Genuine cyclotomic spectra

In this section we give the definition of the ∞-category of genuine cyclotomic spectra and genuine p-cyclotomic spectra. We start with a discussion of genuine p-cyclotomic spectra.

Definition II.3.1. A genuine p-cyclotomic spectrum is a genuineCp^∞-spectrum X together with an equivalence Φp: Φ^CpX−^'!X inCp^∞Sp, where

Φ^CpX∈(Cp^∞/Cp)Sp'Cp^∞Sp

via thepth power mapCp^∞/Cp∼=Cp^∞. The∞-category of genuinep-cyclotomic spectra is the equalizer

Cyc Sp^gen_p = Eq

Cp^∞Sp ^id //

Φ^Cp

//Cp^∞Sp

.

Every genuine p-cyclotomic spectrum (X,Φp) has an associatedp-cyclotomic spec-trum, in the sense of Definition II.1.6(ii). Indeed, there is a functorCp^∞Sp!Sp^BCp∞, and one can compose the inverse mapX!Φ^CpX with the geometric fixed points of the Borel completion Φ^CpX!Φ^CpBC_p∞(X); the corresponding map of underlying spectra withCp^∞-action is aCp^∞-equivariant mapX!X^tCp, where theCp^∞-action on the right is the residualCp^∞/Cp-action via thepth power mapCp^∞/Cp∼=Cp^∞.

Proposition II.3.2. The assignment described above defines a functor Cyc Sp^gen_p −!Cyc Spp.

Proof. As the equalizer is a full subcategory of the lax equalizer, it suffices to con-struct a functor

LEq

Cp^∞Sp ^id //

Φ^Cp

//Cp^∞Sp

−!LEq

Sp^{BCp∞ id} //

−^tCp

//Sp^BCp∞

.

But there is the natural functorCp^∞Sp!Sp^BCp∞ which commutes with the first functor id in the lax equalizer; for the second functor, there is a natural transformation between the composition

Cp^∞Sp−−−−^ΦCp!Cp^∞Sp−!Sp^BCp∞ and the composition

Cp^∞Sp−!Sp^BCp∞−−−−^−tCp!Sp^BCp∞,

by passing to the underlying spectrum in the natural transformation Φ^Cp!Φ^CpBC_p∞. For the identification of −^tCp with the underlying spectrum of Φ^CpBC_p∞, cf. Proposi-tionII.2.13.

In this situation, one always gets an induced functor of lax equalizers, by looking at the definition (DefinitionII.1.4).

Next, we introduce the category of genuine cyclotomic spectra as described in [47,

§2]. For the definition, we need to fix a complete T-universeU; more precisely, we fix U=M

k∈Z i>1

Ck,i,

whereTacts onCk,ivia thek-th power of the embeddingT,!C^×. For this universe, we have an identification

U^Cn=M

k∈nZ i>1

Ck,i,

which is a representation ofT/Cn. Now, there is a natural isomorphism U^Cn−^∼=!U

which sends the summand Ck,i⊆U^Cn for k∈nZ and i>1 identically to the summand Ck/n,i⊆U. This isomorphism is equivariant for the T/C_n∼=T-action given by the nth power. We get functors

Φ^C_Uⁿ:TSp^O−!(T/Cn)Sp^O'TSp^O such that there are natural coherentF-equivalences

Φ^C_U^mΦ^C_Uⁿ−!Φ^C_U^mn,

as in PropositionII.2.12. By inverting F-equivalences, one gets an action of the multi-plicative monoidN>0 of positive integers on the∞-categoryTSp_F.

Definition II.3.3. The ∞-category of genuine cyclotomic spectra is given by the homotopy fixed points ofN>0 on the∞-category ofF-genuineT-equivariant spectra,

Cyc Sp^gen= (TSp_F)^hN>0.

Roughly, the objects of Cyc Sp^gen are objects X∈TSp_F together with equivalences Φ_n:X'Φ^CnX for alln>1 which are homotopy-coherently commutative.

Proposition II.3.4. There is a natural functor Cyc Sp^gen−!Cyc Sp = Sp^BT×Q

p∈PSp^BCp∞

Y

p∈P

Cyc Spp.

Proof. For any p, we have a natural functor Cyc Sp^gen!Cyc Sp^gen_p given by the natural functorTSpF!Cp^∞Sp, remembering only Φp. The induced functor

Cyc Sp^gen!Y

p∈P

Sp^BCp∞

lifts to a functor Cyc Sp^gen!Sp^BTby looking at the underlying spectrum withT-action ofX∈TSp_F.

Remark II.3.5. The functor Cyc Sp^gen!Cyc Sp a priori seems to lose a lot of in-formation: For example, it entirely forgets all non-trivial fixed point spectra and all coherence isomorphisms between the different Φ_n.

In [47, Definition 2.2], the definition of (genuine) cyclotomic spectra is different in that one takes the homotopy fixed points ofN>0 on the category TSp^O, i.e. before inverting weak equivalences (or rather lax fixed points where one requires the structure maps to be weak equivalences). Let us call these objects orthogonal cyclotomic spectra.

Definition II.3.6. An orthogonal cyclotomic spectrum is an object X∈TSp^O to-gether withF-equivalences of orthogonalT-spectra Φ_n: Φ^C_UⁿX!XinTSp^O'(T/C_n)Sp^O for alln>1, such that, for all m, n>1, the diagram

Φ^C_Uⁿ(Φ^C_U^mX) ^' //

Φ^Cn_U (Φm)

Φ^C_U^mnX

Φ_mn

Φ^Cn_U X ^Φn //X

commutes. A map f:X!Y of orthogonal cyclotomic spectra is an equivalence if it is anF-equivalence of the underlying object inTSp^O. Let Cyc Sp^O denote the category of orthogonal cyclotomic spectra.

It may appear more natural to take the morphisms Φ^C_UⁿX!X in the other direc-tion, but the point-set definition of topological Hochschild homology as an orthogonal cyclotomic spectrum actually requires this direction; cf. §III.5 below. We note that Hesselholt–Madsen ask that Φn be an equivalence of orthogonalT-spectra (not merely anF-equivalence), i.e. it also induces an equivalence on the genuineT-fixed points. Also, they ask thatX be a T-Ω-spectrum, where we allow X to be merely a (pre)spectrum.

Our definition follows the conventions of Blumberg–Mandell, [19, Definition 4.7].(²¹) In [19,§5], Blumberg–Mandell construct a “model-∗-category” structure on the category of orthogonal cyclotomic spectra. It is unfortunately not a model category, as the geometric fixed point functor does not commute with all colimits; this is however only a feature of the point-set model. It follows from a result of Barwick–Kan [12] that if one inverts weak equivalences in the model-∗-category of orthogonal cyclotomic spectra, one gets the

∞-category Cyc Sp^gen.

Theorem II.3.7. The morphism N(Cyc Sp^O)!Cyc Sp^gen is the universal functor of ∞-categories inverting the equivalences of orthogonal cyclotomic spectra.

Proof. This follows from [11, Lemma 3.24].

One can also make a similar discussion for p-cyclotomic spectra; however, in this case, our DefinitionII.1.6(ii) differs from previous definitions in that we only require an action of the subgroupCp^∞ ofT. If one changes our definition to aT-action, one could compare DefinitionII.3.1with the definitions ofp-cyclotomic spectra in [47] and [19], as in TheoremII.3.7.

Finally, we can state our main theorem. It uses spectra TC^gen(X, p) for a genuine p-cyclotomic spectrum X and TC^gen(X) for a genuine cyclotomic spectrum X, whose definition we recall in§II.4below. Recall that, by TC(X) (resp. TC(X, p)), we denote the functors defined in DefinitionII.1.8.

Theorem II.3.8. (i) Let X∈Cyc Sp^gen be a genuine cyclotomic spectrum whose underlying spectrum is bounded below. Then,there is an equivalence of spectra

TC^gen(X)'TC(X).

(ii) Let X∈Cyc Sp^gen_p be a genuine p-cyclotomic spectrum whose underlying spec-trum is bounded below. Then, there is an equivalence of spectra

TC^gen(X, p)'TC(X, p).

(²¹) Their commutative diagram in [19, Definition 4.7] looks different from ours, and does not seem to ask for a relation between Φmnand Φm, Φn; we believe ours is the correct one, following [47, Definition 2.2].

Moreover, the forgetful functors Cyc Sp^gen!Cyc Sp and Cyc Sp^gen_p !Cyc Spp are equivalences of ∞-categories when restricted to the respective subcategories of bounded below spectra.