The B¨ okstedt construction

The classical definition of topological Hochschild homology as an orthogonal cyclotomic spectrum relies on a specific construction of the smash product of orthogonal spectra, known as the B¨okstedt construction. In fact, the B¨okstedt construction can also be applied to any finite number of orthogonal spectra, and already the case of a single orthogonal spectrum is interesting, where it leads to a fibrant replacement functor.

B¨okstedt’s construction is based on the following category.

Definition III.4.1. We denote byI the category of finite (possibly empty) sets and injective maps.

IfAis any orthogonal spectrum, one has a natural functor fromIto pointed spaces, taking a finite setI∼={1, ..., i}, i>0, to Ai; here, we make use of the action of the sym-metric group Σi⊆O(i) onAito show that this is, as a functor, independent of the choice of isomorphismI∼={1, ..., i}. More precisely, we send I to the valueA(R^I), whereR^I is equipped with the standard inner product and we use the fact that we can canonically evaluateAon every inner product space as in DefinitionII.2.3.

Note that I is not filtered. However, one has B¨okstedt’s important “approximation lemma”. AsI has an initial object, the statement of the following lemma is equivalent to the similar statement in pointed spaces.

Lemma III.4.2. ([32, Lemma 2.2.3]) Let F:Iⁿ!S be a map to the ∞-category of spaces, x∈Iⁿ be an object, and I_xⁿ⊆Iⁿ be the full subcategory of objects supporting a map from x. Assume that F sends all maps inI_xⁿ tok-connected maps of spaces. Then, the map

F(x)−!colimIⁿF is k-connected.

Intuitively, this says that colim_InFbehaves like a filtered colimit if the maps become sufficiently connective.

The advantage ofI over the category corresponding to the ordered set of positive integers is that I has a natural symmetric monoidal structure, given by the disjoint union. This will be critical in the following.

From now on, as in§II.2above, we use specific strictly functorial models of homotopy colimits taken in (pointed) compactly generated weak Hausdorff topological spaces, as defined in AppendixC, and write hocolim to denote these functors.

Definition III.4.3. Consider the symmetric monoidal category Sp^O of orthogonal spectra. B¨okstedt’s construction is the lax symmetric monoidal functor

B: (Sp^O)^⊗_act−!Sp^O

that takes a finite family of orthogonal spectra (X_i)_i∈I to the orthogonal spectrum whose nth term is

withO(n)-action on Sⁿ, and the natural structure maps.

Here Map_∗denotes the space of based maps. Note that this can indeed be promoted to a functor out of (Sp^O)^⊗_act, i.e. if one has a mapf:I!J of finite sets,X_i, Y_j∈Sp^O for i∈Iandj∈J, and mapsV

i∈f⁻¹(j)X_i!Y_j of orthogonal spectra, one gets induced maps hocolim In-deed, under this map of index categories, the spheres

^

i∈I

S^Ii=^

j∈J

S^Ij

agree (up to canonical isomorphism), and there is a natural map

^

i∈I

(X_i)_Ii−!^

j∈J

(Y_j)_Ij,

by definition of the smash product of orthogonal spectra; cf. [85, §1]. Moreover, B is indeed lax symmetric monoidal: Given orthogonal spectra (Xi)i∈I and (Yj)j∈J, as well as integersmandn, one needs to produceO(m)×O(n)-equivariant maps from the smash product of

and

compatible with the structure maps. The desired transformation is given by smashing the maps.

Now, we need the following result of Shipley, which follows directly from [87, Corol-lary 4.2.4].(²⁴) Here we say that a map (Xi)_i∈I!(Yj)_j∈J in (Sp^O)^⊗_act is a stable equiv-alence if it lies over an equivequiv-alence f:I!J in Fin and the maps Xi!Y_f(i) are stable equivalences for alli∈I.

Theorem III.4.4. The functor B takes stable equivalences to stable equivalences.

Proof. Apart from the comparison between orthogonal and symmetric groups we note that Shipley works with simplicial sets instead of topological spaces. As a result the homotopy colimits in DefinitionIII.4.3 has in her setting always automatically the correct homotopy type. We claim that this is also true in our setting. To see this we take all the spaces

for varying n together, so that we obtain a diagram I^I!Sp^O over which we take the homotopy colimit. The result is the same orthogonal spectrum as in DefinitionIII.4.3, since Bousfield–Kan-type homotopy colimits are computed levelwise. The advantage is that the combined homotopy colimit has the correct stable homotopy type by Propo-sition C.11 of Appendix C. This then lets us apply [87, Corollary 4.2.4] to finish the proof.

Now we use Theorem A.7 from Appendix A to see that the symmetric monoidal

∞-category Sp^⊗_act is the Dwyer–Kan localization of (Sp^O)^⊗_act at the stable equivalences as described above. In particular, if we compose B with the functor NSp^O!Sp, then the results of Appendix A and the preceding theorem imply that it factors over a lax symmetric monoidal functor

B: Sp ^⊗_act−!Sp.

(²⁴) Shipley works with simplicial symmetric spectra, but every orthogonal spectrum gives rise to a symmetric spectrum by taking the singular complex and restriction of symmetry groups. Note that in orthogonal spectra, it is not necessary to apply the level-fibrant replacement functorL⁰.

Theorem III.4.5. There is a unique (up to contractible choice) transformation of lax symmetric monoidal functors from⊗: Sp^⊗_act!SptoB: Sp ^⊗_act!Sp. This natural trans-formation is an equivalence.

Proof. Shipley shows in [87, Proposition 4.2.3 and Corollary 4.2.4] that the functor B: Sp ^⊗_act!Sp is not only lax symmetric monoidal, but actually symmetric monoidal.

Note that this is only true on the level of∞-categories and not on the model. Moreover, the composite functor

Sp⊆Sp^⊗_act−−^B!Sp

is equivalent to the identity by [87, Theorem 3.1.6]. Since this functor is exact and lax symmetric monoidal, it follows from [81, Corollary 6.9 (1)] that there is no choice involved in the equivalence. By PropositionIII.3.2, every such functor is equivalent to⊗.

In other words, we have verified that the B¨okstedt construction is a model for the

∞-categorical tensor product of spectra.

As mentioned in the introduction, the main advantage of the B¨okstedt construction over the usual smash product of orthogonal spectra is its behaviour with respect to geometric fixed points. More precisely, for any orthogonal spectrum X, and integer p>1,(²⁵)there is a natural map

Φ^CpB(X, ..., X)−!B(X) of orthogonal spectra, which is moreover a stable equivalence.

In fact, there is a many-variables version of this statement. Namely, for any finite free Cp-set S, with S=S/Cp, consider a family of orthogonal spectra (X¯s)_¯s∈S. Then, there is a natural Cp-action on the orthogonal spectrum B((Xs¯)s∈S), by acting on the index setS, and a natural map

Φ^CpB((Xs¯)_s∈S)−!B((X¯s)_¯s∈S), which is a stable equivalence.

Let us formalize this statement, including all functorialities. As above, let Free_Cp denote the category of finite freeC_p-sets. Then, our source category is

FreeC_p×Fin(Sp^O)^⊗_act.

Here, objects are pairs of a finite free C_p-set S with quotient S=S/C_p, together with orthogonal spectraX¯sindexed by ¯s∈S. We consider two functors

FreeCp×Fin(Sp^O)^⊗_act−!Sp^O.

(²⁵) Although we writephere, it is not yet required to be a prime.

The first is given by projection to (Sp^O)^⊗_actand composition withB: (Sp^O)^⊗_act!Sp^O. We denote this simply by

B: Free_Cp×_Fin(Sp^O)^⊗_act−!Sp^O, (S,(X_s¯)_{s∈¯ S})7−!B((X_¯s)_¯s∈S).

The other functor first applies the functor

FreeCp×Fin(Sp^O)^⊗_act−!((Sp^O)^⊗_act)^BCp,

for some fixed choice of completeCp-universeU. We denote this second functor by B_p: Free_Cp×_Fin(Sp^O)^⊗_act−!Sp^O,

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

Construction III.4.6. There is a natural transformation Bp−!B: Φ^C_U^pB((X¯s)_s∈S)−!B((Xs¯)_¯s∈S)

By construction of hocolim, theCp-fixed points are given by theCp-fixed points over the hocolim overCp-fixed points in the index category. Note that theCp-fixed points ofI^S are given byI^S. Thus, we have to construct a map

But anyCp-equivariant map induces a map between Cp-fixed points. Moreover, the desired map. It is easy to see that it is functorial.

Finally, we can state the main theorem about the B¨okstedt construction.

Theorem III.4.7. For any

(S,(X¯s)_¯s∈S)∈FreeC_p×Fin(Sp^O)^⊗_act, the natural map

Bp(S,(X¯s)_¯s∈S)−!B(S,(X¯s)_{s∈¯ S}): Φ^C_U^pB((Xs¯)_s∈S)−!B((X¯s)_{s∈¯ S}) is a stable equivalence.

Proof. This is essentially due to Hesselholt and Madsen, [47, Proposition 2.5]. Let us sketch the argument. First, one reduces to the case that allX_s¯are bounded below, in the sense that there is some integer m such that π_iX_¯s,n=0 for i<n−m. Indeed, any orthogonal spectrum X can be written as a filtered colimit of orthogonal spectra X^(m)which are bounded below such that the mapsX_i^(m)!X_i are homeomorphisms for i6m. Indeed, one can consider a truncated version of orthogonal spectra keeping track only of the spaces X_i for i6m; then, the restriction functor has a left adjoint, whose composite with the restriction functor is the identity, and which takes values in bounded

below orthogonal spectra.(²⁶) Now, the statement for general X follows by passage to the colimit of the statement for theX^(m)’s which are bounded below.

Now we have to estimate the connectivity of the map Map_∗

This shows that the homotopy fiber of the first map can be identified with Map_∗

Now, as in the proof of [47, Proposition 2.5], we use that the connectivity of Map_∗(A, B)^Cp is at least the minimum of conn(B^H)−dim(A^H), whereH runs over all subgroups ofC_p,

is a point, while the connectivity of

(²⁶) In the language of orthogonal spectra as pointed continuous functorsO!Top_∗from the cat-egory of finite-dimensional inner product spaces with Thom spaces as mapping spaces, these functors correspond to restriction to the full subcategoryO₆m of spaces of dimension 6m, and the left Kan extension fromO_6mtoO.

is bounded, while the connectivity of

More precisely, let%C_p denote the regular representation of Cp. Then,

dim

In this case, the difference is given by

n+dimV^H−dim%^H_C

Choosing (I_s¯)∈I^Sin the approximation lemma (LemmaIII.4.2) such that n+X

s∈¯ S

(i¯s−m)>nV,

we see that the connectivity of the map

hocolim equivalence in the homotopy colimit overV.