Comparison

Φ^C_U^pqTHH(R)

Φ_pq

Φ^C_U^qTHH(R) ^Φq //THH(R),

one sees that also Φ^C_U^q(Φp) is an equivalence of underlying orthogonal spectra. This implies that Φp is anF-equivalence, as desired.

III.6. Comparison

In this section, we compare the constructions of§III.2and§III.5. A similar comparison between the B¨okstedt model for THH and the cyclic bar construction model for THH (in the model category of [6]) will also appear in [29].

We start with an associative orthogonal ring spectrumR∈Alg(Sp^O), regarded as a functor

R^⊗: Ass^⊗−!(Sp^O)^⊗ of operads. Let

A^⊗:N(Ass^⊗)−!Sp^⊗ denote the associatedE1-algebraA∈Alg_E1(Sp).

First, we have to compare the cyclic spectra computing THH. Recall that in§III.2, it is given by the composite

N(Λ^op)−!N(Ass^⊗_act) ^A

⊗

−−−!Sp^⊗_act−−^⊗!Sp, and in§III.5, it is given by the composite

Λ^op−!Ass^⊗_act ^R

⊗

−−−!(Sp^O)^⊗_act−−−^B!Sp^O,

whereB denotes the B¨okstedt construction. Here, the comparison of these cyclic objects is an immediate consequence of TheoremIII.4.5. To compare the geometric realizations, we combine LemmaIII.5.2and CorollaryB.16to obtain the following result.

Theorem III.6.1. Let R be an orthogonal ring spectrum that is levelwise well-pointed and such that the zeroth component of the unitS⁰!R₀is an h-cofibration. Then, there is a natural (naive)T-equivariant equivalence between the classical construction of

THH(R)using B¨okstedt’s construction as in DefinitionIII.5.1and the ∞-categorical ver-sion THH(A)using the cyclic bar construction as in Definition III.2.3of the associated E1-ring spectrum A. More precisely, there is a commutative square of lax symmetric monoidal functors in which the vertical functors are Dwyer–Kan localizations and

Alg(Sp^O)well⊆Alg(Sp^O)

denotes the subcategory of all orthogonal ring spectra which satisfy the above

well-pointed-ness condition.

It remains to identify the Frobenius maps, so fix a prime p. In §III.5, the Frobe-nius map THH(R)!THH(R)^tCp is the composition of the inverse of the equivalence Φ^C_U^pTHH(R)!THH(R) and the natural map Φ^C_U^pTHH(R)!THH(R)^tCp. Recall that, for the construction of the map Φ^C_U^pTHH(R)!THH(R), the source Φ^C_U^pTHH(R) was modelled by theBCp-equivariant functor

Λ^op_p −!FreeC_p×FinAss^⊗_act ^R

Here and in the following, we lie a little bit, as there are actuallyptimes as many arrows as are drawn (as we are dealing with a Λ^op_p -object), which are however permuted by the BCp-equivariance.

Moreover, the target THH(R) is modelled by theBCp-equivariant functor Λ^op_p −!FreeC_p×FinAss^⊗_act ^R

The natural transformation then comes from the natural transformationB_p^T!BofBCp -equivariant functors

FreeC_p×Fin(Sp^O)^⊗_act−!TSp^O in ConstructionIII.4.6(which isT-equivariant).

Similarly, in §III.2, the Frobenius map THH(A)!THH(A)^tCp was constructed by modelling THH(A) by theBCp-equivariant functor

N(Λ^op_p )−!N(FreeC_p)×_N(Fin)N(Ass^⊗_act) ^A

Moreover, THH(A)^tCp admits a natural map from the realization (in the sense of Propo-sitionB.22) of theBCp-equivariant functor

N(Λ^op_p )−!N(FreeC_p)×_N(Fin)N(Ass^⊗_act) ^A Now, the natural transformation comes from the natural transformationI!Te_p ofBC_p -equivariant functors

N(FreeC_p)×_N(Fin)Sp^⊗_act−!Sp, given by CorollaryIII.3.8.

We compare all different constructions by invoking LemmaIII.3.7. For this, we need to rewrite everything in terms of lax symmetric monoidal functors on

N(Free_Cp)×_N(Fin)Sp^⊗_act.

Unfortunately, one of the functors, namelyB^T_p, is not lax symmetric monoidal, because our version Φ^C_U^p:C_pSp^O!TSp^Oof the geometric fixed points functor is not lax symmetric monoidal.

For this reason, we need to recall a few things about different models of the geometric fixed points functor Φ^Cp:C_pSp!Sp. First, as a consequence of PropositionII.2.14, we can upgrade the functoriality of the geometric fixed points functor Φ^Cp:C_pSp!Sp.

Construction III.6.2. There is a naturalBCp-equivariant symmetric monoidal func-tor

Φ^C_∞^p:CpSp−!Sp.

Proof. We define Φ^C∞^pas the composition of theBC_p-equivariant smashing localiza-tionCpSp!CpSp_>C_p and theBCp-equivariant lax symmetric monoidal functor

−^Cp:C_pSp_>Cp⊆C_pSp−!Sp.

We need to compare this with the models Φ^Cp and Φ^C_U^p used above. Note that Φ^Cp is lax symmetric monoidal but notBCp-equivariant, while Φ^C_U^p:CpSp^O!TSp^O isBCp -equivariant, but not lax symmetric monoidal. Here, U denotes our fixedT-universe, as usual. First, we define a related functor that is bothBCp-equivariant and lax symmetric monoidal.

ConstructionIII.6.3. We construct aBCp-equivariant lax symmetric monoidal func-tor

Φ^C_U,lax^p :C_pSp^O−!TSp^O

as follows. It takes aCp-equivariant orthogonal spectrumX to the orthogonal spectrum Φ^C_U,lax^p (X) whosenth space is given by

where I denotes B¨okstedt’s category of finite sets with injective maps. The structure maps are the evident maps, and there is aT-action given by acting on theVi’s diagonally.

The functor is BCp-equivariant, as the Cp-action on X agrees with the Cp⊆T-action through theVi’s on the fixed points for the diagonal action. The lax symmetric monoidal structure is induced by the map

(I,(Vi)∈ U^I),(J,(Wj)∈ U^J)7−!(ItJ,(Vi, Wj)∈ U^ItJ) on index categories, and the natural maps

X

Construction III.6.4. There is a naturalBCp-equivariant transformation from Φ^C_U^p:C_pSp^O−!TSp^O

to

Φ^C_U,lax^p :C_pSp^O−!TSp^O.

Indeed, this comes from the inclusion of the terms given by a fixed 1-element setI∈I.

Also, this natural transformation induces stable equivalences Φ^C_U^p(X)!Φ^C_U,lax^p (X) of the underlying orthogonal spectra for allX∈CpSp^O. Indeed, the homotopy colimit in the definition of Φ^C_U,lax^p (X) is over homotopy equivalences, so this follows from LemmaIII.4.2.

Construction III.6.5. There is a natural lax symmetric monoidal transformation from

Φ^Cp:CpSp^O−!Sp^O⊆TSp^O, (Xn)n7−!(X(%C_p⊗Rⁿ)^Cp)n, to

Φ^C_U,lax^p :C_pSp^O−!TSp^O.

Indeed, this comes from the inclusion of the term given byI={1, ..., n}∈I and V1=...=Vn=%C_p/R

into the homotopy colimit.

Moreover, this natural transformation induces stable equivalences Φ^Cp(X)!Φ^C_U,lax^p (X)

of the underlying orthogonal spectra for allX∈CpSp^O.

Finally, we can state the desired compatibility between the point-set constructions and the abstract definition.

Proposition III.6.6. The diagram C_pSp^{O Φ}

Cp U,lax //

TSp^O

%%

C_pSp ^Φ

Cp

∞ //Sp //Sp^BT of BC_p-equivariant lax symmetric monoidal functors commutes.

We note that the proof is strictly speaking a construction.

Proof. We first note that, by the results of AppendixA, the composition C_pSp^{O Φ}

Cp U,lax

−−−−−!TSp^O−!Sp^BT

factors uniquely over aBCp-equivariant lax symmetric monoidal functor F:CpSp−!Sp^BT.

This functor has the property that it factors over the localization CpSp_>C_p (as the underlying functor is equivalent to the usual Φ^Cp). Thus, it remains to compare the twoBCp-equivariant lax symmetric monoidal functors

Φ^C_∞^p=−^Cp, F:CpSp_>C_p−!Sp^BT.

But there is a naturalBCp-equivariant lax symmetric monoidal transformation

−^Cp−!Φ^C_U,lax^p :C_pSp^O_Ω−!TSp^O by inclusion ofX(Rⁿ)^Cp=(X(Rⁿ)∧S^Li∈IVi)^Cp intoX(Rⁿ⊕L

i∈IV_i)^Cp; here, C_pSp^O_Ω⊆C_pSp^O

denotes the Cp-Ω-spectra. Passing to the associated BCp-equivariant lax symmetric monoidal transformation CpSp!Sp^BT and restricting to CpSp_>C_p, we get the desired result.

Now, we get a number of relatedBCp-equivariant lax symmetric monoidal functors.

(i) The functor

B_p,lax^T : Free_Cp×Fin(Sp^O)^⊗_act−!TSp^O,

(S,(X_s¯)_{s∈¯ S})7−!Φ^C_U,lax^p B((X_¯s)_s∈S).

(ii) The functor

Bp,∞: FreeC_p×Fin(Sp^O)^⊗_act−!Sp,

(S,(X¯s)_¯s∈S)7−!Φ^C_∞^pB((X¯s)s∈S).

(iii) The functor

B: FreeC_p×Fin(Sp^O)^⊗_act−!Sp^O, (S,(Xs¯)_{s∈¯ S})7−!B((X¯s)_¯s∈S).

(iv) The functor

I:N(FreeC_p)×N(Fin)Sp^⊗_act−!Sp, (S,(X¯s)_{s∈¯ S})7−!O

¯s∈S

X¯s.

(v) The functor

Te_p:N(Free_Cp)×_N(Fin)Sp^⊗_act−!Sp, (S,(X¯s)_{s∈¯ S})7−!

O

s∈S

Xs¯

^tCp

.

Moreover, they are related by a number ofBC_p-equivariant lax symmetric monoidal natural transformations.

(T1) A natural transformation B_p,lax^T !B (after composing B with Sp^O!TSp^O).

This comes from the obvious adaptation of ConstructionIII.4.6.

(T2) A natural transformationB_p,lax^T !B_p,∞(after composingB_p,lax^T withTSp^O! Sp^BT andB_p,∞with Sp!Sp^BT). This comes from PropositionIII.6.6.

(T3) A natural transformation I!B (after composing B with Sp^O!Sp). This comes from TheoremIII.4.5.

(T4) A natural transformationI!Tepfrom Corollary III.3.8.

(T5) A natural transformation B_p,∞!Tep. This comes from Theorem III.4.5 and the naturalBCp-equivariant lax symmetric monoidal transformation

Φ^C_∞^p−!−^tCp:C_pSp−!Sp,

which arises by writing−^tCp as the composition of the (BCp-equivariant, lax symmetric monoidal) Borel completion functor CpSp!CpSpB and Φ^C∞^p. Here, we use that the Borel completion functor is a localization (cf. Theorem II.2.7), and thus automatically BCp-equivariant and lax symmetric monoidal.

The main theorem relating all these functors and natural transformations is now given by the following.

TheoremIII.6.7. The compositions of B_p,lax^T ,Bp,∞,B,IandTepwith the respective

‘forgetful’ BCp-equivariant lax symmetric monoidal functor toSp^BT factor uniquely over BCp-equivariant lax symmetric monoidal and partially exact functors

N(FreeC_p)×_N(Fin)Sp^⊗_act−!Sp^BT.

Moreover, (the image of)Iis initial in the∞-category of BC_p-equivariant lax symmetric monoidal and partially exact functors N(Free_Cp)×_N(Fin)Sp^⊗_act!Sp^BT.

Proof. The factorization follows from the results of AppendixA. The initiality ofI follows from LemmaIII.3.7.

Let us quickly explain how this leads to the desired comparison. We want to show that the natural diagram

commutes. Here, as explained above, all objects arise from certainBCp-equivariant func-torsN(Λ^op_p )!Sp^BT by the mechanism of Proposition B.22(except THH(A)^tCp, which however admits a natural map from such an object, over which everything else factors).

These BCp-equivariant functors N(Λ^op_p )!Sp^BT are obtained by composing the BCp variant of) THH(A)^tCp. In this translation, the upper horizontal map comes from the map B_p^T!B which is the composition of B_p^T!B_p,lax^T and B_p,lax^T !B from (T1) above.

The left vertical map comes from the mapI!B in (T3) above. The right vertical map comes from the composition ofB^T_p!B_p,lax^T ,B_p,lax^T !Bp,∞in (T2) andBp,∞!Tepin (T5) above. Finally, the lower horizontal map comes from the mapI!Tep in (T4) above. By TheoremIII.6.7, the resulting diagram

ofBC_p-equivariant functors commutes. In fact, the outer part of the diagram commutes asBC_p-equivariant lax symmetric monoidal functors as I is initial by TheoremIII.6.7, and the small upper triangle commutesBC_p-equivariantly by construction (on the point-set model). Thus, we have finally proven the following comparison.

Corollary III.6.8. Let R be an orthogonal ring spectrum which is levelwise well-pointed and such that the unit S⁰!R0 is an h-cofibration. Then, under the equiva-lence of Theorem III.6.1the two constructions of cyclotomic structure maps THH(A)! THH(A)^tCp, where A is the object of Alg_E1(Sp) associated with the 1-categorical object R∈Alg(Sp^O),are equivalent, functorially in R.

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