Equivalence of ∞-categories of cyclotomic spectra

We start this section by proving that, for a fixed primep, our∞-category ofp-cyclotomic spectra is equivalent to the∞-category of genuinep-cyclotomic spectra, when restricted to bounded below spectra. The “forgetful” functor

Cyc Sp^gen_p −!Cyc Spp (3)

from Proposition II.3.2 restricts to a functor of bounded below objects on both sides.

By the adjoint functor theorem (and the results about presentability and colimits of the last section), the functor (3) admits a right adjoint functor. We will try to under-stand the right adjoint well enough to prove that this functor induces an equivalence of subcategories of bounded below cyclotomic spectra.

To this end, we factor the forgetful functor (3) as

Cyc Sp^gen_p = Fix_ΦCp(Cp^∞Sp) ^ι //CoAlg_ΦCp(Cp^∞Sp)

U

CoAlg₍₋₎tCp(Sp^BCp∞) = Cyc Spp.

Here, the first functorιis the inclusion and the second functor U takes the underlying naive spectrum. See also the constructionII.3.2 of the forgetful functor. Both functors in this diagram admit right adjoints by the adjoint functor theorem, the right adjoint R of the first functor was discussed in the last section, and the right adjoint B of the second functorU will be discussed now.

Recall the notion of Borel complete spectra and the Borel completion from Theo-remII.2.7. The following lemma is the non-formal input in our proof, and a consequence of the Tate orbit lemma (LemmaI.2.1).

Lemma II.6.1. Let X be a Borel complete object in C_p∞Spwhose underlying spec-trum is bounded below. Then, the object Φ^CpX∈Cp^∞Sp is also Borel complete. In particular,the canonical map

Φ^CpB_Cp∞Y−−^'!B_Cp∞(Y^tCp)

is an equivalence for every bounded below spectrum Y with Cp^∞-action.

Proof. AsX is bounded below, we get from PropositionII.4.6 a pullback square X^Cpn //

(Φ^CpX)^Cpn−1

X^hCpn //(X^tCp)^hCpn−1,

where we want to remind the reader of the discussion following PropositionII.2.13 for the definition of the right-hand map. To show that Φ^CpX is Borel complete, we have to verify that the right-hand map is an equivalence. This now follows, since the left-hand side is an equivalence by assumption.

Lemma II.6.2. Let (X, ϕp)∈Cyc Spp be a bounded below cyclotomic spectum. Then, the right adjointB: Cyc Spp!CoAlg_ΦCp(Cp^∞Sp)of the forgetful functor applied to(X, ϕp) is given by the Borel complete spectrumBC_p∞X∈Cp^∞Spwith the coalgebra structure map

B_Cp∞ϕ_p:B_Cp∞X−!B_Cp∞(X^tCp)'Φ^Cp(B_Cp∞X).

The counit U B(X, ϕ)!(X, ϕ)is an equivalence.

Proof. We have to check the universal mapping property. Thus, let (Y,Φp) be a Φ^Cp-coalgebra. Then, the mapping space in the ∞-category CoAlg_ΦCp(Cp^∞Sp) from (Y,Φp) to (BC_p∞X, BC_p∞ϕp) is given by the equalizer of the diagram

Map_C

p∞Sp(Y, B_Cp∞X)

(Φ_p)∗Φ^Cp

//

(B_Cp∞ϕp)∗

//Map_C

p∞Sp(Y, B_Cp∞X^tCp),

as we can see from Proposition II.1.5(ii). But, since the Borel functor BC_p∞ is right adjoint to the forgetful functorCp^∞Sp!Sp^BCp∞, this equalizer can be rewritten as the equalizer of

Map_SpBCp∞(Y, X)

(ϕY,p)∗(−^tCp) //

(ϕp)∗

//Map_SpBCp∞(Y, X^tCp),

where we have written (Y, ϕY,p) for the corresponding naivep-cyclotomic spectrum. This last equalizer is again, by PropositionII.1.5(ii), the mapping space between the cyclo-tomic spectra (Y, ϕY,p) and (X, ϕp) in the∞-category Cyc Spp. This shows the claim.

TheoremII.6.3. The forgetful functor Cyc Sp^gen_p !Cyc Spp induces an equivalence between the subcategories of those objects whose underlying non-equivariant spectra are bounded below.

Proof. We have to show that the composition Uι: Cyc Sp^gen_p −!Cyc Sp_p

is an equivalence of∞-categories when restricted to full subcategories of bounded below objects. As we have argued before both functorsU as well as ι have right adjoints B and R, and thus the composition has a right adjoint. We show that the unit and the counit of the adjunction are equivalences. First, we observe that the functorUιreflects equivalences. This follows since equivalences of genuine p-cyclotomic spectra can be detected on underlying spectra, as they can be detected on geometric fixed points.

Thus, it is sufficient to check that the counit of the adjunction is an equivalence, i.e. the map

U ιRB(X)−!X

is an equivalence of spectra for every bounded below spectrumX∈Cyc Spp. This follows from LemmaII.6.2and TheoremII.5.6.

Remark II.6.4. With the same proof one also gets an equivalence between genuine and naive∞-categories ofp-cyclotomic spectra which support aT-action (as opposed to

aCp^∞-action); see RemarkII.1.3. This variant of the genuinep-cyclotomic∞-category is in fact equivalent to the underlying∞-category of the model-∗-category ofp-cyclotomic spectra considered by Blumberg–Mandell, as shown by Barwick–Glasman [11] (cf. Theo-remII.3.7above). We have decided for the slightly different∞-category ofp-cyclotomic spectra, since theCp^∞-action is sufficient to get TC(−, p), and this avoids the completion issues that show up in the work of Blumberg–Mandell [19].

Remark II.6.5. With the same methods (but much easier, since Lemma II.6.1 is almost a tautology in this situation) we also get an unstable statement, namely that a genuineC_p^∞-spaceX with an equivalence of C_p^∞-spaces X−^∼!X^Cp is essentially the same as a naiveC_p^∞-space together with a C_p^∞-equivariant mapX!X^hCp.

Now, we prove the global analogue of the statement above, namely that our ∞-category of cyclotomic spectra is equivalent to the ∞-category of genuine cyclotomic spectra when restricted to bounded below spectra. Similar to the p-primary case we consider the subcategories of Cyc Sp^gen and Cyc Sp of objects whose underlying spectra are bounded below. The “forgetful” functor

Cyc Sp^gen−!Cyc Sp (4)

from PropositionII.3.4takes bounded below objects to bounded below objects. By the adjoint functor theorem and the results of the last section the forgetful functor (4) admits a right adjoint functor. We factor the functor (4) as

Cyc Sp^gen= Fix_(ΦCp)p∈P(TSp_F) ^ι //CoAlg_(ΦCp)p∈P(TSp_F)

U

CoAlg₍₋tCp)p∈P(Sp^BT)'Cyc Sp.

Here, the first functorιis the inclusion and the second functor U takes the underlying naive spectrum. The equivalence

CoAlg₍₋tCp)p∈P(Sp^BT)'Cyc Sp

follows from LemmaII.5.8. We now want to understand the right adjoint functor of the functorU.

LemmaII.6.6. LetX be a Borel complete object in TSp_F whose underlying spectrum is bounded below. Then, for every prime p, the spectrum Φ^CpX∈TSp_F is also Borel complete.

Proof. We know that the underlying spectrum withT-action of Φ^CpX is given by X^tCp∈Sp^BT. As in TheoremII.2.7, we consider the associated Borel complete spectrum B(X^tCp)∈TSp_F. There is a morphism Φ^CpX!B(X^tCp) as the unit of the adjunction.

We need to show that this map is an equivalence, i.e. that it is an equivalence on all geometric fixed points for finite subgroupsH⊆T. ForH being a cyclicp-group this has already been done in Lemma II.6.1. Thus, we may restrict our attention to subgroups H that haveq-torsion for some primeq6=p. In this case, it follows from the next lemma that Φ^HΦ^CpX'Φ^HfX'0, whereHe={h∈T:h^p∈H}.

If X is an algebra object, it follows from Corollary II.2.8 that the map Φ^CpX! B(X^tCp) is a map of algebras. Since geometric fixed points are lax symmetric monoidal, we get that also the map

Φ^HΦ^CpX−!Φ^HB(X^tCp)

is a map of algebras. Since the source is zero, this also implies that the target is zero.

Now, since every Borel complete spectrumXis a module over the Borel complete sphere, it follows that, for all Borel complete spectraX, the spectrum Φ^HB(X^tCp) is zero, and thus the claim.

LemmaII.6.7. Let X be a Borel completeG-spectrum for some finite groupGwhich is not a p-group for some prime p. Then, Φ^G(X)'0.

Proof. AsX is a module over the Borel complete sphere spectrum, we may assume thatX is the Borel complete sphere spectrum. In this case, Φ^GX is anE∞-algebra, and it suffices to see that 1=0∈π0Φ^GX. There is a map of E∞-algebrasS^hG=X^G!Φ^GX, and there are norm maps X^H!X^G for all proper subgroups H G whose composite X^H!X^G!Φ^GX is homotopic to zero. Let I⊆π0S^hG be the ideal generated by the images of the norm mapsπ0X^H!π0X^G. It suffices to see that I contains a unit. For this, note that there is a natural surjective map

π0S^hG−!π0Z^hG=Z

whose kernel lies in the Jacobson radical. More precisely, we can write S^hG= lim

n (τ₆nS)^hG, and the map

π0(τ₆nS)^hG−!Z

is surjective with nilpotent and finite kernel. By finiteness of allπ₁(τ_6nS)^hG, we get π₀S^hG= lim

n π₀(τ_6nS)^hG.

Now, if an elementα∈π0S^hGmaps to a unit inZ, then it maps to a unit in allπ0(τ₆nS)^hG, and therefore is a unit inπ0S^hG.

Now, note that the transfer mapsX^H=S^hH!X^G=S^hGsit in commutative diagrams π₀S^hH //

π₀S^hG

Z

[G:H] //Z.

If G is not a p-group, then (by the existence of p-Sylow subgroups) the ideal of Z generated by [G:H] for proper subgroups H of G is given by Z. By the above, this implies thatIcontains a unit, as desired.

Recall that the functor

U: CoAlg_(ΦCp)p∈P(TSp_F)−!CoAlg₍₋tCp)p∈P(Sp^BT)'Cyc Sp has a right adjoint. The next lemma is analogous to LemmaII.6.2.

Lemma II.6.8. Let X∈Cyc Sp be a bounded below cyclotomic spectrum. Then, the right adjoint B: Cyc Sp!CoAlg_(ΦCp)p∈P(TSp_F)applied toX has underlying object given by the Borel complete spectrum B_TX∈TSp_F, and the counit U B(X, ϕ)!(X, ϕ) is an equivalence.

Proof. Consider the Borel complete spectrumB_TX with the maps B_TX−!B_T(X^tCp)'Φ^CpB_TX,

where we have used LemmaII.6.6. We also note that we have Φ^CpΦ^CqB_TX'0 for distinct primespandq, by LemmaII.6.7. Thus, by LemmaII.5.8(i), this equipsB_TX with the structure of an object in CoAlg_(ΦCp)_p∈P(TSp_F). Moreover, for any other object Y in CoAlg_(ΦCp)_p∈P(TSp_F), we get, by LemmaII.5.8(ii), the equivalence of mapping spaces

Map_CoAlg

(ΦCp)p∈P

(TSpF)(Y, B_TX) 'Eq Map_TSpF(Y, B_TX) ////Q

p∈PMap_TSpF(Y, B_T(X^tCp)) 'Eq Map_SpBT(U Y, X) ////Q

p∈PMap_SpBT(U Y, X^tCp) 'Map_{Cyc Sp}(U Y, X).

Theorem II.6.9. The forgetful functor Cyc Sp^gen!Cyc Sp induces an equivalence between the subcategories of those objects whose underlying non-equivariant spectra are bounded below.

Proof. As in the proof of Theorem II.6.3, this follows from Theorem II.5.13 and LemmaII.6.8.

Remark II.6.10. From TheoremII.6.9 and the definition TC(X)=map_{Cyc Sp}(S, X), we deduce that TC(X)'map_{Cyc Sp}gen(S, X) for a genuine cyclotomic spectrumXbounded below. We also know that this is equivalent to Goodwillie’s integral TC, by Theo-remII.4.11. Since the bounded-below part of the∞-category Cyc Sp^genis equivalent to the bounded-below part of the∞-category underlying the model-∗-category of Blumberg and Mandell, by TheoremII.3.7we deduce that the mapping spectrum in their category is also equivalent to Goodwillie’s integral TC. Blumberg and Mandell have only shown this equivalence after p-completion and not integrally. In this sense, our result refines their result. It would be interesting to see a proof of this fact in the language of [19]. We have been informed that such a discussion will be given in forthcoming work of Calvin Woo.