorically, and that all limits and colim-its are formed in the appropriate ∞ -category. Moreover, we use | X • | as a shorthand for colim ∆ op X • . Regarding other special limits, we often write pt for the terminal object of an N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 5 ∞ -category. Moreover, we recall that every cocomplete ∞ -category is tensored over S andwe use the symbol ⊗ to refer this tensoring. Acknowledgments.
Despite lacking firm foundations until more recently, equivariant el-liptic cohomology is by now an old and diverse subject, going back to the 1980s and ad-mitting applications across much of modern mathematics and physics. Many people havecontributed to the subject and their contributions are too numerous to name. But we’dlike to thank Matthew Ando, John Greenlees, and Jacob Lurie in particular for shaping ourthinking about equivariant elliptic cohomology. We furthermore thank Thomas Nikolausfor his input at the beginning of this project, Viktoriya Ozornova and Daniel Schäppi foruseful conversations about categorical questions and Stefan Schwede for his comments onan earlier version.The authors would like to thank the Isaac Newton Institute for Mathematical Sciencesfor support and hospitality during the programme “Homotopy harnessing higher structures”when work on this paper was undertaken. This work was supported by EPSRC grantnumber EP/R014604/1. Additionally, the authors would like to thank the MathematicalSciences Research Institute for providing an inspiring working environment during the pro-gram “Higher categories and categorification”.2.
Orbispaces
In this section, we will introduce the particular framework of global unstable homotopytheory we will work in: orbispaces. These were first introduced in [GH07], but we chooseto give a (mostly) self-contained and ∞ -categorical treatment. While philosophically ourapproach is similar to [GH07], there are certain technical differences and we refer to Ap-pendix B for a precise comparison. A different approach is taken in [Sch18], where unstableglobal homotopy theory is based on orthogonal spaces, which is in the same spirit as usingorthogonal spectra to model stable global homotopy theory. The orbispace approach of[GH07] and the orthogonal space approach have been shown to be in equivalent in [Sch20]and [Kör18]. Another valuable source on orbispaces is [Rez14], though beware that he usesthe term ‘global spaces’ for our orbispaces.We will construct the ∞ -category S Orb of orbispaces in a three step process. We will firstdefine an ∞ -category Stk ∞ of topological stacks, then define Orb as the full subcategoryon the orbits [pt /G ] for G compact Lie and lastly define S Orb as presheaves on
Orb . Thisrequires first recalling basic concepts about topological stacks, which we will do next.We write
Top for the cartesian closed category of compactly generated weak Hausdorfftopological spaces. We will also be interested in the cartesian closed -category TopGpd of topological groupoids, and we will typically write X • , Y • ,... for topological groupoids,viewed as simplicial topological spaces. Since we will also consider topological stacks, wewill additionally need to equip the category
Top with the Grothendieck topology; althoughthe usual “open cover” topology would work just as well for us, for consistency we choosethe “étale” topology (covers are generated by open covers and covering spaces; see [GH07]for details). The topological groupoid
Map
TopGpd ( Y • , X • ) of maps has as its objects space the space of functorsand as morphism space the space of natural transformations. Their definition is analogous to that forclassical groupoids, e.g. the space of functors is the evident subspace of the product of Map
Top ( Y , X ) and Map
Top ( Y , X ) . Here the mapping spaces are equipped with the compact-open topology. This is a specialcase of the cartesian closedness of internal categories [nLa20]. DAVID GEPNER AND LENNART MEIER
We write
Gpd for the -category of groupoids. This is an example of (2 , -category, i.e.a -category whose -morphisms are invertible. We will view (2 , -categories implicitly as ∞ -categories via their Duskin nerve and refer to [GHN17, Appendix A] and [Lur20, Section2.3 009P] for details. If we allow weak (2 , -categories, an ∞ -category arises as a Duskinnerve if and only if it is a -category in the sense of [Lur09a, Section 2.3.4], i.e. the innerhorn liftings are unique in dimensions greater than [Dus02, Theorem 8.6].For us, a topological stack will mean a sheaf of groupoids on Top in the sense of highercategory theory. In order to avoid universe issues we must index our stacks on a small fullsubcategory of
Top ′ ⊂ Top . We suppose that it contains up to homeomorphism all countableCW-complexes and furthermore that it is closed under finite products and subspaces; fortechnical reasons, we will also demand that every object of
Top ′ is paracompact. Theparticular choice of indexing category is not very relevant for our purposes. We will assumein doubt that the object and morphism space of every topological groupoid is in Top ′ . Definition 2.1. A topological stack is a functor X : Top ′ op → Gpd which satisfies the sheafcondition; that is, for all T ∈ Top ′ and all coverings p : U → T in Top ′ , the canonical map X ( T ) / / lim { X ( U ) / / / / X ( U × T U ) / / / / / / X ( U × T U × T U ) } is an equivalence. The -category of stacks is the full subcategory Stk ⊂ Pre Stk = Fun(Top ′ op , Gpd) of the -category of prestacks (that is, presheaves of groupoids on topological spaces).Our actual interest is in an ∞ -categorical quotient of the -category of topological stacks,which we denote Stk ∞ and refer to as the ∞ -category of “stacks modulo homotopy”. Theidea is to view the -category of topological stacks as enriched over itself via its cartesianclosed (symmetric) monoidal structure, and then to use the realization functor |−| : Stk → S to change enrichment (see [GH07] for a more precise formulation along this line). What thisessentially amounts to is imposing a homotopy relation upon the mapping spaces, whichmotivates our implementation of this idea. Definition 2.2.
The ∞ -category Stk ∞ (respectively, Pre Stk ∞ ) is the quotient of the -category Stk of topological stacks (respectively, the -category Pre Stk of topologicalprestacks) induced by the action of the standard cosimplicial simplex ∆ • Top : ∆ → Top . Heuristically, this ∞ -category has the same objects, with mapping spaces Map(
X, Y ) ≃ | Map
Fun(Top op , Gpd) ( X × ∆ • Top , Y ) | . Composition uses that geometric realizations commute with finite products and the diagonalmap ∆ • Top → ∆ • Top × ∆ • Top . See Construction A.6 for details of this construction.
Remark 2.3.
The definition is motivated by the following observation: If we take thequotient of the -category of CW-complexes by the action of the cosimplicial simplex ∆ • Top : ∆ → Top , we obtain the ∞ -category S of spaces. Indeed: Taking homwise Sing of the topological category of CW-complexes, we obtain a simplicial category C , which isequivalent to the simplicial category of Kan complexes. Moreover, the quotient categorywe are considering is by definition colim ∆ op N C • , where C n is the category obtained fromtaking the n -simplices in each mapping space and the colimit is taken in Cat ∞ . That thisagrees up to equivalence with N coh C ≃ S can be shown analogously to Corollary B.3. Here and in the following we regard topological spaces as topological stacks via their representablefunctors.
N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 7
Remark 2.4.
The assignment X Map
Stk ∞ (pt , X ) is one way to assign a homotopy typeto a topological stack. For alternative treatments we refer to [GH07], [Noo12] and [Ebe09];geometric applications have been given in [Car16]. Given a presentation of a topologicalstack by a topological groupoid, all of these sources show the equivalence of the homotopytype of the stack with the homotopy type of the topological groupoid. As we will do so aswell in Proposition 2.11, our approach is compatible with the cited sources.An especially important class of stacks for us will be orbit stacks. Definition 2.5.
A topological stack X is an orbit if there exists a compact Lie group G and an equivalence of topological stacks between X and the stack quotient [pt /G ] . Remark 2.6.
Given a compact Lie group G , we write B G for the stack of principal G -bundles. This is an orbit stack, as a choice of basepoint induces an equivalence of topologicalstacks [pt /G ] ≃ B G , and all choices of basepoint are equivalent. We will typically write B G for an arbitrary orbit stack; strictly speaking, however, one cannot recover the group G from the stack B G without a choice of basepoint, in which case G ≃ Ω B G .While the identification X ≃ [pt /G ] requires the choice of a basepoint pt → X , one cancharacterize the orbit stacks alternatively, without having to choose a basepoint of X : theyare the smooth and proper Artin stacks which admit representable surjections pt → X fromthe point.The reason we are primarily concerned with orbit stacks is that they generate the ∞ -category of orbispaces [GH07]. One should think of an orbispace as encoding the various“fixed point spaces” of a topological stack, just as the equivariant homotopy type of a G -space is encoded by the fixed point subspaces, ranging over all (closed) subgroups H of G . Since topological stacks are typically not global quotient stacks, one cannot restrict tosubgroups of one given group, but simply indexes these spaces on all compact Lie groups,regarded as orbit stacks. These are the objects of the ∞ -category Orb of orbit stacks.
Definition 2.7.
The global orbit category
Orb is the full subcategory of the ∞ -category of Stk ∞ on those objects of the form B G , for G a compact Lie group. Definition 2.8. An orbispace is a functor Orb op → S . The ∞ -category of orbispaces is the ∞ -category S Orb = Fun(Orb op , S ) of functors Orb op → S . Remark 2.9.
The Yoneda embedding defines a functor
Stk ∞ → S Orb , X ( B G Map
Stk ∞ ( B G, X )) . In this way, we may view every stack as an orbispace, viewed through the lens of the orbitstacks. In particular, when later defining cohomology theories on orbispaces, these will alsodefine cohomology theories of topological stacks.In order to understand the ∞ -category of orbispaces, we must first understand the map-ping spaces in Orb or more generally mapping spaces in
Stk ∞ . As almost any computationabout stacks, this is done by choosing presentations of the relevant stacks: given a topolog-ical groupoid X • , we denote the represented prestack the same way and obtain a stack X †• by stackification. We view X • as a presentation of the associated stack. The example wecare most about is the presentation { G →→ pt } of B G . The theory in [GH07] allows for an arbitrary family of groups, but we will only consider the case of thefamily of compact Lie groups. For morphisms between orbit stacks we allow all continuous homomorphisms,as these are automatically smooth [Fer98, Theorem 3.7.1].
DAVID GEPNER AND LENNART MEIER
One thing a presentation allows us to compute is the homotopy type of a stack, based onthe homotopy type of a topological groupoid: for a topological groupoid X • , we considerthe space | X • | arising as the homotopy colimit colim ∆ op X • or equivalently as the fat real-ization of X • (cf. Appendix B). While the following two results are not strictly necessary tounderstand Orb , they are reassuring pieces of evidence that our notion of mapping spacesbetween topological stacks is “correct”.
Lemma 2.10.
Let X • be a topological groupoid. There are natural weak equivalences ofspaces | X • | ≃ | Map
Pre Stk (∆ ∗ Top , X • ) | ≃ Map
Pre Stk ∞ (pt , X • ) . Proof.
The space of maps from ∆ n to X is represented by the simplicial set (Sing X • ) n .Thus, | Map
TopGpd (∆ ∗ , X • ) | ≃ || (Sing X • ) ∗ || , where the inner realization is with respect to • and the outer with respect to ∗ . Here weuse that the space associated with a simplicial set K • agrees with | K • | , where we view each K n as a discrete space.To obtain the result, we just have to use the natural equivalence | (Sing X n ) ∗ | → X n for every n and that the order of geometric realization does not matter (as these are justcolimits over ∆ op ).The second is equivalence is by definition. (cid:3) Proposition 2.11.
Let X • be a topological groupoid. Then there is a natural equivalence | X • | ≃ Map
Stk ∞ (pt , X † ) .Proof. We recall from [GH07, Section 2.5 and 3.3] that there is a class of fibrant topologicalgroupoids with the property that if Y • is fibrant, the associated prestack is already a stack.Moreover, there is a fibrant replacement functor X • → fib( X • ) , which is both a homotopyequivalence and a categorical equivalence, that is: it is an equivalence in the topologicalcategory of topological groupoids (obtained by only considering the space of functors, notof natural transformations) and in the -category of topological groupoids (obtained byforgetting the topology on the mapping groupoids).From the first property, we see that | X • | → | fib( X • ) | is an equivalence; from the secondproperty that X †• → fib( X • ) † ≃ fib( X • ) is an equivalence in Stk and hence in
Stk ∞ . Thus,we can assume that X • is fibrant. In this case, Stk ∞ (pt , X †• ) agrees with PreStk ∞ (pt , X • ) and we can apply the previous lemma. (cid:3) We are now ready to calculate the mapping spaces in
Orb . In the proof we will use aconcept of interest in its own right, namely action groupoids: Given a topological group G and a G -space X , the action of G on X may be encoded via a topological groupoid, calledthe action groupoid , which we will denote { G × X →→ X } , where one of the arrows is theprojection and the other one the action. Proposition 2.12.
For compact Lie groups G and H , there is a chain of equivalences Map
Orb ( B H, B G ) ≃ | Map
TopGpd ( H →→ pt , G →→ pt) | ≃ Map
Lie ( H, G ) hG , which is natural in morphisms of compact Lie groups. Here, the action of G on the homo-morphisms is by conjugation. We stress that the double bars stand here for an iterated geometric realization and not for a fat geometricrealization.
N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 9
Proof.
By Proposition B.6, there is a natural equivalence
Map
Orb ( B H, B G ) ≃ | Map
TopGpd ( H →→ pt , G →→ pt) | . One easily calculates that the mapping groupoid
Map
TopGpd ( H →→ pt , G →→ pt) is isomorphicto the action groupoid { Map
Lie ( H, G ) × G →→ Map
Lie ( H, G ) } , with the action given byconjugation. As the realization of an action group is the homotopy quotient, the resultfollows. (cid:3) Remark 2.13.
An alternative approach to prove Proposition 2.12 would be to use fibranttopological groupoids again as in the proof of Proposition 2.11. And indeed there is no reasonto expect in general an equivalence between
Map
Stk ∞ ( X †• , Y †• ) and | Map
TopGpd ( X • , Y • ) | if Y • is not fibrant and one should rather consider derived mapping groupoids between topologicalgroupoids as in [GH07]. The special feature we used in the preceding proposition is thatthe object space of the source is contractible. Remark 2.14.
Denote by
CptLie the topological category of compact Lie groups, whichwe view as an ∞ -category. We claim that the association G
7→ B G defines an equivalence CptLie → Orb ∗ , where B G is equipped with the base point coming from the inclusion ofthe trivial group into G .To see this, first consider the functor CptLie → TopGpd which sends G to { G →→ pt } .Composing with the functors TopGpd → Stk → Stk ∞ , we observe that the image of CptLie lies in the full subcategory
Orb ⊂ Stk ∞ . Since pt is an initial and terminal object of CptLie ,we see that
CptLie is naturally a pointed ∞ -category, and the functor CptLie → Orb therefore factors through the projection
Orb ∗ → Orb . The equivalence
CptLie ≃ Orb ∗ nowfollows from direct calculation: The diagram Map
CptLie ( H, G ) / / (cid:15) (cid:15) Map
CptLie ( H, G ) hG / / (cid:15) (cid:15) BG (cid:15) (cid:15) Map
Orb ∗ ( B H, B G ) / / Map
Orb ( B H, B G ) / / Map
Orb (pt , B G ) defines a morphism of fiber sequences in which the middle and rightmost vertical maps areequivalences. It follows that the leftmost vertical map is also an equivalence.We conclude this section with a couple of observations about the relationship betweenorbispaces and G -spaces. Above we have defined for a topological group G and a G -space X an action groupoid . This is functorial in morphisms of G -spaces: given a second G -space Y and a G -equivariant map Y → X , we obtain a morphism of action groupoids { G × Y →→ Y } →{ G × X →→ X } by applying f : Y → X on the level of objects and (id G , f ) : G × Y → G × X onthe level of morphisms. Moreover, the resulting functor Top G → TopGpd is compatible withthe topological enrichement. However, it is not fully faithful unless G is trivial, as morphismsof topological groupoid need not be injective on stabilizer groups. This is remedied byinsisting that the functors are compatible with the projection to { G →→ pt } . Proposition 2.15.
The action groupoid functor
Top G → TopGpd , for any topological group G , factors through the projection TopGpd / { G →→ pt } → TopGpd , and the induced functor
Top G → TopGpd / { G →→ pt } is fully faithful (even as a functor of topologically enriched categories). Proof.
The space of topological groupoid morphisms is the subspace of the product
Map
Top ( Y, X ) × Map
Top ( G × Y, G × X ) consisting of those maps which are compatible with the groupoid structure as well as theprojection to { G →→ pt } . These conditions are exactly what is needed to ensure that foran allowed pair ( f, h ) , the map h is determined by f and f is G -equivariant: Indeed,writing Map
Top ( G × Y, G × X ) ∼ = Map Top ( G × Y, G ) × Map
Top ( G × Y, X ) , we see that themap G × Y → G must be the projection in order to be compatible with the map downto { G →→ pt } . Moreover, in order to be compatible with the groupoid structure, the map G × Y → X must factor as the composite of the projection G × Y → Y followed by f andthe map f must be equivariant. It is straightforward to verify that the topologies agree. (cid:3) Corollary 2.16.
Given a compact Lie group G , the ∞ -category associated with the topo-logical category Orb G fully faithfully embeds into Orb / B G onto those morphisms that arerepresentable, i.e. correspond to inclusions of subgroups. Corollary 2.17.
The induced functor S G ≃ P (Orb G ) → P (Orb / B G ) ≃ P (Orb) / B G is fully faithful, with essential image the representable maps X → B G , i.e. those maps suchthat X × B G pt is a space. Recall that given a G -space X , we can define the stack quotient [ X/G ] as the stackificationof the action groupoid or, equivalently, as the classifying stack for G -principal bundleswith a G -map to X . Every G -homotopy equivalence induces an equivalence between thecorresponding stack quotients in Stk ∞ and thus sending a G -CW-complex X to [ X/G ] defines a functor S G → Stk ∞ . Proposition 2.18.
For every compact Lie group G , the functor S G → Stk ∞ → S Orb , whichsends X to [ X/G ] , preserves colimits.Proof. We need to show that
Map
Stk ∞ ( B H, [ − /G ]) : S G → S preserves colimits for everycompact abelian Lie group H . Letting H vary, this implies indeed that the functor S G → Stk ∞ → S Orb , X [ X/G ] preserves colimits.Let X be a G -space and Φ : B H → B G be a map in Stk ∞ , which after choice of basepointscorresponds to a homomorphism ϕ : H → G . As in the proof of Lemma B.5, one showsusing that all principal bundles on ∆ n are trivial that the natural maps Map
TopGpd / ( G →→ pt) ( H × ∆ n →→ ∆ n , G × X →→ X ) → Map
Stk / B G ( B H × ∆ n , [ X/G ]) are equivalences of (discrete) groupoids for all n . After geometrically realizing the re-sulting simplicial objects, the right hand side becomes Map (Stk ∞ ) / B G ( B H, [ X/G ]) , whilewe can identify the left hand side with the space of ϕ -equivariant maps pt → X (usingessentially the argument of Proposition 2.15). This in turn is the same as the im( ϕ ) -fixed points of X . As a colimit of G -spaces induces a colimit of im( ϕ ) -fixed points, thefunctor Map (Stk ∞ ) / B G ( B H, [ − /G ]) : S G → S presrves thus colimits. As it is the fiber of Map
Stk ∞ ( B H, [ − /G ]) → Map
Stk ∞ ( B H, B G ) over Φ , we see that Map
Stk ∞ ( B H, [ − /G ]) pre-serves colimits as well. (cid:3) N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 11 Abelian group objects in ∞ -categories If C is an ∞ -category with finite products, we can consider the ∞ -category Ab( C ) ofabelian group objects in C . We caution the reader that “abelian group object” is meant inthe strict sense as opposed to the more initial notion of commutative (that is, E ∞ ) groupobject. In order to make this precise, it is convenient to invoke the language of universalalgebra. We refer the reader to [Lur09a, Section 5.5.8] or [GGN15, Appendix B] for relevantfacts about Lawvere theories in the context of ∞ -categories.Let T Ab denote the Lawvere theory of abelian groups, which we regard as an ∞ -category.That is, T opAb ⊂ Ab = Ab(Set) is the full subcategory of abelian groups (in sets) consistingof the finite free abelian groups. A skeleton of T Ab is given by the ∞ -category with objectset N and (discrete) mapping spaces Map( q, p ) ∼ = Map Ab ( Z p , Z q ) ∼ = ( Z q ) × p . Definition 3.1.
Let C be an ∞ -category. The ∞ -category Ab( C ) of abelian group objectsin C is the ∞ -category Fun Π ( T opAb , C ) of product preserving functors from T opAb to C .In particular, if C is the ordinary category of sets, then Ab( C ) ≃ Ab recovers the ordinarycategory of abelian groups. However, we will be primarily interested in the case in which C is an ∞ -topos, such as the ∞ -category S of spaces, or sheaves of spaces on a site. Remark 3.2.
It is important to note that
Ab( S ) is not equivalent to the ∞ -category of(grouplike) E ∞ -spaces since T Ab is a full subcategory of the ordinary category of abeliangroups. Should one need to consider this ∞ -category, one would instead start with theLawvere theory T E ∞ of (grouplike) E ∞ -spaces. Note also that T Ab is equivalent to thecategory of lattices. See [Lur18a, Section 1.2] for further details on (strict) abelian groupobjects in ∞ -categories. Example 3.3.
Topological abelian groups give examples of abelian group objects in S .To make this relationship respect the simplicial enrichment of the category of topologicalabelian groups, we first need to investigate the relationship between abelian group objectsin a simplicial category and its associated ∞ -category. Thus let C be a simplicial category.Denote by Ab( C ) the simplicial category of product preserving functors from T opAb to C .Denoting the coherent nerve by N coh , there is a natural functor N coh Ab( C ) → Ab( N coh C ) constructed as follows: Taking the adjoint of the coherent nerve of the evaluation map T opAb × Fun( T opAb , C ) → C gives a map N coh (Fun( T opAb , C )) → Fun( T opAb , N coh ( C )) . Restrictingto product preserving functors gives the desired functor N coh Ab( C ) → Ab( N coh C ) .Let us denote by TopAb ∞ the ∞ -category arising via the coherent nerve from the sim-plicial category whose objects are topological abelian groups and whose mapping spacesare the singular complexes of spaces of homomorphisms. From the last paragraph we get afunctor TopAb ∞ → Ab( S ) .We are interested in the full subcategories of TopAb ∞ on compact abelian Lie groups CptLie ab and the slightly bigger full subcategory FGLie ab on products of tori and finitelygenerated abelian groups. We claim that the functor FGLie ab → Ab( S ) is fully faithful.To that purpose recall from [Lur18a, Remark 1.2.10] that there is an equivalence B ∞ : Ab( S ) → Mod cn Z to connective Z -module spectra, preserving homotopy groups. Thus, for a discrete group A ,we have B ∞ ( A ) = A and B ∞ ( T n ) = Σ Z n . The claim thus reduces to an easy calculationof mapping spaces in Mod cn Z . Example 3.4.
Every object in
Orb ab has naturally the structure of an abelian groupobject. Indeed, the classifying stack functor B : CptLie → Stk ∞ preserves products andevery abelian compact Lie group defines an abelian group object in the category of compactLie groups. Proposition 3.5.
Let C be a cartesian closed presentable ∞ -category. Then the ∞ -category Ab( C ) of abelian group objects in C is symmetric monoidal in such a way that the free abeliangroup functor C →
Ab( C ) extends to a symmetric monoidal functor.Proof. This is true for
C ≃ S , the ∞ -category of spaces, and the general case follows fromthe equivalence Ab( C ) ≃ Ab( S ) ⊗ C [GGN15, Proposition B.3], where the tensor product isthat of presentable ∞ -categories [Lur12, Section 4.8.1]. (cid:3) We recall the notion of a preoriented abelian group object from [Lur09b].
Definition 3.6.
Let X be a presentable ∞ -category. A preoriented abelian group objectin X is an abelian group object A ∈ Ab( X ) equipped with a morphism of abelian groupobjects B T ⊗ pt → A . The ∞ -category of preoriented abelian group objects in X is theunder ∞ -category Ab( X ) B T ⊗ pt / . We will denote it by PreAb( X ) .Equivalently, a preoriented abelian group object is an abelian group object A ∈ Ab( X ) equipped with a morphism of abelian group spaces B T → Map(pt , A ) or, again equivalently,an object A ∈ Ab( X ) with a pointed map S → Map(pt , A ) . Note that these latter twodescriptions actually make sense in an arbitrary ∞ -category with finite products. Construction 3.7.
Morally, the elliptic cohomology groups for a compact abelian Lie group G are related to the sheaf cohomology groups of the elliptic curve and other related abelianvarieties. In order to make this precise, we invoke a stacky version of Pontryagin duality,which we refer to as Picard duality , since it is computed by considering the space of linebundles on a given orbit stack. More precisely, it is given by the functor
Map
Orb ( − , B T ) : (Orb ab ) op → Ab( S ) B T / = PreAb( S ) , X b X which sends the abelian orbit stack X to the preoriented abelian group space B T ≃ Map(pt , B T ) → Map( X, B T ) . Here the preorientation is induced from the projection X → pt , and the abelian group structure on the mapping space Map( X, B T ) is inducedpointwise from that on B T .By Proposition 2.12, we have d B G ≃ B b G × B T , where b G is the Pontryagin dual, i.e. thespace of homomorphisms from G to T .The following proposition is not used in our construction of equivariant elliptic cohomol-ogy, but shows that Picard duality does not lose any information. Proposition 3.8.
The Picard duality functor
Map( − , B T ) : (Orb ab ) op → PreAb( S ) is fully faithful.Proof. We have a morphism of augmented simplicial ∞ -categories (Orb ab ) op (cid:15) (cid:15) (Orb ab ∗ ) op o o (cid:15) (cid:15) (Orb ab2 ∗ ) op o o o o (cid:15) (cid:15) (Orb ab3 ∗ ) op o o o o o o (cid:15) (cid:15) · · · o o o o o o o o Ab( S ) B T / Ab( S ) o o Ab( S ) T / o o o o Ab( S ) T × T / o o o o o o · · · o o o o o o o o N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 13 in which we’ve identified
Map
Orb ab n ∗ (pt , B T ) ≃ T n − . Here
Orb ab n ∗ ≃ Orb ab ∗ × Orb ab · · · × Orb ab Orb ab ∗ denotes the iterated fibered product, which weview as the ∞ -category of “ n -pointed objects.”By Remark 2.14 the ∞ -category Orb ab ∗ is equivalent to CptLie ab . From this fact andProposition 2.12, one can show more generally inductively for every n ≥ that Orb ab n ∗ isequivalent to the full subcategory of FGLie ab / Z n − on Lie groups of the form G × Z n − for G compact abelian. Under this equivalence, the vertical arrows given by Picard duality areinduced by the composition FGLie ab , op → FGLie ab → Ab( S ) , where the first functor is Pontryagin duality and thus an equivalence and the second functoris fully faithful by Example 3.3. We thus see that the vertical maps (except, possibly, forthe one on the far left) are fully faithful.Observe now that the horizontal maps are quotient functors (see Appendix A for details).It follows that the left hand vertical map is also fully faithful (cf. the description of mappingspaces in the proof of Proposition A.3). (cid:3) Our next goal is to associate to any preoriented abelian group object in some X a functor PreAb( S ) → X ; precomposing this functor with Picard duality will allow us to define afunctor from orbispaces to X . The key input is the following categorical fact, of which partswere already proven in [GGN15, Proposition B.3]. Proposition 3.9.
For any presentable ∞ -category X the functors Ab( X ) → Fun R (Ab( S ) op , X ) , A ( B Map
Ab( X ) ( B ⊗ pt , A )) and PreAb( X ) → Fun R (PreAb( S ) op , X ) , A ( B Map
PreAb( X ) ( B ⊗ pt , A )) are equivalences.Proof. We obtain the first equivalence by chasing through the equivalences
Ab( X ) ≃ Ab(Fun R ( X op , S )) ≃ Fun π, R ( T opAb × X op , S ) ≃ Fun R ( X op , Ab( S )) ≃ Fun L ( X , Ab( S ) op ) op ≃ Fun R (Ab( S ) op , X ) Here the first functor is induced by the Yoneda embedding and
Fun π,R stands for functorspreserving products in the first variable and defining a right adjoint functor X op → S forevery fixed object in T opAb .Next we observe that there is a natural equivalence PreAb( X ) ≃ Ab( X ) × X X ∗ , where the functor X ∗ → X is the forgetful functor and Ab( X ) → X takes A to Ω Map X (pt , A ) .We compute PreAb( X ) ≃ Ab( X ) × X X ∗ ≃ Fun R ( X op , Ab( S )) × Fun R ( X op , S ) Fun R ( X op , S ∗ ) ≃ Fun R ( X op , PreAb( S )) ≃ Fun R (PreAb( S ) op , X ) . Here we use the chain of equivalences
Fun R ( X op , S ∗ ) ≃ Fun R ( X op , S ) ∗ ≃ X ∗ . (cid:3) Combining the preceding proposition with Construction 3.7 leads to one of the mainconstructions of this article.
Construction 3.10.
Let X be a presentable ∞ -category. Precomposition with Picardduality turns the second functor in Proposition 3.9 into a functor PreAb( X ) → Fun(Orb ab , X ) , A ( X Map
PreAb( X ) (pt ⊗ b X, A )) . This functor is also functorial in X in the following sense: Given a functor F : X → Y preserving finite limits, we obtain commutative square
PreAb( X ) / / F ∗ (cid:15) (cid:15) Fun(Orb ab , X ) F ∗ (cid:15) (cid:15) PreAb( Y ) / / Fun(Orb ab , Y ) Indeed: The choice of a base point gives a splitting of the Picard dual of an object in
Orb ab as a product of a finitely generated abelian group G (with trivial preorientation) and B T (with tautological preorientation). Thus Map
PreAb( X ) (pt ⊗ ( G × B T ) , A ) can be identifiedwith the cotensor A [ G ] = Map Ab( X ) (pt ⊗ G, A ) , which can be computed from A as a finitelimit.Given A ∈ PreAb( X ) , left Kan extension along Orb ab ⊂ Orb defines a functor
Orb → X .As S Orb is the presheaf ∞ -category on Orb , i.e. its universal cocompletion, this extendsessentially uniquely to a colimit-preserving functor S Orb → X . More precisely, we obtain afunctor
PreAb( X ) → Fun L ( S Orb , X ) . This construction is natural in X with respect to geometric morphisms, i.e. functors pre-serving colimits and finite limits. Remark 3.11.
The preceding construction is closely related to [Lur09b, Proposition 3.1],whose proof was only outlined though and did not contain the notion of Picard duality.
Example 3.12.
We take X = S Orb and A = B T , with the preorientation given by thecanonical identification of B T with Map
Orb (pt , B T ) . The corresponding functor Orb ab →S Orb sends any X to its double Picard dual, which is easily seen to be naturally equivalentto X again. Thus, the functor Orb ab → S Orb is just the natural inclusion as representablepresheaves. We obtain a functor S Orb → S
Orb by left Kan extending this inclusion alongitself. This functor is not the identity, but we claim it rather to be the colocalization functorgiven by the composite S Orb ι ∗ −→ Fun(Orb ab , op , S ) ι ! −→ S Orb , N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 15 where we denote by ι : Orb ab → Orb the inclusion and by ι ! the left adjoint to restrictionof presheaves.This claim is a special case of the following more general claim: Let F : C → D be a functorof ∞ -categories. Denoting by Y C : C →
Fun( C op , S ) the Yoneda embedding, we claim thatthe natural transformation Lan F Y C → F ∗ Y D of functors D →
Fun( C op , S ) , induced by Y C → F ∗ Y D F , is an equivalence. In the case that C = pt and F corresponds to an object d ∈ D , this boils down to the natural equivalence colim D F/e pt ≃ Map D ( d, e ) , where D F/e is the comma category of F and pt e −→ D . For the general case, we note that it suffices toshow that i ∗ Lan F Y C → i ∗ F ∗ Y D is an equivalence for every functor i : pt → C . This naturaltransformation fits into a commutative square Lan
F i ( Y pt ) ≃ (cid:15) (cid:15) ≃ / / ( F i ) ∗ Y D ≃ i ∗ F ∗ Y D Lan F Lan i Y pt ≃ (cid:15) (cid:15) Lan F i ∗ Y C ≃ / / i ∗ Lan F Y C O O and is thus an equivalence as the upper horizontal and lower left vertical morphisms areequivalences by the special case treated above and the upper left vertical and the lowerhorizontal morphism are equivalences by general properties of left Kan extensions (as i ∗ preserves colimits).4. Elliptic curves, formal completions and orientations
The goal of this section is to define the notion of an oriented elliptic curve over a non-connective spectral Deligne–Mumford stack, following the ideas of Lurie in [Lur09b] and[Lur18b]. In the affine case, an orientation of an elliptic curve E → Spec R consists of anequivalence of the formal completion b E with the formal group Spf R B T , providing a purelyspectral analogue of the algebro-topological definition of an elliptic cohomology theory (see[Lur09b, Definition 1.2] or [AHS01, Definition 1.2]). While the notion of a preorientationsuffices for the definition of equivariant elliptic cohomology, one only expects good propertiesif one actually starts with an oriented elliptic curve. Setting up the necessary definitions inthe non-connective spectral Deligne–Mumford case will occupy the rest of this section.Our first goal will be to define an ∞ -topos of sheaves on a given non-connective spectralDeligne–Mumford stack. Since simply defining it to be sheaves on the site of all morphismsinto our given stack would run into size issues, we impose the following finiteness condition.Here and in the following, we will use for a non-connective spectral Deligne–Mumford stack M = ( M , O M ) the shorthand τ ≥ M for ( M , τ ≥ O M ) . Definition 4.1.
A non-connective spectral Deligne–Mumford stack M = ( M , O M ) is called quasi-compact if every cover ` U i → ∗ in M has a finite subcover. A morphism f : N → M is called quasi-compact if for every étale morphism Spec R → M the pullback N × M Spec R is quasi-compact (cf. [Lur16, Definition 2.3.2.2]). A morphism f : N → M of spectral Deligne–Mumford stacks is called almost of finitepresentation if it is locally almost of finite presentation in the sense of [Lur16, Definition4.2.0.1] and quasi-compact.We call a morphism f : N → M of non-connective spectral Deligne–Mumford stacks al-most of finite presentation if f is the pullback of a morphism N ′ → τ ≥ M almost of finitepresentation along M → τ ≥ M , where N ′ is also spectral Deligne–Mumford. Example 4.2.
Every map
Spec R → M into a non-connective spectral Deligne–Mumfordstack is quasi-compact as affine non-connective spectral schemes are quasi-compact by[Lur16, Propositions 2.3.1.2] and we assumed all non-connective spectral Deligne–Mumfordstacks to be quasi-separated.Moreover, étale morphisms are always almost locally of finite presentation (as followse.g. by [Lur16, Corollary 4.1.3.5]). Thus, every étale morphism Spec R → M (and moregenerally every quasi-compact étale morphism) is almost of finite presentation. Definition 4.3.
We define the big étale site of a spectral Deligne–Mumford stack M to bethe full sub- ∞ -category of spectral Deligne–Mumford stacks over M that are almost of finitepresentation; coverings are given by jointly surjective étale morphisms. We define Shv( M ) to be the ∞ -category of space-valued sheaves on the big étale site of M .If M is more generally a non-connective spectral Deligne–Mumford stack, we define Shv( M ) to be Shv( τ ≥ M ) . Remark 4.4.
Our insistence that all our spectral Deligne–Mumford stacks are quasi-separated and locally noetherian is connected to the finiteness conditions in our definitionof the big étale site. Milder finiteness conditions in Definition 4.3 would result in milderconditions on our stacks.
Remark 4.5.
Given a spectral Deligne–Mumford stack M = ( M , O M ) , there is a geometricmorphism between M and Shv( M ) , i.e. a left adjoint ι ∗ : M →
Shv( M ) (preserving finitelimits) and a right adjoint ι ∗ : Shv( M ) → M . This is induced by an inclusion ι of sites fromthe small étale site M ´ et , consisting of all quasi-compact étale morphisms into M , into the bigétale site. Here, one uses the equivalence Shv( M ´ et ) ≃ M , which one reduces by a hypercoverargument to the affine case (cf. [Lur09a, Proposition 6.3.5.14]), where it is essentially thedefinition used in [Rez19b, Section 9] to define the étale spectrum of an E ∞ -ring. We lastlynote that the non-connective case reduces to the spectral case as the relevant sites and topoido not change by passing to connective covers. Notation 4.6.
We will often use the notation pt for the terminal object in the ∞ -topos Shv( M ) ; this is represented by id τ ≥ M : τ ≥ M → τ ≥ M in the big étale site of τ ≥ M .For Definition 4.3 to be well-behaved, we need the following lemma. Lemma 4.7.
Given a non-connective spectral Deligne–Mumford stack M , its big étale siteis an (essentially) small ∞ -category.Proof. Recall from [Lur12, Proposition 7.2.4.27(4)] that if for a connective E ∞ -ring A aconnective E ∞ - A -algebra B is almost of finite presentation, each of its truncations τ ≤ n B isa retract of τ ≤ n A ′ for a compact object A ′ ∈ CAlg A . As the ∞ -category of compact objectsin CAlg A is essentially small and B ≃ lim n τ ≤ n B , we see that the sub- ∞ -category of CAlg A of those connective algebras almost of finite presentation is essentially small, too. We recall that a spectral Deligne–Mumford stack is a non-connective spectral Deligne–Mumford stackwhose structure sheaf is connective.
N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 17
For the general case, we may assume that M is a spectral Deligne–Mumford stack. Choos-ing a hypercover by disjoint unions of affines reduces to the case M = Spec A . By definitionwe see that for every étale map Spec B → Z for Z almost of finite presentation over Spec A ,the A -algebra B must be almost of finite presentation over A . Thus, there is up to equiva-lence only a set of possible hypercovers of some Z almost of finite presentation over Spec A such that each stage is a finite union of affines. As we can recover Z as the geometricrealization of the hypercover and Z is quasi-compact by [Lur16, Propositions 2.3.1.2 and2.3.5.1], this proves the lemma. (cid:3) Next we want to define the notion of an elliptic curve.
Definition 4.8.
Let M be a non-connective spectral Deligne–Mumford stack. An ellipticcurve E over M is an abelian group object in the ∞ -category of non-connective spectralDeligne–Mumford stacks over M such that(1) E → M is flat in the sense of [Lur16, Definition 2.8.2.1],(2) τ ≥ E → τ ≥ M is almost of finite presentation and proper in the sense of [Lur16,Definition 5.1.2.1],(3) For every morphism i : Spec k → τ ≥ M with k an algebraically closed (classical)field the pullback i ∗ E → Spec k is a classical elliptic curve. Remark 4.9.
One can show that given an elliptic curve E over M , the connective cover τ ≥ E is an elliptic curve over τ ≥ M and that this procedure provides an inverse of the base changefrom τ ≥ M to M . Thus, we obtain an equivalence between the ∞ -categories of elliptic curvesover M and over τ ≥ M (cf. [Lur18a, Remark 1.5.3]).We will view an elliptic curve E over M as an abelian group object in Shv( M ) by usingthe functor of points of τ ≥ E .We fix from now on a non-connective spectral Deligne–Mumford stack M and an ellipticcurve p : E → M over M . A preorientation of E is a preorientation of E as an abelian groupobject in Shv( M ) , i.e. a morphism of abelian group objects B T ⊗ pt → E or, equivalently, B T → Ω ∞ E ( M ) . The map B T ⊗ pt → E factors through the completion b E of E . Definition 4.10.
We define the formal completion b E ∈ Ab(Shv( M )) of E as follows: Denoteby U the complement of the unit section in τ ≥ E . For every Y → τ ≥ M we let b E ( Y ) ⊂ ( τ ≥ E )( Y ) be the full sub- ∞ -groupoid on those maps Y → τ ≥ E such that the fiber product Y × τ ≥ E U is empty.The formal completion of an elliptic curve is an example of a formal hyperplane . Torecall this notion from [Lur18b], we have first to discuss the cospectrum of a commutativecoalgebra (cf. [Lur18b, Construction 1.5.4]). Definition 4.11.
For an E ∞ -ring R , define the ∞ -category cCAlg R of commutative coal-gebras over R as CAlg(Mod op R ) op . In the case that R is connective, we define the cospectrum cSpec( C ) ∈ Fun(CAlg cn R , S ) of a coalgebra C ∈ cCAlg R by cSpec( C )( A ) = Map cCAlg A ( A, C ⊗ R A ) . More generally for a spectral Deligne–Mumford stack and a coalgebra
C ∈ cCAlg M = CAlg(QCoh( M ) op ) op , If the base is affine, our definition coincides with what Lurie calls strict elliptic curves. As we do notconsider non-strict elliptic curves in this article, we drop the adjective. we define the relative cospectrum cSpec M ( C ) ∈ Shv( M ) by cSpec M ( f : N → M ) = Map cCAlg N ( O N , f ∗ C ) . Given a discrete ring R , an important example of an R -coalgebra is the continuous dualof R J t , . . . , t n K , which we denote by Γ R ( n ) as it coincides as an R -module with the dividedpower algebra on n generators. The cospectrum cSpec Γ R ( n ) coincides with Spf R J t , . . . , t n K on discrete rings. Definition 4.12.
Let M be a spectral Deligne–Mumford stack. An object F ∈ Shv( M ) iscalled a formal hyperplane if it is of the form cSpec M ( C ) for some coalgebra C ∈ cCAlg M that is smooth , i.e. • locally, π C is of the form Γ R ( n ) for some discrete ring R and some n ≥ , and • the canonical map π C ⊗ π O M π k O M → π k C is an isomorphism for all k ∈ Z . Lemma 4.13.
For every spectral Deligne–Mumford stack, the cospectrum functor definesan equivalence between the ∞ -category of smooth coalgebras on M and formal hyperplaneson M Proof.
The claim is equivalent to cSpec M being fully faithful on smooth coalgebras. Oneeasily reduces to the case M = Spec A for a connective E ∞ -ring A . As M is locally noetherianby assumption, we can further assume that π A is noetherian and π i A is finitely generatedover π A .[Lur18b, Proposition 1.5.9] proves that cSpec is fully faithful on smooth A -coalgebras asa functor into Fun(CAlg cn , S ) . In contrast, we need to show fully faithfulness as a functorinto Fun(CAlg cn , afp A , S ) , where the afp stands for almost of finite presentation. Tracingthrough the proof in [Lur18b], we need to show that Spf is fully faithful as a functor into
Fun(CAlg cn , afp A , S ) on adic E ∞ -rings that arise as duals of smooth coalgebras.Let R and S be adic E ∞ -rings. By [Lur16, Lemma 8.1.2.2] we can find a tower · · · → R → R → R in CAlg cn R such that colim i Spec R i → Spec R factors over an equivalence colim i Spec R i ≃ Spf R in Fun(CAlg cn , S ) and every R i is almost perfect as a R -module. We claim that R i is almost of finite presentation over A if R is the dual of a smooth coalgebra C over A . By [Lur12, Proposition 7.2.4.31], this is equivalent to π R i being a finitely generated π A -algebra and π k R i being a finitely generated π R i -module for all k . By the definition ofsmooth coalgebras, π R is of the form ( π A ) J t , . . . , t n K for some n and π k R ∼ = π k A ⊗ π A π R .By [Lur12, Proposition 7.2.4.17], π k R i is finitely generated over π R . The claim followsas Spec R i → Spec R factors over Spf R by construction and thus a power of the ideal ( t , . . . , t n ) is zero in R i .Now let R and S be duals of smooth coalgebras. In particular, we see that R is completeso that R = lim i R i in adic E ∞ -rings (cf. [Lur16, Lemma 8.1.2.3]). Map
CAlg cn , ad A ( S, R ) ≃ lim i Map
CAlg cn , ad A ( S, R i ) ≃ lim i (Spf S )( R i ) ≃ lim i Map
Fun(CAlg cn , afp A , S ) (Spec R i , Spf S ) ≃ Map
Fun(CAlg cn , afp A , S ) (Spf R, Spf S ) In the third step we have used the Yoneda lemma. (cid:3)
N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 19
Example 4.14.
For every elliptic curve E over a non-connective spectral Deligne–Mumfordstack M , the formal completion b E is a formal hyperplane over τ ≥ M [Lur18b, Proposition7.1.2]. Definition 4.15.
We call an E ∞ -ring R complex periodic if it is complex orientable andZariski locally there exists a unit in π R . For a complex periodic E ∞ -ring R , the Quillenformal group b G Q R is defined as cSpec( τ ≥ ( R ⊗ B T )) . More generally, for a locally complexperiodic non-connective spectral Deligne–Mumford stack M , the Quillen formal group b G Q M is defined as cSpec M ( τ ≥ ( O M ⊗ B T )) .By [Lur18b, Theorem 4.1.11] the Quillen formal group is indeed a formal hyperplane.Moreover the functor T opAb → Formal Hyperplanes , M cSpec M ( τ ≥ ( O M ⊗ B c M )) equips it with the structure of an abelian group object.Recall that a preorientation of an elliptic curve gives us a morphism B T ⊗ pt → b E , i.e.a T -equivariant morphism pt → b E . As both pt and b E are formal hyperplanes on τ ≥ M ,this is by Lemma 4.13 obtained from a T -equivariant morphism O M → C of commutativecoalgebras on τ ≥ M . As O M ⊗ B T ≃ ( O M ) h T , we obtain a morphism O M ⊗ B T → C andhence a morphism b G Q M → b E in the case that M is locally complex periodic. Definition 4.16.
Let M be locally complex periodic and p : E → M a preoriented ellipticcurve over M . We say that E oriented if the morphism b G Q M → b E is an equivalence.Having such an orientation will force a version of the Atiyah–Segal completion theoremto hold in equivariant elliptic cohomology.5. Equivariant elliptic cohomology
We are now ready to give a definition of the equivariant elliptic cohomology theory asso-ciated to a preoriented elliptic curve p : E → M , which we fix throughout the section. Here, M denotes a non-connective spectral Deligne–Mumford stack. Construction 5.1.
To the preoriented elliptic curve p : E → M we can associate the ∞ -topos X = Shv( M ) of space-valued sheaves on the big étale site of M . As E defines apreoriented abelian group object in X , Construction 3.10 yields a colimit-preserving functor Ell : S Orb → X , where we leave the dependency on E implicit. Example 5.2.
By construction, we have
Ell( B T ) = E and Ell( B C n ) = E [ n ] , the n -torsionin the elliptic curve.We would like to specialize to a theory for G -spaces for a fixed G by a pushforwardconstruction. The key will be the following algebro-geometric observation. Proposition 5.3.
Let H ⊂ G be an inclusion of compact abelian Lie groups. Then theinduced morphism f : Ell( B H ) → Ell( B G ) is affine.Proof. We claim first that for a surjection h : A → B of finitely generated abelian groupsand a classical elliptic curve E over a base scheme S , the map Hom(
B, E ) → Hom(
A, E ) is a closed immersion and hence affine. As the kernel of h is a sum of cyclic groups and a composition of closed immersions is a closed immersion, we can assume that ker( h ) is cyclic.The pushout square B A o o O O ker( h ) O O o o induces a pullback square Hom(
B, E ) (cid:15) (cid:15) / / Hom(
A, E ) (cid:15) (cid:15) S / / Hom(ker( h ) , E ) . Furthermore, S → Hom(ker( h ) , E ) is a closed immersion as it is a right inverse of thestructure morphism Hom(ker( h ) , E ) → S and the latter is a separated morphism. (Seee.g. [MO14, Lemma 2.4] for this criterion for closed immersions.) Thus, its pullback Hom(
B, E ) → Hom(
A, E ) is also a closed immersion and thus affine.Recall that Ell( B G ) ≃ Hom( b G, E ) , with b G denoting the Pontryagin dual, and that aninclusion H ⊂ G induces a surjection b G → b H . Thus, we know from the preceding paragraphthat the underlying map of Ell( B H ) → Ell( B G ) is affine. We can moreover assume that thebase of the elliptic curve E is an affine derived scheme Spec R . By [Lur18a, Remark 1.5.3],the elliptic curve E is based changed from Spec τ ≥ R . Thus we can assume additionally that R is connective and hence that Ell( B G ) and Ell( B H ) are spectral schemes (and not moregeneral non-connective ones).Let Spec A → Ell( B G ) be a map from an affine and let ( X , O X ) denote the pullback Spec A × Ell( B G ) Ell( B H ) . As a pullback of spectral schemes, ( X , O X ) is a spectral schemeagain, and the underlying scheme of ( X , O X ) is the pullback of the underlying scheme ( X , π O X ) , hence affine. Using [Lur16, Corollary 1.1.6.3], we conclude that ( X , O X ) isaffine. (cid:3) Construction 5.4.
Let G be a compact abelian Lie group and Orb G its orbit ∞ -category,which we identify using Corollary 2.16 with the full subcategory of Orb / B G on morphisms B H → B G inducing an injection H → G . We consider the functor Orb G → QCoh(Ell( B G )) op , G/H f ∗ O Ell( B H ) , where f : Ell( B H ) → Ell( B G ) is the map induced by the inclusion H ⊂ G . As f is affine bythe preceding proposition, f ∗ sends indeed quasi-coherent sheaves to quasi-coherent sheavesby [Lur16, Proposition 2.5.11]. The functor from Orb G extends to a colimit-preservingfunctor E ll G : S G → QCoh(Ell( B G )) op , where S G ≃ Fun(Orb op G , S ) denotes the ∞ -category of G -spaces. As the target is pointed,this functor factors canonically to define a reduced version f E ll G : S G ∗ → QCoh(Ell( B G )) op . For us, the most important case is G = T . As Ell( B T ) = E , we obtain in this case afunctor from T -spaces to quasi-coherent sheaves on E . If desired, we can take homotopygroups to obtain an equivariant cohomology theory with values in quasi-coherent sheaveson the underlying classical elliptic curve of E . This is the kind of target we are used to from N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 21 the classical constructions of equivariant elliptic cohomology, for example by Grojnowski[Gro07].Next we want to compare our two constructions of equivariant elliptic cohomology func-tors. To that purpose we want to recall two notions. First, given any (non-connective)spectral Deligne–Mumford stack N and any Z ∈ Shv( N ) , we consider the restriction O Z of O N to Shv( N ) / Z and define Mod O Z as the ∞ -category of modules over it. Second, for any F ∈
Mod O Z , we can consider the O N -module f ∗ F , associated with f : Z → N and definedby f ∗ F ( U ) = F ( U × N Z ) , where N = pt is the final object in Shv( N ) . Lemma 5.5.
Assume that M is locally -periodic. For G a compact abelian Lie group and X ∈ S G , there is a natural equivalence E ll G ( X ) ≃ Q ( f ∗ O Ell([
X/G ]) ) , where f : Ell([ X/G ]) → Ell( B G ) denotes the map induced by X → pt and Q is the rightadjoint to the inclusion QCoh(Ell( B G )) ⊂ Mod O Ell( B G ) . If X is finite, f ∗ O Ell([
X/G ]) isalready quasi-coherent so that Q ( f ∗ O Ell([
X/G ]) ) agrees with f ∗ O Ell([
X/G ]) .Proof. Throughout this proof we will abbreviate
Ell( B G ) to Z . The key is to check that thefunctor P : S G → Mod op O Z , X f ∗ O Ell([
X/G ]) preserves colimits. This suffices as Q preserves limits and hence the composite QP : S G → QCoh( Z ) op , X
7→ Q ( f ∗ O Ell([
X/G ]) ) preserves colimits as well. As this functor agrees on orbits with E ll G , it agrees thus on allof S G . Moreover, QCoh( Z ) is closed under finite limits, implying the last statement.To show that P preserves colimits we reinterpret the pushforward using A , i.e. thespectrum of the free E ∞ -ring on one generator. For any map h : Y → Z with Y ∈ Shv( M ) ,there is a natural equivalence of Ω ∞ h ∗ O Y with the sheaf ( U → Z ) Ω ∞ Γ( O Y × Z U ) ≃ Map U ( Y × Z U, U × A ) . This implies that the functor from
Shv( M ) / Z to Shv( Z ) op , sending h : Y → Z to Ω ∞ h ∗ O Y ,preserves all colimits.We claim that Ω ∞ : Mod O Z → Shv( Z ) is conservative. Indeed: By assumption M islocally of the form Spec R for a -periodic E ∞ -ring R . As Z = Ell( B G ) maps to M , thesame is true for Z . Since the conservativity of Ω ∞ can be checked locally, we just have toshow that Ω ∞ : Mod R → S is conservative if R is -periodic and this is obvious.As Ω ∞ : Mod O Z → Shv( Z ) also preserves limits, we see that the functor Shv( M ) / Z → Mod op O Z , ( h : Y → Z ) h ∗ O Y preserves colimits. Moreover, since Ell : S Orb → Shv( M ) preserves colimits, the same is truefor ( S Orb ) / B G → Shv( M ) / Ell( B G ) . Applying Proposition 2.18 finishes the proof. (cid:3) This right adjoint exists because
QCoh(Ell( B G )) is presentable and the inclusion preserves all colimitsby [Lur16, Proposition 2.2.4.1]. Strictly speaking, Lurie uses here a different definition of Mod O , namelyjust modules in sheaves of spectra on the small étale topos. But as follows from Remark 4.5, pullback definesa functor from this to our version of Mod O , which preserves all colimits. Last we want to speak about the functoriality of equivariant elliptic cohomology in theelliptic curve. Thus denote for the moment the elliptic cohomology functors based on E by Ell E and E ll E G . Proposition 5.6.
Let f : N → M be a morphism of non-connective spectral Deligne–Mumford stacks that is almost of finite presentation, E be a preoriented elliptic curve over M , and f ∗ E the pullback of E to N . Then there are natural equivalences f ∗ Ell E ≃ Ell f ∗ E in Fun( S Orb , Shv( N )) and f ∗ E ll E G ≃ E ll f ∗ E G in Fun(( S G ) fin , op , QCoh( f ∗ Ell E ( B G ))) for all compact abelian Lie groups G .Proof. Write f ∗ : Shv( N ) → Shv( M ) for the induced functor on big étale topoi, which admitsa left adjoint f ∗ : Shv( M ) → Shv( N ) preserving finite limits. The preoriented abelian groupobject E induces a left adjoint functor Ell E : S Orb → Shv( M ) , which we can postcomposewith f ∗ to obtain a functor f ∗ Ell E : S Orb → Shv( M ) → Shv( N ) .By Construction 3.10, f ∗ Ell E is indeed the functor S Orb → Shv( N ) associated with f ∗ E .The functor f ∗ E ll E G : ( S G ) fin , op → QCoh( f ∗ Ell E ( B G )) preserves all finite limits. Thus,we only have to provide natural equivalences f ∗ E ll E G ( G/H ) ≃ E ll f ∗ E G ( G/H ) for all closedsubgroups H ⊂ G . These are indeed provided by applying the following commutative squareto the structure sheaf of Ell E ( B H ) : QCoh(Ell E ( B H )) pullback / / pushforward (cid:15) (cid:15) QCoh(Ell f ∗ E ( B H )) pushforward (cid:15) (cid:15) QCoh(Ell E ( B G )) pullback / / QCoh(Ell f ∗ E ( B G )) This commutative square in turn is associated by [Lur16, Proposition 2.5.4.5] to the pullbackdiagram
Ell f ∗ E ( B H ) / / (cid:15) (cid:15) Ell E ( B H ) (cid:15) (cid:15) Ell f ∗ E ( B G ) / / Ell E ( B G ) that we obtain from the first claim together with the fact that the sheaf in Shv( N ) rep-resented by the pullback N × M Ell E ( B H ) agrees with the pullback sheaf f ∗ Ell E ( B H ) andsimilarly for G . (cid:3) Remark 5.7.
We do not claim that the functor
Ell defines the “correct” version of ellipticcohomology for arbitrary orbispaces. For a general X ∈ S Orb there is for example noreason to believe that
Ell( X ) is a nonconnective spectral Deligne–Mumford stack, not evena formal one. It should be thus seen more like a starting point to obtain a reasonablegeometric object. For example, given an abelian compact Lie group G and a G -space Y , wehave seen how to recover E ll G ( Y ) from Ell([
Y /G ]) . Taking Spec (if Y is finite) or a suitableversion of Spf (if Y is infinite) seems to be a reasonable guess for the “correct” geometricreplacement of Ell([
Y /G ]) in these cases. How to perform such a geometric replacement of Ell( X ) for a general orbispace X remains unclear to the authors at the time of writing. N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 23 Eilenberg–Moore type statements
The goal of this section is to recall and extend results of Eilenberg–Moore, Dwyer andBousfield about the homology of fiber squares and to rephrase parts of them in terms ofcospectra. Applied to classifying spaces of abelian compact Lie groups, this will be a keystep to proving symmetric monoidality properties of equivariant elliptic cohomology.Throughout this section, we will consider a (homotopy) pullback diagram M / / (cid:15) (cid:15) X (cid:15) (cid:15) Y / / B of spaces. We assume that for every b ∈ B , the fundamental group π ( B, b ) acts nilpotentlyon the integral homology of the homotopy fiber Y b .We denote by C ( X, B, Y ) the cobar construction, i.e. the associated cosimplicial objectwhose n -th level is X × B × n × Y . This cosimplicial object is augmented by M . Theorem 6.1 (Bousfield, Dwyer, Eilenberg–Moore) . Under the conditions above, the aug-mentation of the cobar construction induces a pro-isomorphism between the constant tower H ∗ ( M ; Z ) and the tower ( H ∗ (Tot m C ( X, B, Y )); Z ) m ≥ .Proof. This follows from Lemmas 2.2 and 2.3 and Section 4.1 in [Bou87]. (cid:3)
The following corollary is essentially also already contained in [Bou87].
Corollary 6.2.
The augmentation of the cobar construction induces a pro-isomorphismbetween the constant tower E ∗ ( M ) and the tower ( E ∗ (Tot m C ( X, B, Y ))) m ≥ for any boundedbelow spectrum E . In particular, lim m E ∗ (Tot m C ( X, B, Y )) = 0 , and the natural map E ∗ ( M ) → lim m E ∗ (Tot m C ( X, B, Y )) is an isomorphism.Proof. In the last theorem, we can replace Z by arbitrary direct sums of Z and get ourclaim for E any shift of the associated Eilenberg–MacLane spectrum. By the five lemma inpro-abelian groups, we obtain the statement of this corollary for all truncated spectra E .The group E k ( Z ) coincides with ( τ ≤ k E ) k ( Z ) for every space Z . This implies the result. (cid:3) We would like to pass from an isomorphism of pro-groups to an equivalence of spectra. Inthe following, let R an arbitrary E ∞ -ring spectrum. Note that Tot m coincides with the finitelimit lim ∆ ≤ m and it thus commutes with smash products. We deduce that the R -homologyof the Tot m of the cobar construction coincides with the homotopy groups of the spectrum Tot m R ⊗ ( X × B ו × Y ) ≃ Tot m (( R ⊗ X ) ⊗ R ( R ⊗ B ) ⊗ R • ⊗ R ( R ⊗ Y )) . Note that as before we view all spaces here as unpointed so that R ⊗ X has the same meaningas R ⊗ Σ ∞ + X .We recall the cotensor product: Given morphisms C ′ → C and C ′′ → C in cCAlg R , thecotensor product C ′ (cid:3) C C ′′ is the fiber product in cCAlg R . As cCAlg R = CAlg(Mod op R ) op ,the usual formula for the pushout of commutative algebras, i.e. the relative tensor product,translates into the formula C ′ (cid:3) C C ′′ ≃ lim ∆ C ′ ⊗ R C ⊗ R • ⊗ R C ′′ , which we can also use to define the cotensor products for arbitrary comodules. Thus, lim ∆ C ( X, B, Y ) ≃ ( R ⊗ X ) (cid:3) R ⊗ B ( R ⊗ Y ) . In this language, Corollary 6.2 implies: Corollary 6.3. If R is bounded below, the augmentation map R ⊗ M → ( R ⊗ X ) (cid:3) R ⊗ B ( R ⊗ Y ) is an equivalence. Corollary 6.4. If R is connective and R ⊗ X is in the thick subcategory of R ⊗ B in R ⊗ B -comodules, then the map cSpec( R ⊗ M ) → cSpec( R ⊗ X ) × cSpec( R ⊗ B ) cSpec( R ⊗ Y ) is an equivalence.Proof. Given A ∈ CAlg cn R , the functor coMod( R ⊗ B ) → coMod( A ⊗ B ) , N A ⊗ R N preserves colimits and is in particular exact. Thus, ( R ⊗ X ) (cid:3) R ⊗ B ( R ⊗ Y ) ⊗ R A ≃ ( A ⊗ X ) (cid:3) A ⊗ B ( A ⊗ Y ) , using that R ⊗ X is in the thick subcategory of R ⊗ B . It follows that cSpec(( R ⊗ X ) (cid:3) R ⊗ B ( R ⊗ Y ))( A ) ≃ Map cCAlg A ( A, ( A ⊗ X ) (cid:3) A ⊗ B ( A ⊗ Y )) ≃ (cSpec( R ⊗ X ) × cSpec( R ⊗ B ) cSpec( R ⊗ Y ))( A ) . (cid:3) Example 6.5.
Let R be the connective cover of a complex oriented and -periodic E ∞ -ring,with chosen isomorphism π R B T ∼ = ( π R ) J t K . The short exact sequence → ( π R ) J t K t −→ ( π R ) J t K → π R → of topological π R -modules dualizes to a short exact sequence of comodules → π R → Γ π R (1) → Γ π R (1) → . As for any R -free ( R ⊗ B T ) -comodule C the map π Map coMod( R ⊗ B T ) ( C, R ⊗ B T ) → Map Γ π R (1) ( π C, Γ π R (1)) is an isomorphism, we obtain a cofiber sequence R → R ⊗ B T → R ⊗ B T → Σ R of ( R ⊗ B T ) -comodules. This implies that R is in the thick subcategory of R ⊗ B T in ( R ⊗ B T ) - comod . Using the last corollaries, we obtain equivalences R ⊗ BC n ≃ −→ R ⊗ B T (cid:3) R ⊗ B T R cSpec( R ⊗ BC n ) ≃ −→ cSpec( R ⊗ B T ) × cSpec( R ⊗ B T ) Spec R, (1)where the map R ⊗ B T → R ⊗ B T is multiplication by n on B T . The first equivalenceshows in particular that R ⊗ BC n is also in the thick subcategory of R ⊗ B T . Thus weobtain from the last corollary an equivalence cSpec( R ⊗ ( BC n × B T BC m )) ≃ −→ cSpec( R ⊗ BC n ) × cSpec R ⊗ B T cSpec( R ⊗ BC m ) . Note that (1) also implies that cSpec( R ⊗ BC n ) ≃ Hom( C n , cSpec( R ⊗ B T )) . N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 25 Elliptic cohomology is symmetric monoidal
Given an oriented elliptic curve E over a (locally complex periodic) non-connective spec-tral Deligne–Mumford stack M , our aim is to prove the following theorem. Theorem 7.1. If X and Y are finite T -spaces, the natural map E ll T ( X ) ⊗ O E E ll T ( Y ) → E ll T ( X × Y ) is an equivalence. We first note that it suffices to prove the claim in the case where X and Y are T -orbits,as E ll T sends finite colimits to finite limits and these commute with the tensor product.Moreover, by the base change property Proposition 5.6 (in conjuction with Example 4.2)we can always assume that M ≃ Spec R , where R is a complex-orientable and -periodic E ∞ -ring.As a first step, we will reformulate our claim into a statement about affine morphismsusing the following lemma. Lemma 7.2.
Let S be a non-connective spectral Deligne–Mumford stack and Aff S the ∞ -category of affine morphisms U → S . Then the functor Aff op S → CAlg(QCoh( X )) , ( U f −→ S ) f ∗ O U is an equivalence. We denote the inverse by Spec S .Proof. The proof is analogous to [Lur16, Proposition 2.5.1.2]. (cid:3)
Given a representable morphism X → B T (meaning that X × B T pt is a space), we define Ell T ( X ) as Spec E E ll T ( X × B T pt) . If X is the image of a finite space along the embedding S → S
Orb , we observe that we have an equivalence
Ell T ( X ) ≃ Spec R X because the unit section Spec R → E is affine. Noting that Ell T ( B C m ) = E [ m ] and Ell T ( B T ) = E , our claim reduces to the following lemma. Lemma 7.3.
The canonical map (2)
Ell T ( B C m × B T B C n ) → E [ m ] × E E [ n ] . is an equivalence for all m, n ≥ .Proof. We first assume that m and n are relatively prime. One computes B C m × B T B C n ≃ T /C mn ∼ = T . Thus the source in (2) is equivalent to
Spec R T . We will construct next an equivalence(3) Spec( τ ≥ R ) T → ( τ ≥ E )[ m ] × τ ≥ E ( τ ≥ E )[ n ]; base changing along Spec R → Spec( τ ≥ R ) shows that (2) is an equivalence as well.Given a morphism f : X → τ ≥ E from a spectral scheme, the map X → X × τ ≥ E b E is anequivalence if the image of f is contained in the image of the unit section; this follows from b E → τ ≥ E being a monomorphism. As we can observe on underlying schemes, the image of ( τ ≥ E )[ m ] × τ ≥ E ( τ ≥ E )[ n ] → τ ≥ E is indeed contained in the unit section. Thus, ( τ ≥ E )[ m ] × τ ≥ E ( τ ≥ E )[ n ] ≃ ( τ ≥ E )[ m ] × τ ≥ E ( τ ≥ E )[ n ] × τ ≥ E b E ≃ b E [ m ] × b E b E [ n ] . As the orientation provides an equivalence b E ≃ cSpec( τ ≥ R ⊗ B T ) , we can further identifythis fiber product with cSpec( τ ≥ R ⊗ BC m ) × cSpec( τ ≥ R ⊗ B T ) cSpec( τ ≥ R ⊗ BC n ) ≃ cSpec( τ ≥ R ⊗ B C m × B T B C n ) ≃ cSpec( τ ≥ R ⊗ T ) using Example 6.5. The computation cSpec( τ ≥ R ⊗ T )( A ) = cCAlg A ( A, A ⊗ T ) ≃ CAlg A ( A T , A ) ≃ CAlg τ ≥ R (( τ ≥ R ) T , A ) , for A ∈ CAlg cn τ ≥ R shows that cSpec( τ ≥ R ⊗ T ) ≃ Spec( τ ≥ R ) T . This provides the equiva-lence (3).For m and n general, let d be their greatest common divisor. Choose relatively prime k and l such that m = kd and n = ld and consider the diagram Ell T ( B C m × B T B C n ) Ell T ( B C k × B T B C l ) E [ n ] × E E [ m ] E [ k ] × E E [ l ]Ell T ( B T ) Ell T ( B T ) E E [ d ][ d ] The front square is easily seen to be a fiber square. Concerning the back square, we have B C k × B T B C l ≃ T /C kl ∼ = T . Moreover, the resulting map T → B T has to factor throughthe point as there are no non-trivial T -principal bundles on T . Thus, the back squaredecomposes into a rectangle: Ell T ( T × B C d ) / / (cid:15) (cid:15) Ell T ( T ) (cid:15) (cid:15) Ell T ( B C d ) (cid:15) (cid:15) / / Ell T (pt) (cid:15) (cid:15) Ell T ( B T ) [ d ] / / Ell T ( B T ) The lower square is cartesian by definition and for the cartesianity of the upper square onejust has to observe that
Ell T ( T × B C d ) ≃ Spec Ell T ( B C d ) T and Ell T ( T ) ≃ Spec R T ; thusthe backsquare is cartesian. Now it remains to observe that three of the diagonal arrowsare equivalences, either by definition or the above and hence the arrow Ell T ( B C m × B T B C n ) → E [ n ] × E E [ m ] is an equivalence as well. (cid:3) N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 27 Elliptic cohomology and equivariant spectra
The goal of this section is to connect our treatment of equivariant elliptic cohomology tostable equivariant homotopy theory.We let E denote an oriented spectral elliptic curve over a non-connective spectral Deligne–Mumford stack M . For the next lemma, we fix the following notation: We denote by ρ thetautological complex representation of T = U (1) and by e n : E [ n ] ֒ → E the inclusion of the n -torsion. Lemma 8.1.
Applying f E ll T to the cofiber sequence ( T / T [ n ]) + → S → S ρ ⊗ n results in acofiber sequence (4) ( e n ) ∗ O E [ n ] ← O E ← O E ( − e n ) . The O E -module O E ( − e n ) is invertible.Proof. As e n is affine, we can compute π ∗ ( e n ) ∗ O E [ n ] as ( e n ) ∗ π ∗ O E [ n ] (cf. [Mei18, Lemma2.8], where the separatedness can be circumvented, e.g. by the use of hypercovers). Here,we abuse notation to denote by e n also the map of underlying classical stacks. As theunderlying map of the multiplication map [ n ] : E → E is flat, the underlying stack of E [ n ] isprecisely the n -torsion in the underlying elliptic curve of E .By definition, the map f E ll T ( S ) → f E ll T ( T / T [ n ] + ) agrees with the canonical map O E → ( e n ) ∗ O E [ n ] . By the above, the map π O E → ( e n ) ∗ π O E [ n ] is surjective as the underlyingmap of E [ n ] → E is a closed immersion. Moreover, by Proposition 5.6 we can reduce tothe universal case of the universal oriented elliptic curve over M = M orEll (see [Lur18b]),where both source and target of O E → ( e n ) ∗ O E [ n ] are even-periodic and thus the map O E → ( e n ) ∗ O E is surjective on homotopy groups in all degrees. In particular, its fiber f E ll T ( S ρ ⊗ n ) is also even-periodic and its π agrees with the kernel of π O E → ( e n ) ∗ π O E [ n ] ,i.e. the ideal sheaf associated with the underlying map of e n . As this sheaf is invertible, sois f E ll T ( S ρ ⊗ n ) . (cid:3) Using the universal property of equivariant stabilization, we can deduce that E ll T factorsover finite T -spectra. More precisely, we denote by S T , fin the ∞ -category of finite T -spaces(i.e. the closure of the orbits under finite colimits) and by Sp T ,ω the compact objects in T -spectra and obtain the following statement: Proposition 8.2.
We have an essentially unique factorization S T , fin() + (cid:15) (cid:15) E ll T & & ▼▼▼▼▼▼▼▼▼▼▼ S T , fin ∗ Σ ∞ (cid:15) (cid:15) f E ll T / / QCoh( E ) op Sp T ,ω . E ll T rrrrr Proof.
By Theorem 7.1 the functor E ll T and hence also the functor f E ll T is symmetricmonoidal. Thus the universal property from Corollary C.8 applies once we have checkedthat f E ll T sends every representation sphere to an invertible object and every finite T -spaceto a dualizable object. The first follows from Lemma 8.1 as every T -representation is a sumof tensor powers of ρ . For the second it suffices to show that f E ll T ( T / T [ n ] + ) is dualizable for every n . This follows again from Lemma 8.1 as it provides a cofiber sequence with f E ll T ( T / T [ n ] + ) and two invertible quasi-coherent sheaves. (cid:3) In the following construction we will explain how to obtain a genuine T -spectrum from T -equivariant elliptic cohomology and also sketch the analogous process for other compactabelian Lie groups. Construction 8.3.
Let Sp ωG → Sp op be a finite colimit preserving functor. As Sp G ≃ Ind(Sp ωG ) , this factors over a colimit preserving functor F : Sp G → Sp op , which we can alsoview as a right adjoint Sp op G → Sp . As Sp G is stable, we have Sp G ⊗ Sp ≃ Sp G , where ⊗ denotes the tensor product of presentable ∞ -categories [Lur12, Proposition 4.8.2.18].Thus the Yoneda functor Sp G → Fun R (Sp op G , Sp) is an equivalence and we see that F isrepresentable by a G -spectrum R . By definition, F (Σ ∞ G/H + ) agrees with the mappingspectrum from Σ ∞ G/H + to R , i.e. with the fixed point spectrum R B G . Note here that weuse the notation R B H for what traditionally would usually be denoted R H , the reason forwhich is two-fold: First, it fits well with our philosophy that the fixed points should reallybe associated with the stack B H rather than the group H (and could be viewed as themapping spectrum from B H to R ; cf. [Sch18, Theorem 4.4.3]). Second, it avoids possibleconfusion between the H -fixed points of R and the cotensor R H , where H is viewed as atopological space.In our case, G will be T and F the composition of E ll T : Sp ω T → QCoh( E ) op with theglobal sections functor Γ : QCoh( E ) → Sp and we obtain a representing T -spectrum R . Byconstruction R B H agrees with Γ E ll T ( T /H ) and in particular the underlying spectrum of R agrees with the global sections Γ( O M ) . Moreover, we see that more generally for every finite T -space X , the spectrum Map T (Σ ∞ X, R ) of T -equivariant maps is equivalent to Γ E ll T ( X ) .We remark that a similar, but slightly more complicated argument also will construct a T n -spectrum for elliptic cohomology if n > , the only essential difference being the necessityto modify Lemma 8.1. Denoting this T n -spectrum also by R , it is true by definition thatthe fixed points R B T n are equivalent to the global sections of O E × M n . As every compactabelian Lie groups embeds into a torus, we get by restriction more generally an equivariantelliptic cohomology spectrum for any compact abelian Lie group.9. The circle-equivariant elliptic cohomology of a point
Before we continue with elliptic cohomology, we need to recall the degree-shifting transferin equivariant homotopy theory. Recall from Corollary C.11 the equivalence of Σ ∞ + G ⊗ S − L with the Spanier–Whitehead dual D Σ ∞ + G for the tangent representation L of an arbitrarycompact Lie group G . Taking the dual of the map G + → S induces thus a map S → D Σ ∞ G + ≃ S − L ⊗ Σ ∞ G + . Mapping into a G -spectrum X and taking G -fixed points results in a further map res Ge ( S L ⊗ X ) ≃ Map G ( S − L ⊗ Σ ∞ G + , X ) → X B G , called the degree-shifting transfer .As in the last section we will fix an oriented elliptic curve E over a non-connective spectralDeligne–Mumford stack M . In the last section, we constructed a T -spectrum R with fixedpoints R B T = Γ( O E ) and whose underlying spectrum is Γ( O M ) . Leaving out res B T e tosimplify notation, the degree-shifting transfer thus takes the form of a map Σ R → R B T .Furthermore, the projection p : E → M induces a map R → R B T , which may be seen asrestriction along T → { e } . We are now ready for our main calculation. N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 29
Theorem 9.1.
Restriction along T → { e } and degree shifting shifting transfer define anequivalence R ⊕ Σ R ≃ −→ R B T Proof.
By the Wirthmüller isomorphism Corollary C.11, we can identify the dual of thecofiber sequence Σ ∞ T + → S → Σ ∞ S ρ with Σ − Σ ∞ T + ← S ← Σ ∞ S − ρ . As E ll T is symmetric monoidal on finite T -spectra by Theorem 7.1, it preserves duals. Thus,applying E ll T and taking global sections produces a cofiber sequence(5) Σ R → R B T = Γ( O E ) → Γ( O E ( e )) , where e : M → E is the unit section and we use Lemma 8.1 for the identification of the lastterm. Essentially by definition, the first map is the degree shifting transfer.We will compute Γ( O E ( e )) by identifying p ∗ O E ( e ) . Denoting by e : M → E the un-derlying classical morphism of e , we have already argued in the proof of Lemma 8.1 thatwe can reduce to the case where π ∗ O M is even and thus π ∗ O E ( e ) is concentrated in evendegrees and the restriction of π O E ( e ) to the classical locus is O E ( e ) .By [Del75, §1] R p ∗ O E ( e ) = 0 and the morphism O M → p ∗ O E ( e ) is an isomorphism.By the flatness of E → M and the projection formula, R p ∗ ( π k O E ( e )) ∼ = R p ∗ ( O E ( e ) ⊗ O E p ∗ π k O M ) ∼ = R p ∗ O E ( e ) ⊗ O M π k O M vanishes as well. Moreover, the higher derived images vanish as E is smooth of relativedimension . Thus the relative descent spectral sequence R s π t O E ( e ) ⇒ π t − s p ∗ O E ( e ) is concentrated in the zero-line. We see that the map O M → p ∗ O E → p ∗ O E ( e ) is an equiv-alence and thus taking global sections shows that R → R B T → Γ( O E ( e )) is an equivalenceas well. Hence (5) takes the form of a split cofiber sequence Σ R → R B T → R. (cid:3) Remark 9.2.
Intuitively the equivalence Γ( O E ) ≃ R ⊕ Σ R corresponds to the fact thatan elliptic curve (say, over a field) has only non-trivial H and H , both of rank . Theappearance of a non-trivial H -term yields to a peculiar behaviour of the completion map R B T ≃ R ⊕ Σ R → R B T . If R is concentrated in even degrees, we see that the map factorsover the standard map R → R B T (thus is not injective in any sense, in contrast to thesituation for equivariant K -theory). At least implicitly, this factorization is a key ingredientfor the equivariant proofs of the rigidity of the elliptic genus as in [Ros01]. Corollary 9.3.
There is an equivalence n M k =0 M P k ( n ) Σ k R → Γ( O E × n ) = R B T n , where P k ( n ) runs over all k -element subsets of { , . . . , n } and E × n stands for the n -fold fiberproduct over M . One can obtain the relative descent spectral sequence e.g. by applying π ∗ p ∗ to the Postnikov tower of O E ( e ) and observing that for a sheaf F concentrated in degree t , we have π t − s p ∗ F ∼ = R s π t F . Proof.
Consider the oriented elliptic curve E n → E n − over E n − given by projection to thefirst ( n − coordinates, i.e. the pullback of E → M along E n − → M . Applying Theorem 9.1,we see that Γ( O E × n ) ≃ Γ( O E × ( n − ) ⊕ ΣΓ( O E × ( n − ) Induction yields the result. (cid:3)
Example 9.4.
Let M = M or Ell be the moduli stack of oriented elliptic curves and E be theuniversal oriented elliptic curve over it (see [Lur18b, Proposition 7.2.10] and [Mei18, Section4.1] for definitions). The stack M or Ell can be thought of as the classical moduli stack of ellipticcurves equipped with the Goerss–Hopkins–Miller sheaf O top of E ∞ -ring spectra. In this case, R = Γ( O top ) is by definition the spectrum TMF of topological modular forms. Our theoryspecializes to define T n -spectra with underlying spectrum TMF and the previous corollarycomputes its T n -fixed points. Let us comment about the fixed points for other compactabelian Lie groups.In [Mei18] the second-named author computed the G -fixed points of TMF for G a finiteabelian group after completing at a prime l not dividing the group order, namely as a sumof shifts of \ TMF (3) if p = 2 , of \ TMF (2) if l = 3 and of [ TMF p if l > . In the latter case,the result can actually be strengthened as follows: Denote by p : E → M Ell the underlyingmap of classical Deligne–Mumford stacks of p : E → M or Ell . By [Mei18, Lemma 4.7], themorphism
Hom( b G, E ) → M Ell is finite and flat. Thus ( p ) ∗ O Hom( b G, E ) is a vector bundleon M Ell . After localizing at a prime l > , [Mei15, Theorem A] implies that this vectorbundle splits into a sum of ω ⊗ i = π i O top . A descent spectral sequence argument showsthat TMF B G ( l ) = Γ( O Hom( b G, E ) ) ( l ) splits indeed into shifts of TMF ( l ) , without the assumptionthat l does not divide | G | . In contrast if l = 2 or and it divides the order of the group,the computation of TMF B G ( l ) is still wide open, as far as we know, even if G is C or C .More generally, we can consider TMF B ( G × T n ) for a finite abelian group G . This coincideswith the global sections of the structure sheaf of Hom( b G, E ) × M E n . Denoting the pullbackof E to Hom( b G, E ) by E ′ this agrees with the n -fold fiber product of E ′ over Hom( b G, E ) .Thus, Corollary 9.3 implies TMF B ( G × T n ) ≃ n M k =0 M P k ( n ) Σ k TMF B G . Corollary 9.5.
The
TMF -module
TMF B H = Γ( O Ell( B H ) ) of H -fixed points is dualizablefor every abelian compact Lie group H .Proof. Let M = M or Ell and E as in the example above. We can split H ∼ = G × T , where G isfinite and T a torus of dimension n . As seen in the previous example, TMF B A splits as asum of suspensions of TMF B G and so it suffices to show that TMF B G is dualizable. Arguingas in [Mei18, Proposition 2.13] this follows from Hom( b G, E ) → M Ell being finite and flat,with notation as in the previous example. (cid:3)
Remark 9.6.
We actually conjecture that
TMF B G is dualizable for every compact Liegroup. (Strictly speaking, we have not defined TMF B G if G is not abelian, but this we seeas the lesser problem.) Crucial evidence is given by Lurie’s results on the finite group case,which we will summarize.Every oriented spectral elliptic curve E over an E ∞ -ring A gives rise to a P -divisiblegroup G over A in the sense of [Lur19, Definition 2.6.1] by [Lur19, Section 2.9]. By [Lur19, N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 31
Notation 4.0.1] this in turn gives rise to a functor A G : T op → CAlg A , where T denotesthe full subcategory of S on the spaces BH for H a finite abelian group, and [Lur19,Construction 3.2.16] allows to extend this to a functor on H -spaces for all finite groups H .We conjecture that the spectrum Lurie associates to H y pt is equivalent to our spectrum Γ( O Ell( B H ) ) and moreover that these are the H -fixed points of a global spectrum A (e.g. inthe sense of [Sch18]) – in any case, we denote the spectrum constructed by Lurie by A B H .Lurie’s Tempered Ambidexterity Theorem implies that this spectrum is a dualizable (andeven self-dual) A -module as follows: Lurie defines for every finite group H an ∞ -category LocSys G ( B H ) . For H = e , we have LocSys G (pt) ≃ Mod( A ) and the pushforward f ∗ A B H ofthe “constant local system” along f : B H → pt corresponds to A B H under this equivalence.Moreover, the Tempered Ambidexterity Theorem [Lur19, Theorem 1.1.21] identifies thiswith f ! A B H , where f ! denotes the left adjoint to the restriction functor. We compute Map A ( A B H , A ) ≃ Map A ( f ! A B H , A ) ≃ Map
LocSys G ( B H ) ( A B H , f ∗ A ) ≃ f ∗ A B H ≃ A B H . Appendix A. Quotient ∞ -categories In this appendix we collect some basic results concerning quotient ∞ -categories. Recallthat a quotient of an ordinary category D is a category C obtained by identifying morphismsin C by means of an equivalence relation which is suitably compatible with composition.A quotient functor q : D → C is necessarily essentially surjective, as C is typically taken tohave the same objects as D , though nonisomorphic objects of D may become isomorphicin C . A standard example (and one which is important for our purposes) is the homotopycategory of spaces, which is a quotient of the ordinary category of Kan complexes by thehomotopy relation.For some of the arguments in this appendix, it will be convenient to regard Cat ∞ ⊂ Fun(∆ op , S ) as a reflective subcategory of the ∞ -category of simplicial spaces. From this point of view,many different simplicial spaces C • give rise to the same ∞ -category C , even if we assumethat the simplicial space satisfies the Segal condition. The standard simplicial model of C isthe simplicial space C • with C n = Map(∆ n , C ) ; up to equivalence, this is the only complete Segal space model of C . However, we will have occasion to consider other (necessarily notcomplete) models as well, especially those models which arise from restricting the space ofobjects along a given map. Note that to compute the geometric realization of a simplicialobject ∆ op → Cat ∞ , we may first choose any lift to Fun(∆ op , S ) , where the realization iscomputed levelwise, and then localize the result (which amounts to enforcing the Segal andcompleteness conditions). Construction A.1.
Let C • : ∆ op → S be a simplicial space and let f : D → C be amorphism in S . Let i : ∆ ≤ → ∆ denote the inclusion of the terminal object [0] , andconsider the unit map C • → i ∗ i ∗ C • ≃ i ∗ C . Define a simplicial space D • : ∆ op → S as thepullback D • / / (cid:15) (cid:15) C • (cid:15) (cid:15) i ∗ D / / i ∗ C . We also refer to this simplicial space as f ∗ C • . Observe that, more or less by construction, if C • is additionally a Segal space, then D • is also a Segal space, and moreover that if S and T are any pair of points in D , then Map D • ( S, T ) ≃ Map C • ( f ( S ) , f ( T )) . In particular, D • → C • is fully faithful, and it is essentially surjective if f is π -surjective(and conversely if C • is additionally complete).Thus, if f : S → C is any π -surjective map of spaces, then the simplicial space f ∗ C • obtained by restriction is a simplicial space model of C which satisfies the Segal condition,though it will not be complete unless f is an equivalence (cf. [Rez01, Theorem 7.7]). Inparticular, we can take S to be discrete and see that every Segal space is equivalent toone with a discrete space of objects. Moreover, given an essentially surjective functor q : D → C , we may choose a π -surjection f : S → D and replace q by the equivalentmap f ∗ D → ( qf ) ∗ C , which is the identity on the (discrete) spaces of objects. If q is notessentially surjective, we can factor it into an essentially surjective functor and an inclusion,thus obtaining a Segal space model that is an injection on discrete spaces of objects. Definition A.2.
A morphism of ∞ -categories q : D → C is a quotient functor if it isessentially surjective and π Map D ( s, t ) → π Map C ( s, t ) is surjective for all pairs of objects s and t of D .In other words, q : D → C is a quotient functor if all objects and arrows of C lift to objectsand arrows of D , respectively. Proposition A.3.
A functor q : D → C is a quotient if and only if the augmented simplicialdiagram · · · →→→→ D × C D × C D →→→ D × C D →→ D → C is a colimiting cone.Proof.
First suppose that the canonical map |D × C • | → C is an equivalence. In particular,it is essentially surjective, so that π |D × C • | → π C is surjective. Note that π D surjectsonto π |D × C • | . This can be seen for example by using a Segal space model where D → C is injective on discrete spaces of objects and thus Lemma A.4 implies that |D × C • | can beviewed as a Segal space with the same set of objects as D . Thus, the composite π D → π C is surjective as well and it follows that D → C is essentially surjective. We now showthat
D → C is also surjective on mapping spaces. By the essential surjectivity, the map q : D → C admits a Segal space model which is the identity on (discrete) spaces of objects S ,in which case, by Lemma A.4 below, the realization of D × C • can be computed levelwise. Butthen Map C ( A, B ) is the realization of Map D ×C• ( A, B ) , and consequently π Map D ×C• ( A, B ) surjects onto π Map C ( A, B ) .Conversely, suppose that q : D → C is a quotient functor, and consider the comparisonmap |D × C • | → C . Since π D → π C is surjective, we may again suppose without lossof generality that C and D admit Segal space models with same discrete space of objects S , in which case it follows that π |D × C • | ∼ = π |D × C • | ∼ = π C . In particular, |D × C • | → C is essentially surjective. To see that it is fully faithful, we use that for any π -surjectivemap X → Y of spaces, the induced map | X × Y • | → Y is an equivalence (see e.g. [Lur09a,Corollary 7.2.1.15]). We thus compute the mapping spaces as follows: Map |D ×C• | ( A, B ) ≃ | Map D ×C• ( A, B ) |≃ | Map D ( A, B ) × Map C ( A,B ) • |≃ Map C ( A, B ) (cid:3) N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 33
Lemma A.4.
Suppose given a simplicial Segal space D • : ∆ op → Fun
Seg (∆ op , S ) with aconstant and discrete space of objects; that is, the functor ∆ op → Fun
Seg (∆ op , S ) → S obtained by composing D • with evaluation at [0] is constant with value some discrete space S ∈ τ ≤ S . Then |D • | ∈ Fun(∆ op , S ) is again a Segal space with object space S . Moreover,for any pair of objects s , s ∈ S , there is an equivalence Map |D • | ( s , s ) ≃ | Map D • ( s , s ) | .Proof. The space of objects of |D • | is again S , which is discrete. Using the fact thatgeometric realization commutes with colimits and finite products, we see that for < k < n , Map(∆ n , |D • | ) ≃ a ( s ,...,s n ) ∈ S × [ n ] Map |D • | ( s , s ) × · · · × Map |D • | ( s n − , s n ) ≃ Map( N n , |D • | ) , where N n ≃ ∆ ` ∆ ` · · · ` ∆ ∆ denotes the n -spine. It follows that |D • | satisfies theSegal condition. The last statement is the case n = 1 , which is similar, but strictly easier,as term involving the 1-spine N = ∆ is irrelevant. (cid:3) Proposition A.5.
Let D • : ∆ op → Cat ∞ be a functor with colimit C ≃ |D • | . Then theprojection D → C is a quotient functor.Proof. Similar to the first part of the proof of Proposition A.3. (cid:3)
Let C be an ∞ -category with finite limits and let X ∈ C be an object. We can define anew ∞ -category C [ X ] in which the mapping spaces Map C [ X ] ( S, T ) ≃ Map C ( X × S, T ) are X -indexed families of maps. More precisely, we consider a functor µ X : C → C givenby multiplication with X . Since X ≃ µ X (pt) , this factors through the slice C /X , and wefurther factor this functor as a composite C → C [ X ] ⊂ C /X . Here C [ X ] ⊂ C /X denotes the full subcategory consisting of those objects in the image of µ X , and (for ease of notation) we will simply denote these objects by their name in C . Thisapproach has the advantage that composition is well-defined ( C [ X ] is a full subcategory of C /X ), and we have an equivalence Map C [ X ] ( S, T ) = Map C /X ( X × S, X × T ) ≃ Map C ( X × S, T ) . To see that this construction is natural in X , we use an auxiliary construction. Considerthe target projection p : Fun(∆ , C ) → C . It is a cartesian fibration via pullback, since C admits finite limits, and the straightening of this fibration is a functor C op → Cat ∞ , whichsends X to C /X . In particular, given a simplicial set I and a functor f : I → C , we obtainthe desired functor I op → Cat ∞ by precomposing C op → Cat ∞ with f op . Construction A.6.
Let D be an ∞ -category with finite products equipped with a cosim-plicial object ∆ D : ∆ → D . Let D n := D [∆ n D ] ⊂ D / ∆ n D , so that we obtain a simplicial ∞ -category D • , and set C = |D • | . Then the projection q : D ≃ D → C is a quotientfunctor. Appendix B. Comparison of models for the global orbit category
The goal of this appendix is to compare our definition of the global orbit category
Orb with the construction in [GH07]. This is especially important because the latter one has beencompared to other models of unstable global homotopy theory in [Kör18] and [Sch20]. As this appendix will involve point set considerations, we will use hocolim to denote a homotopycolimit in a relative category or, equivalently, a colimit in the associated ∞ -category.The model of [GH07] of the global orbit category is based on the notion of the fatgeometric realization of a simplicial space, which we denote by || − || . In contrast to theusual geometric realization, || X • || is always a model for the homotopy colimit hocolim ∆ op X • .Indeed, Segal has shown in [Seg74, Appendix A] that the fat realization is always homotopyinvariant and equivalent to the usual realization on good spaces and in particular on Reedycofibrant ones; on these it is well-known that the geometric realization is a model of thehomotopy colimit.One usually defines the fat realizations like the geometric realizations, but one onlytakes identifications along face and not along degeneracy maps. We will use the equivalentdefinition as the geometric realization of the simplicial space X • that arises as the left Kanextension of X • along ∆ inj → ∆ , where ∆ inj stands for the subcategory of ∆ of injections.Concretely, we have X n = ` f : [ n ] ։ [ k ] X k ; given α : [ m ] → [ n ] , we factor the composite to [ k ] as h ◦ g : [ m ] ։ [ l ] ֒ → [ k ] and get α ∗ ( f, x ) = ( g, h ∗ x ) .Homotopically, fat realization is strong monoidal: Segal showed in [Seg74, Appendix A]that the canonical map || X × Y || → || X || × || Y || is a weak equivalence and || pt || = ∆ ∞ iscontractible. We claim moreover that fat realization is lax monoidal on the nose: The map || X × Y || → || X ||×|| Y || admits a retract r , which is induced by a retract r : X × Y → X × Y of the canonical map: Given a pair of ( f : [ n ] ։ [ k ] , x ∈ X k ) and ( g : [ m ] ։ [ l ] , y ∈ Y k ) ,there is a unique surjection h : [ n ] ։ [ p ] such that f and g factor over h and p is minimalwith this property. We define r (( f, x ) , ( g, y )) as the pair of h and the pair of images of x and y in ( X × Y ) p .For a simplicial diagram C • in TopCat whose functor of objects is constant, applyingthe fat realization to the mapping spaces thus defines a topological category ||C • || . Theanalogous construction also works for simplicial diagrams in simplicial categories. Definition B.1.
Consider the category of topological groupoids as enriched over itself.Define
Orb
Gpd as the full enriched subcategory on the groupoids { G →→ pt } for G compactLie. We define the Gepner–Henriques global orbit category Orb ′ as the topological category || Orb
Gpd || (cf. [GH07, Remark 4.3]). Concretely, we have Map
Orb ′ ( H, G ) = || Map
TopGpd ( H →→ pt , G →→ pt) || if we identify its objects with the corresponding compact Lie groups.To compare Orb ′ with our definition of Orb , we have to investigate the precise homotopicalmeaning of taking homwise fat realization of a simplicial diagram of topological categories.For us, a functor between topological or simplicial categories is a (Dwyer–Kan) equivalence if it is a weak equivalence on all mapping spaces and an equivalence on homotopy categories.Homwise geometric realization and singular complex define inverse equivalences between therelative categories
TopCat and sCat . As the latter can be equipped with the Bergner modelstructure, we see that both have all homotopy colimits. We moreover recall the Rezk modelstructure on
Fun(∆ op , sSet) from [Rez01]. Proposition B.2.
Let C • be a simplicial diagram in TopCat whose functor of objects isconstant. Then ||C • || is a model for the homotopy colimit hocolim ∆ op C • in topological cate-gories. Furthermore, the analogous statement is true for simplicial categories.Proof. Observe first that homwise geometric realization and singular complex define anequivalence between the relative categories
TopCat and sCat . As the geometric realization
N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 35 functor sSet → Top commutes with fat realization of simplicial objects, it suffices to showthe statement for simplicial categories.We will use the nerve functor N : sCat → Fun(∆ op , sSet) defined as follows: given asimplicial category C , the zeroth space of N C is the set of objects, the first space is thedisjoint union over all mapping spaces and the higher ones are disjoint unions of productsof composable mapping spaces. To analyze this functor, we introduce the intermediatecategory SeCat of Segal precategories, i.e. objects of
Fun(∆ op , sSet) whose zeroth space isdiscrete. In [Ber07], Bergner writes N as a composite sCat R −→ SeCat I −→ Fun(∆ op , sSet) . The functor R is a right Quillen equivalence for a certain model structure on SeCat . Thefunctor I is a left Quillen equivalence from a different model structure with the same weakequivalences and all objects cofibrant to the Rezk model structure on Fun(∆ op , sSet) . Thuswe see that N induces an equivalence of relative categories between fibrant simplicial cate-gories and Fun(∆ op , sSet) with the Rezk equivalences. In particular, N preserves and reflectshomotopy colimits on fibrant simplicial categories.As fat realization commutes with disjoint unions and is homotopically strong symmetricmonoidal, we see that N intertwines the homwise fat realization in sCat with the levelwisefat realization in Fun(∆ op , sSet) up to levelwise equivalence of simplicial spaces. Moreover,levelwise fat realization is a homotopy colimit for levelwise equivalences of simplicial spacesand hence also for Rezk equivalences (as the Rezk model structure is a left Bousfield localiza-tion of the Reedy model structure). We deduce that homwise fat realization is a homotopycolimit in simplicial categories. (cid:3) Corollary B.3.
Let C • be a simplicial diagram C • in TopCat whose functor of objects isconstant. Then N coh Sing ||C • || is equivalent hocolim ∆ op N coh Sing C • , where Sing and || − || are formed on the level of mapping (simplicial) spaces.Proof.
Applying the singular complex to the mapping spaces of a topological category resultsin a fibrant simplicial category. The coherent nerve N coh preserves homotopy colimits offibrant simplicial categories (as the right derived functor of N coh defines an equivalence of ∞ -categories between simplicial categories and quasi-categories). As Sing also commuteswith homotopy colimits, the result follows from the preceding proposition. (cid:3)
Our main interest in simplicial diagrams of topological categories lies in their associationwith topological groupoids (and thus to
Orb ′ ): We can associate to a topological groupoid X • a simplicial topological space and following [GH07] we denote by || X • || the fat realizationof this space. On the level of categories we obtain from a category enriched in topologicalgroupoids a simplicial diagram in topological categories with a constant set of objects. Weobtain the following further corollary. Corollary B.4.
Let C be a category enriched in topological groupoids. Let ||C|| be thetopological category obtained by applying || − || homwise. Then the associated ∞ -category N coh Sing ||C|| can alternatively be computed as follows: Apply
Sing homwise to C to obtaina simplicial diagram in groupoid-enriched categories, take the associated ∞ -categories andtake their homotopy colimit. Recall that
Orb
Gpd is the category enriched in topological groupoids, which is the fullsubcategory of
TopGpd on those objects of the form { G →→ pt } for G compact Lie. Applying Sing to the mapping groupoids results in a simplicial object
Orb
Gpd • in groupoid-enrichedcategories whose mapping groupoids in level n are Map
TopGpd ( G × ∆ n →→ ∆ n , H →→ pt) . There is a corresponding simplicial object
Orb
Stk • whose n -th groupoid enriched categoy hasmapping groupoids Map
Stk (∆ n ×B G, B H ) . Stackification provides a map Orb
Gpd • → Orb
Stk • of simplicial groupoid-enriched categories. Lemma B.5.
The map
Orb
Gpd • → Orb
Stk • defines in each level on each mapping groupoidan equivalence.Proof. Let H and G be compact Lie groups. We have to show that Map
TopGpd ( G × ∆ n →→ ∆ n , H →→ pt) → Map
Stk (∆ n × B G, B H ) is an equivalence of groupoids. As B H is a stack and stackification is left adjoint, thetarget is equivalent to Map
Pre Stk ( G × ∆ n →→ ∆ n , B H ) . This groupoid is the groupoid of H -principal bundles on { G × ∆ n →→ ∆ n } : an object is a principal H -bundle on ∆ n with anisomorphism between the two pullbacks to G × ∆ n satisfying a cocycle condition. As everyprincipal H -bundle on ∆ n is trivial, we may up to equivalence replace this groupoid by thesubgroupoid where we require the H -principal bundle on ∆ n to be equal to H × ∆ n . Thissubgroupoid is precisely Map
TopGpd ( G × ∆ n →→ ∆ n , H →→ pt) . Indeed, the set of objects is Map(∆ n , Hom(
G, H )) and the set of morphisms is Map(∆ n , Hom(
G, H ) × H ) . (cid:3) Recall that we can view every groupoid-enriched category as a quasi-category via theDuskin nerve N Dusk ; this is isomorphic to taking the homwise nerve and applying thecoherent nerve. Note further by construction there is for each n a fully faithful embedding N Dusk
Orb
Stk n → Stk[∆ n ] , with [∆ n ] as in the preceding appendix. These define a morphismof simplicial diagrams and thus a fully faithful embedding hocolim ∆ op N Dusk
Orb
Stk • → Stk ∞ with image Orb . By composing this with the equivalence of the preceding lemma, we obtainan equivalence hocolim ∆ op N Dusk
Orb
Gpd • → Orb . By Corollary B.4 the source can be identified with N coh Sing Orb ′ , i.e. with the ∞ -categoryassociated with Orb ′ . Thus we obtain the goal of this appendix: Proposition B.6.
Stackification induces an equivalence between
Orb and the ∞ -categoryassociated with Orb ′ . Appendix C. The ∞ -category of G -spectra The aim of this appendix is to introduce the ∞ -category of equivariant spectra, compareit to the theory of orthogonal spectra and deduce from Robalo’s thesis a universal propertyof this ∞ -category. Other ∞ -categorical treatments of G -spectra for (pro)finite groups G include [Bar17] and [Nar16].C.1. G -spaces and G -spectra. We first review a construction of ∞ -categories from -categorical data: A relative category is a category with a chosen class of morphisms, called weak equivalences , containing all identities and closed under composition. To every relativecategory C , we can consider an ∞ -category L C , which universally inverts all weak equiva-lences. We recall the following functorial construction: Let core : Cat ∞ → S be the rightadjoint of the inclusion (i.e. the maximal ∞ -subgroupoid) and consider the resulting functor Cat ∞ → Cat ∆[1] ∞ , C 7→ (core( C ) → C ) . This functor preserves limits and is accessible (as the same is true for core ) and thus theadjoint functor theorem implies that it admits a left adjoint L ∞ : Cat ∆[1] ∞ → Cat ∞ . The N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 37 universal property of the adjunction implies that
C ≃ L ∞ ( ∅ ⊂ C ) → L ∞ ( W ⊂ C ) is a local-ization at W in the sense of [Cis19, Definition 7.1.2]. Let RelCat denote the (2 , -category ofrelative categories, weak equivalence preserving functors and natural isomorphisms. Viewing (2 , -categories as ∞ -categories as before, we obtain a composite functor RelCat → Cat ∆[1] → Cat ∆[1] ∞ L ∞ −−→ Cat ∞ , which we define to be L .From now on let G be a compact Lie group. We denote by Top G the relative categoryof (compactly generated, weak Hausdorff) topological spaces with G -action, where weakequivalences are detected on fixed points for all subgroups H ⊂ G . We denote by S G = L Top G the associated ∞ -category. Elmendorf’s theorem implies that S G ≃ Fun(Orb G , S ) ,where Orb G denotes the orbit category. Likewise there are pointed versions: Calling a G -space well-pointed if all its fixed points are well-pointed, we denote by Top G ∗ the relativecategory of well-pointed topological G -spaces and by S G ∗ ≃ Fun(Orb G , S ∗ ) the associated ∞ -category.Given an orthogonal G -representation V , we denote by S V its one-point compactificationand by Σ V the functor − ∧ S V . This defines a homotopical functor Top G ∗ → Top G ∗ . Indeed,by [Sch18, Proposition B.1(iii)], we have ( S V ∧ X ) H ∼ = S V H ∧ X H . This reduces to thenon-equivariant version, where it is well-known (see also [Sch18, p.258]). We remark thatthe induces functor S G ∗ → S G ∗ agrees with the tensor − ⊗ S V in the pointwise monoidalstructure on Fun(Orb G , S ∗ ) .We fix in the following an orthogonal G -representation U . In the main body of this article,this will always be a complete universe, i.e. U is countably-dimensional and contains up toisomorphism every countable direct sum of finite-dimensional orthogonal G -representations.(This always exists as e.g. shown in [Sch18, p.20].)Informally speaking, we obtain the ∞ -category Sp G U of G -spectra by inverting all sub-representation of U on S G ∗ . More precisely, let Sub U be the poset of finite-dimensionalsubrepresentations of U . We define a (pseudo-)functor T : Sub U → RelCat sending each V to the subcategory Top G ∗ . Given an inclusion V ⊂ W , we denote by W − V the orthogonalcomplement of V in W and define the relative functor T ( V ⊂ W ) as Top G ∗ Σ W − V −−−−→ Top G ∗ .In particular, this induces a functor LT : Sub U → Cat ∞ .We note that S G ∗ is by its characterization as a functor category presentable and for each V ⊂ W , the functor LT ( V ⊂ W ) is a left adjoint with right adjoint Ω W − V . As by Illman’stheorem, S W − V has the structure of a finite G -CW-complex, Ω W − V : S G ∗ → S G ∗ preservesfiltered colimits (as can, for example, be shown by induction over the cells) and hence Σ W − V preserves compact objects. Thus, LT can be seen as taking values in the ∞ -category Cat ω ∞ of compactly generated ∞ -categories and compact-object preserving left adjoints. (We referto [Heu15, Appendix A] for a quick overview of compactly generated ∞ -categories.) Definition C.1.
The ∞ -category Sp G U of G -spectra is the colimit over LT in Cat ω ∞ . If U isa complete universe, we write Sp G for Sp G U . We denote the map S G ∗ → Sp G U correspondingto the subspace ⊂ U by Σ ∞ .By [Heu15, Lemma A.4], we can write the compact objects in Sp G U as the idempotentcompletion of colim Sub U ( S G ∗ ) ω , where ( S G ∗ ) ω are the compact objects and the colimit istaken in ∞ -categories. Thus, Sp G U agrees with the ind-completion of this colimit [Lur09a,Proposition 5.5.7.8]. If U is countably-dimensional, we can find a cofinal map N → Sub U and furthermore the map from underlying graph of N to the nerve of N is cofinal (e.g. using[Lur09a, Corollary 4.1.1.9]), making this colimit particularly easy to understand.Another perspective on this colimit is as the limit over Sub op U of the diagram T R with T R ( V ⊂ W ) = ( S G ∗ Ω V ⊥ −−−→ S G ∗ ) (see [Lur09a, Proposition 5.5.7.6, Remark 5.5.7.7]). Thisway we also see that the colimit defining Sp G U agrees with the corresponding colimit in the ∞ -category Pr L of presentable ∞ -categories.C.2. Comparison to orthogonal spectra.
We want to compare our definition of G -spectra with the (maybe more traditional) definition via orthogonal spectra. Thus let Sp GO be the relative category of G -objects in orthogonal spectra, whose weak equivalences are thestable equivalences with respect to a complete universe U (which we will fix from now on).We refer to [HHR16], [Sch14], [MM02] and [Sch18] for general background on equivariantorthogonal spectra. We will use the stable model structure from [MM02, Section III.4]. Lemma C.2.
The ∞ -category L Sp GO is compactly generated.Proof. The ∞ -category L Sp GO has all colimits and these can be computed as homotopycolimits in Sp GO [Lur09a, Section 4.2.4]. By [Lur09a, Proposition 5.4.2.2] it thus suffices toobtain a set of compact objects generating Sp GO under filtered homotopy colimits.To that purpose we first choose a set C ′ of pointed G -spaces such that every pointed finite G -CW complex is isomorphic to an object in C ′ . We define C as the set of all Σ − V Σ ∞ X for X ∈ C ′ and V an orthogonal subrepresentation of the universe U . Here, Σ − V Σ ∞ X = F V ( X ) with F V being the left adjoint to the evaluation ev V from Sp GO to pointed G -spaces. As in[HHR16, B.4.3], we can pick a cofinal map N → Sub U with images V n and an arbitraryorthogonal G -spectrum E to form zig-zags Σ − V n Σ ∞ E ( V n ) ≃ ←− Σ − V n +1 Σ ∞ Σ V n +1 − V n E ( V n +1 ) → Σ − V n +1 Σ ∞ E ( V n +1 ) , all mapping to E . Pasting the zig-zags and taking the homotopy colimit produces anequivalence hocolim Σ − V n Σ ∞ E ( V n ) → E (as one sees by taking homotopy groups). Each E ( V n ) in turn can be written as a filtered colimit in S G ∗ of finite G -CW complexes and thuseach Σ − V n Σ ∞ E ( V n ) as a filtered colimit of objects in C . (cid:3) We remark that Σ ∞ : Top G ∗ → Sp GO is homotopical by [Sch18, Proposition 3.1.44]. Thus Σ ∞ descends to a functor S G ∗ → L Sp GO . Lemma C.3.
Given
X, Y ∈ S
G,ω ∗ , the zig-zag colim V ∈ Sub U Map S G ∗ (Σ V X, Σ V Y ) (cid:15) (cid:15) Map L Sp GO (Σ ∞ X, Σ ∞ Y ) / / colim V ∈ Sub U Map L Sp GO (Σ ∞ Σ V X, Σ ∞ Σ V Y ) consists of equivalences.Proof. The horizontal map is an equivalence as Σ ∞ commutes with Σ V and the latter definesan equivalence on L Sp GO .For the other equivalence recall that the fibrant objects in the stable model structureare the G - Ω -spectra, i.e. the orthogonal G -spectra Z such that the adjoint structure maps Z ( V ) → Ω W Z ( V ⊕ W ) are G -equivalences for all representations V, W . One checks that theorthogonal G -spectrum (Σ V Y ) ′ given by (Σ V Y ) ′ n = hocolim V ′ ∈ Sub U Ω V ′ Σ V ′ + n Σ V Y defines N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 39 a fibrant replacement for Σ ∞ Σ V Y . Moreover, we can assume that X is a G -CW complexesso that Σ ∞ Σ V X is cofibrant. Thus, we can compute Map L Sp GO (Σ ∞ Σ V X, Σ ∞ Σ V Y ) as themapping space in the topological category of orthogonal G -spectra between Σ ∞ Σ V X and (Σ V Y ) ′ . As Σ ∞ is adjoint to taking the zeroth space, this agrees with Map S G ∗ (Σ V X, hocolim V ′ ∈ Sub U Ω V ′ Σ V ′ Σ V Y ) ≃ colim V ′ ∈ Sub U Map S G ∗ (Σ V ′ ⊕ V X, Σ V ′ ⊕ V Y ) . The map colim V ∈ Sub U Map S G ∗ (Σ V X, Σ V Y ) → colim V,V ′ ∈ Sub U Map S G ∗ (Σ V ′ ⊕ V X, Σ V ′ ⊕ V Y ) is clearly an equivalence. (cid:3) Proposition C.4.
There is an equivalence Sp G ≃ L Sp GO .Proof. Analogously to the functor T : Sub U → RelCat considered before, we can consider a(pseudo-)functor T ′ : Sub U → RelCat that sends each V to Sp GO and each inclusion V ⊂ W to Σ W − V , with W − V the orthogonal complement of V in W . (The functors Σ W − V : Sp GO → Sp GO are indeed homotopical by [Sch18, Proposition 3.2.19].) As Σ W − V : Sp GO → Sp GO definesan equivalence of associated ∞ -categories, we can identify the colimit of LT ′ : Sub U → Cat ω ∞ with L Sp GO .As Σ W − V Σ ∞ is canonically isomorphic to Σ ∞ Σ W − V , we see that Σ ∞ defines a naturaltransformation T ⇒ T ′ . Taking colimits of the associated natural transformation LT ⇒ LT ′ (with values in Cat ω ∞ ), we obtain a functor F : Sp G → L Sp GO in Cat ω ∞ .By definition, precomposing F with the map ι V : S G ∗ → Sp G associated with a subrep-resentation V ⊂ U , sends some X ∈ S G ∗ up to equivalence to Σ − V Σ ∞ X (as all choicesof V -fold desuspensions are equivalent). As such orthogonal spectra generate L Sp GO viacolimits, Sp G has all colimits and F preserves them, we see that F is essentially surjective.As Sp G is compactly generated, every object in Sp G is a filtered colimit of objects of theform ι V ( X ) with X ∈ S G ∗ compact. As a mapping space in a filtered colimit of ∞ -categoriesis just the filtered colimit of mapping spaces, we can identify the map Map Sp G ( ι V ( X ) , ι V ( Y )) → Map L Sp GO ( F ι V ( X ) , F ι V ( Y )) with the natural map from colim V ⊂ W ∈ Sub U Map S G ∗ (Σ W − V X, Σ W − V Y ) to colim V ⊂ W ∈ Sub U Map L Sp GO (Σ ∞ Σ W − V X, Σ ∞ Σ W − V Y ) ≃ Map L Sp GO (Σ − V Σ ∞ X, Σ − V Σ ∞ X ) , which is an equivalence by the previous lemma. Thus, F is fully faithful. (cid:3) C.3.
Symmetric monoidal structures and universal properties.
Next, we want todeduce a symmetric monoidal universal property for Sp G . Recall to that purpose that anobject X of a symmetric monoidal ∞ -category is called symmetric if the cyclic permutationmap acting on X ⊗ X ⊗ X is homotopic to the identity. Lemma C.5.
Let G be a compact Lie group and V an orthogonal G -representation. Then S V is symmetric in G -spaces. One easy way of seeing this is by identifying the mapping space between X and Y in some ∞ -category C with the fiber product pt × C ∂ ∆1 C ∆ in Cat ∞ , where the map pt → C ∂ ∆ classifies ( X, Y ) . As both ∂ ∆ and ∆ are compact in Cat ∞ , the result follows. Proof.
We can write V as L i W ⊕ n i i , where the W i are irreducible. By Schur’s lemma thegroup of G -equivariant automorphisms Aut G ( V ) of V is isomorphic to Q i GL n i k i , where k i = R , C or H . Thus, π Aut G V is a finite product of Z / . As the cyclic permutation σ : V ⊕ → V ⊕ is of order , we see that σ is in the path-component of the identityin Aut G ( V ) . Taking one-point compactifications, we see that the cyclic permutation of S V ∧ S V ∧ S V is homotopic to the identity. (cid:3) Recall that a symmetric monoidal ∞ -category is presentably symmetric monoidal if itis presentable and the tensor products commutes in both variables with colimits. (Equiv-alently, it is a commuative algebra in the ∞ -category Pr L of presentable ∞ -categories.)Robalo shows in [Rob15, Proposition 2.9 and Corollary 2.20] the following: Theorem C.6.
Let C be a presentably symmetric monoidal category and X ∈ C a symmetricobject. Let Stab X C be the colimit of C ⊗ X −−→ C ⊗ X −−→ · · · in Pr L . Then C →
Stab X C refines to a symmetric monoidal functor and for any other pre-sentably symmetric monoidal ∞ -category D and a symmetric monoidal left adjoint F : C →D , sending X to an invertible object, there is a essentially unique factorization over a sym-metric monoidal left adjoint Stab X C → D . Moreover, the resulting square C / / F (cid:15) (cid:15) Stab X C z z t t t t t t (cid:15) (cid:15) D / / Stab F ( X ) D consists of two commutative triangles.Proof. Everything except for the last statement is directly contained in the cited statementsof [Rob15]. For the last statement, we have to open up slightly the box of proofs. In[Rob15, Proposition 2.1] Robalo shows the existence of a left adjoint L ⊗C ⊗ ,X to the inclusionof presentably stable symmetric monoidal ∞ -categories under C where X acts invertibly toall presentably stable symmetric monoidal ∞ -categories under C . The symmetric monoidalfunctor F induces a commutative diagram C F (cid:15) (cid:15) / / L ⊗C ⊗ ,X ( C ) ≃ / / (cid:15) (cid:15) | | ① ① ① ① ① L ⊗C ⊗ ,X (Stab X C ) ≃ / / (cid:15) (cid:15) Stab X C (cid:15) (cid:15) D ≃ / / L ⊗C ⊗ ,X ( D ) ≃ / / L ⊗C ⊗ ,X (Stab F ( X ) D ) ≃ / / Stab F ( X ) D The functor
Stab X C → D is the composite of the inverses of the two upper horizontalarrows and the diagonal arrow. The statement follows from the commutativity of the twotriangles in the diagram, which follows in turn from the triangle identity of adjoints. (cid:3)
This theorem does not directly apply to our situation, as we do not invert just a singleobject, but S V for all (irreducible) finite-dimensional representations V . We neverthelessobtain the following: Corollary C.7.
The functor Σ ∞ : S G ∗ → Sp G refines to a symmetric monoidal functor.Moreover, for any symmetric monoidal left adjoint F : S G ∗ → D into a presentably symmetricmonoidal ∞ -category D such that F ( S V ) is invertible for every irreducible G -representation N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 41 V , there is an essentially unique symmetric monoidal left adjoint F : Sp G → D with anequivalence F Σ ∞ ≃ F .Proof. Choose irreducible orthogonal G -representations V , V , . . . such that every irre-ducible G -representation is isomorphic to exactly one of these. Then Sp G can be identifiedwith the directed colimit S G ∗ → Stab S V S G ∗ → Stab S V Stab S V S G ∗ → · · · in Pr L .The forgetful functor from presentably symmetric monoidal ∞ -categories under S G ∗ to Pr L preserves filtered colimits. Inductively, we see by the previous theorem that all the mappingspaces from Stab S Vn · · · Stab S V S G ∗ to D , computed in presentably symmetric monoidal ∞ -categories under S G ∗ , are contractible, and thus the same is true if we go the colimit. (cid:3) The result we actually use will rather be a version for small categories. For this wedenote by S G, fin ∗ the subcategory of S G ∗ of finite G -spaces , i.e. the subcategory generatedby finite colimits from the orbits G/H + for all closed subgroups H ⊂ G . Moreover, Sp G,ω denotes the ∞ -category of (retracts of) finite G -spectra, i.e. the compact objects in Sp G . Itis easy to see that the functor Σ ∞ : S G ∗ → Sp G restricts to a functor Σ ∞ : S G, fin ∗ → Sp G,ω .Analogously, we can define an ∞ -category S G, fin of unpointed finite G -CW-complexes andobtain a functor S G, fin → S G, fin ∗ by adjoining a disjoint base point. The composition ofthese functors will be denoted by Σ ∞ + . Corollary C.8.
Let D be a presentably symmetric monoidal ∞ -category and F : S G, fin ∗ →D op be a symmetric monoidal functor that preserves finite colimits, sends every object toa dualizable object and every representation sphere to an invertible object. Then F factorsover Σ ∞ : S G, fin ∗ → Sp G,ω to produce a finite colimit preserving functor Sp G,ω → D op .Proof. Denote by D dual the dualizable objects in D . Dualizing yields a symmetric monoidalfunctor F ′ : S G, fin ∗ → D dual , op → D dual ⊂ D . The functor F ′ factors over S G, fin ∗ → Ind( S G, fin ∗ ) ≃ S G ∗ , yielding a colimit-preserving sym-metric monoidal functor S G ∗ → D . Applying the previous corollary, this factors over afunctor Sp G → D , which we can restrict again to Sp G,ω . The compact objects agree withthe dualizable objects and thus dualizing once more yields the result. (cid:3)
We want to compare the symmetric monoidal structure defined on Sp G above with thesymmetric monoidal structure coming from the category Sp GO of orthogonal spectra. Tothat purpose we want to recall how to pass from a symmetric monoidal model categoryto a symmetric monoidal ∞ -category. This was already discussed in [Lur12, Section 4.1.7]and [NS18, Appendix A], but we sketch a more elementary treatment suggest to us byDaniel Schäppi. The starting point is a symmetric monoidal relative category ( C , W , ⊗ ) ,i.e. a symmetric monoidal category with a subcategory of weak of equivalences W such that c ⊗ d → c ′ ⊗ d is a weak equivalence if c → c ′ is. We will assume that W contains all objectsand satisfies -out-of- ; this implies in particular that W contains all isomorphisms. Animportant class of examples is the category of cofibrant objects in a symmetric monoidalmodel category.As sketched in [Seg74, Section 2], every symmetric monoidal category defines a functor Γ op → Cat , sending to the terminal category and satisfying the Segal condition. In caseof a symmetric monoidal relative category, this lifts to a functor Γ op → RelCat into the category of relative categories and weak-equivalence preserving functors. This satisfies theSegal conditions both on underlying categories and categories of weak equivalences. Thelocalization functor L : RelCat → Cat ∞ preserves products [Cis19, Proposition 7.1.13].Thus, composing our functor Γ op → RelCat with L we obtain a functor from Γ op into Cat ∞ satisfying the Segal conditions and thus defining a symmetric monoidal ∞ -category (see[Lur12, Remark 2.4.2.2, Proposition 2.4.2.4]).Going back to equvariant homotopy theory, we consider again the stable model structureof [MM02, Section III.4] on Sp GO . The stable model structure satisfies the pushout-productaxiom with respect to the smash product [MM02, Proposition III.7.5] and its unit S iscofibrant – thus its category Sp G, cof O of cofibrant objects defines a symmetric monoidalrelative category and L Sp G, cof O obtains the structure of a symmetric monoidal ∞ -category. Proposition C.9.
The equivalence in Proposition C.4 refines to a symmetric monoidalequivalence between Sp G with the symmetric monoidal structure from Corollary C.7 and L Sp G, cof O ≃ L Sp GO with the symmetric monoidal structure induced by the smash product.Proof. By definition of the generating cofibrations in [MM02, Section III.2] one observesthat the suspension spectrum functor Σ ∞ : Top G ∗ → Sp GO preserves cofibrant objects, wherewe consider on Top G ∗ the model structure where fibrations and weak equivalences are definedto be those maps that are fibrations and weak equivalences, respectively, on fixed pointsfor all subgroups. As Σ ∞ is strong symmetric monoidal, we see that the resulting functor S G ∗ → L Sp GO is strong symmetric monoidal again (since it defines a natural transformationof functors from Γ op ).As the smash product on Sp GO commutes in each variable with colimits and colimits in L Sp GO can be computed as homotopy colimits in Sp GO , the induces symmetric monoidalstructure on L Sp GO commutes with colimits in both variables. As moreover Σ ∞ sends allrepresentation spheres to invertible objects, the universal property Corollary C.7 yields asymmetric monoidal functor Sp G → L Sp G, cof O . The last part of Theorem C.6 implies thatthe underlying functor of ∞ -categories agrees up to equivalence with the equivalence wehave constructed in Proposition C.4 (once restricted to cofibrant objects). (cid:3) C.4.
The Wirthmüller isomorphism.
As a last point we want to state the Wirthmüllerisomorphism. Note to that purpose that for every compact Lie group G and every closedsubgroup H ⊂ G the restriction functor Sp G → Sp H has two adjoints, which we denote by G + ⊗ H − and Map H ( G + , − ) . Actually, we need these adjoints only on the level of homotopycategories, where they are well-known, e.g. by comparing to orthogonal spectra. Theorem C.10 (Wirthmüller) . Let L = T eH G/H be the tangent G -representation. Thenthere is for every X ∈ Sp H an equivalence G + ⊗ H X → Map H ( G + , S L ⊗ X ) . References include [May03] and [Sch18, Section 3.2]. (While the latter constructs in(3.2.6) only the Wirthmüller map on the level of homotopy groups, it is clear the sameconstruction actually defines it as a transformation of homology theories. As the G -stablehomotopy category is a Brown category [HPS97, Example 1.2.3b, Section 4.1], this can belifted to a map in Sp G .) The special case we need is the following: Corollary C.11.
Let L be the tangent representation of G . Then there is an equivalencebetween Σ ∞ G + ⊗ S − L and the Spanier–Whitehead dual D Σ ∞ G + . This special case can also be seen as special cases of equivariant Atiyah–Duality [LMSM86,Theorem III.5.1].
N EQUIVARIANT TOPOLOGICAL MODULAR FORMS 43
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