Functoriality of the Tate diagonal

As explained in the last section, we need to equip the Tate diagonal ∆p with stronger functoriality, and we also need a version in many variables. We start by noting that, by TheoremI.3.1, the functor

Tp: Sp−!Sp,

X7−!(X⊗...⊗X)^tCp,

acquires a canonical lax symmetric monoidal structure. We can now also make the natural transformation ∆p:idSp!Tp into a lax symmetric monoidal transformation.

Proposition III.3.1. There is a unique lax symmetric monoidal transformation

∆p: idSp−!Tp.

The underlying transformation of functors is given by the transformation from Defini-tion III.1.4.

Proof. This follows from [81, Corollary 6.9 (1)], which states that the identity is initial among exact lax symmetric monoidal functors from Sp to Sp.

Now, for the construction of the cyclotomic structure maps, we need to construct a natural transformation between the twoBCp-equivariant functors

N(Free_Cp)×_N(Fin)Sp^⊗_act−!Sp,

which are given by

I: (S,(X_s¯)_{s∈¯ S})7−!O

¯s∈S

X_¯s

and

Tep: (S,(X¯s)_{s∈¯ S})7−! O

s∈S

Xs

^tCp

,

respectively. If one trivializes a freeCp-set toS=S×Cp, the transformation is given by O

¯s∈S

X_¯s−!O

¯s∈S

(X_¯s⊗...⊗X_¯s)^tCp−! O

s∈S

X_¯s ^tCp

where the first map is the tensor product of ∆p for all ¯s∈S, and the second map uses that−^tCp is lax symmetric monoidal. One needs to see that this map does not depend on the chosen trivialization ofS.

In fact, to get this natural transformation I!Tep, we will do something stronger.

Namely, we will make both functors lax symmetric monoidal, and then show that there is a unique lax symmetric monoidal transformation. For this, we have to recall a few results about lax symmetric monoidal functors.

First, cf. Appendix A, we recall that, for any symmetric monoidal ∞-category C (more generally, for any ∞-operad O), one can form a new symmetric monoidal ∞-categoryC_act^⊗ whose underlying ∞-category is given by

C_act^⊗ =C^⊗×N(Fin∗)N(Fin),

where Fin is the category of finite (possibly empty) sets, and the functor Fin!Fin_∗adds a disjoint base point. The fiber ofC_act^⊗ over a finite setI∈Fin is given byC^I. Here, the symmetric monoidal structure onC_act^⊗ is given by a “disjoint union” type operation; it takes (Xi)_i∈I∈C^Iand (Xj)_j∈J∈C^Jto (Xk)_k∈ItJ∈C^ItJ. There is a natural lax symmetric monoidal functorC!C_act^⊗ whose underlying functor is the inclusion into the fiber ofC_act^⊗ over the 1-element set.

Proposition III.3.2. Let C and D be symmetric monoidal ∞-categories. Restric-tion along C ⊆C_act^⊗ is an equivalence of ∞-categories

Fun⊗(C_act^⊗ ,D)'Funlax(C,D).

Proof. This is a special case of [71, Proposition 2.2.4.9].

Note in particular that the identityC!Ccorresponds to a functor⊗:C_act^⊗ !C, which is informally given by sending a list (X1, ..., Xn) of objects inCtoX1⊗...⊗Xn. Another way to construct it is by noting thatC_act^⊗ !N(Fin) is a cocartesian fibration, andN(Fin) has a final object. This implies that the colimit of the corresponding functorN(Fin)! Cat∞ is given by the fiber C=C_h1i^⊗ over the final object h1i∈N(Fin), and there is a natural functor from C_act^⊗ to the colimit C; cf. [69, Corollary 3.3.4.3]. Moreover, the functor⊗:C_act^⊗ !C can also be characterised as the left adjoint to the inclusionC ⊆C_act^⊗ ; see RemarkIII.3.4 below.

In the situation of PropositionIII.3.2, it will be useful to understand lax symmetric monoidal functors fromC^⊗_actas well. For this, we use the following statement.

Lemma III.3.3. Let C and D be symmetric monoidal ∞-categories. Then, the canonical inclusion

Fun⊗(C_act^⊗ ,D)⊆Funlax(C_act^⊗ ,D) admits a right adjoint R. The composition

Funlax(C_act^⊗ ,D)−−−^R!Fun_⊗(C^⊗_act,D)−−−^∼!Funlax(C,D) is given by restriction along the lax symmetric monoidal functor C!C_act^⊗ .

Proof. We will use the concrete proof of [71, Proposition 2.2.4.9] in which it is shown that the composition

Fun⊗(C_act^⊗ ,D)⊆Funlax(C_act^⊗ ,D) ⁱ

∗

−−!Funlax(C,D)

is an equivalence of∞-categories. To do this, let us writeq: (C_act^⊗ )^⊗!NFin_∗for the sym-metric monoidal ∞-category whose underlying ∞-category is C_act^⊗ . Then, Lurie proves the following two facts:

(i) Every lax symmetric monoidal functorC!Dwhich is considered as an object of Fun_N(Fin∗)(C^⊗,D^⊗) admits a relative left Kan extension to a functor in

Fun_N(Fin∗)((C_act^⊗ )^⊗,D^⊗).

(ii) A functor in Fun_N(Fin∗)((C_act^⊗ )^⊗,D^⊗) is symmetric monoidal precisely if it is the relative left Kan extension of its restriction toC^⊗⊆(C_act^⊗ )^⊗.

This finishes Lurie’s argument using [69, Proposition 4.3.2.15]. In particular, we see that the inclusion Fun_⊗(C_act^⊗ ,D)⊆Funlax(C_act^⊗ ,D) is equivalent to the relative left Kan extension functor

i_!: Fun_lax(C,D)−!Fun_lax(C^⊗_act,D).

This functor is left adjoint to the restriction functor

i^∗: Funlax(C_act^⊗ ,D)!Funlax(C,D)

as an argument similar to [69, Proposition 4.3.2.17] shows: the compositioni^∗i!is equiv-alent to the identity giving a candidate for the unit of the adjunction. Then one uses [69, Lemma 4.3.2.12] to verify that the induced map on mapping spaces is an equivalence to get the adjunction property.

Remark III.3.4. For a symmetric monoidal ∞-category C the full, lax symmetric monoidal inclusionC ⊆C_act^⊗ admits a symmetric monoidal left adjoint Las we will show now. From this, one can deduce that, in the situation of LemmaIII.3.3, the right adjoint

R: Funlax(C_act^⊗ ,D)−!Fun_⊗(C_act^⊗ ,D)'Funlax(C,D) admits a further right adjoint given by precomposition withL.

Let us start by proving that the underlying functor admits a left adjoint. To this end, assume we are given an object ¯c∈C_act^⊗ ⊆C^⊗. The object lies over some finite pointed set (equivalent to) hni. There is a unique active morphism hni!h1i. Now choose a cocartesian liftf: ¯c!c inC^⊗ covering this morphism inNFin_∗. Then, the morphism f lies inC_act^⊗ and the object c lies inC ⊆C_act^⊗ . Moreover, f is initial among morphisms in C_act^⊗ from ¯cto an object inC ⊆C_act^⊗ . This follows since every such morphism has to cover the active morphismhni!h1iand from the defining property of cocartesian lifts. As a result, we find that c is the reflection of ¯c into C ⊆C_act^⊗ . Since this reflection exists for every ¯c∈C^⊗, the inclusionC ⊆C_act^⊗ admits a left adjointL. On a more informal level, the object ¯c is given by a listc1, ..., cn of objects ofC. Then,c=L(¯c) is given by the tensor productc1⊗...⊗cn.

Now, we show that the left adjoint L is a symmetric monoidal localization. To this end, we have to verify the assumptions of [71, Proposition 2.2.1.9] given in [71, Definition 2.2.1.6 and Example 2.2.1.7]. Thus, we have to show that, for every morphism f: ¯c!d¯in C^⊗ such that Lf is an equivalence in C and every object ¯e, the morphism L(f⊕¯e) is an equivalence in C, where ⊕ is the tensor product in C_act^⊗ . Unwinding the definitions, this amounts to the following: we have that ¯c is given by a list c₁, ..., c_n, d¯by a list d₁, ..., d_m and ¯e by a list e₁, ..., e_r. Then, the induced morphismL(f) is, by assumption, an equivalencec₁⊗...⊗cn!d₁⊗...⊗dm. But then clearly also the morphism L(f⊗¯e):c₁⊗...⊗c_n⊗e₁⊗...⊗e_r!d₁⊗...⊗d_m⊗e₁⊗...⊗e_ris an equivalence inC.

We remark that the left adjointL∈Fun_⊗(C_act^⊗ ,C) corresponds, under the equivalence Fun_⊗(C_act^⊗ ,C)−^∼!Fun_lax(C,C), to the identity functor.

Remark III.3.5. In fact, Proposition III.3.2 and Lemma III.3.3 hold true if C (or rather C^⊗) is replaced by any ∞-operad O^⊗. Let us briefly record the statements, as we will need them later. As before, D is a symmetric monoidal ∞-category. Let O^⊗ be an ∞-operad with symmetric monoidal envelope O_act^⊗ =O^⊗×_N(Fin∗)N(Fin). There is a natural map of∞-operads O^⊗!(O^⊗_act)^⊗. Restriction along this functor defines an equivalence

Fun⊗(O^⊗_act,D)'Alg_O(D),

where the right-hand side denotes the ∞-category of O-algebras in D; equivalently, of

∞-operad mapsO^⊗!D^⊗. Moreover, the full inclusion Fun_⊗(O_act^⊗ ,D)−!Fun_lax(O_act^⊗ ,D)

admits a right adjoint given by the composition Funlax(O_act^⊗ ,D)!Alg_O(D)'Fun_⊗(O_act^⊗ ,D), where the first functor is restriction alongO^⊗!(O^⊗_act)^⊗.

Recall that we want to construct a natural transformation between twoBCp-equivariant functors

N(FreeC_p)×N(Fin)Sp^⊗_act−!Sp, which are given by

I: (S,(Xs¯)_¯s∈S)7−!O

s∈¯ S

Xs¯

and

Tep: (S,(Xs¯)_{s∈¯ S})7−! O

s∈S

Xs

^tCp

, respectively. More precisely, the first functor

I:N(FreeC_p)×_N(Fin)Sp^⊗_act−!Sp

is the symmetric monoidal functor given by the composition of the projection to Sp^⊗_act and⊗: Sp^⊗_act!Sp.

For the second functor, we first have to construct a symmetric monoidal functor N(FreeCp)×N(Fin)Sp^⊗_act−!(Sp^⊗_act)^BCp,

(S,(X_¯s)_¯s∈S)7−!(S,(X_¯s)_s∈S).

Proposition III.3.6. For any symmetric monoidal ∞-category C and any integer p>1,there is a natural symmetric monoidal functor

N(Free_Cp)×_N(Fin)C_act^⊗ −!(C_act^⊗ )^BCp, (S,(X_s¯)_{s∈¯ S})7−!(S,(X_s¯)_s∈S).

Proof. We observe that, as in the proof of Lemma III.3.7below, Fun⊗(N(FreeC_p)×N(Fin)C_act^⊗ ,D) = Fun(N(TorC_p),Funlax(C,D))

for any symmetric monoidal∞-categoryD(here applied toD=(C_act^⊗ )^BCp), where Tor_Cp is the category of C_p-torsors. This can be constructed as the composite of the lax symmetric monoidal embeddingC!C_act^⊗ and a functor

N(Tor_Cp)−!Fun_⊗(E,E^BCp) = Fun(BC_p,Fun_⊗(E,E))

which exists for any symmetric monoidal∞-category E (here, applied to E=C_act^⊗ ). For this, note that Fun_⊗(E,E) is itself symmetric monoidal (using the pointwise tensor product), and so the identity lifts to a unique symmetric monoidal functor from the symmetric monoidal envelope N(Fin)^' of the trivial category. Restricting the functor N(Fin)^'!Fun_⊗(E,E) toCp-torsors gives the desired symmetric monoidal functor

N(TorC_p)−!Fun(BCp,Fun_⊗(E,E)).

Composing the resulting symmetric monoidal functor N(FreeCp)×N(Fin)Sp^⊗_act−!(Sp^⊗_act)^BCp

with the lax symmetric monoidal functors⊗: (Sp^⊗_act)^BCp!Sp^BCp and−^tCp: Sp^BCp!Sp, we get the desired lax symmetric monoidal functor

Te_p:N(Free_Cp)×_N(Fin)Sp^⊗_act−!Sp.

In fact, the lax symmetric monoidal functorsIandTepareBCp-equivariant for the natural action on the source (acting on the setS), and the trivial action on the target. This is clear for I, and forTep, the only critical step is the functor −^tCp: Sp^BCp!Sp, where it follows from the uniqueness results in TheoremI.4.1.

Now, consider the ∞-category of all lax symmetric monoidal functors Funlax(N(FreeC_p)×_N(Fin)Sp^⊗_act,Sp).

We call a functorF:N(FreeC_p)×N(Fin)Sp^⊗_act!Sppartially exact if, for every Cp-torsor S, the induced functorF(S,−): Sp!Sp, obtained by restriction to

{S}×_{∗}Sp⊆N(Free_Cp)×N(Fin)Sp^⊗_act,

is exact. This leads to the following lemma, which will also be critical to comparing our new construction with the old construction of THH.

LemmaIII.3.7. The functor Iis initial among all lax symmetric monoidal functors N(Free_Cp)×N(Fin)Sp^⊗_act−!Sp

which are partially exact.

Proof. By Lemma III.3.3, or rather the ∞-operad version of Remark III.3.5, it is enough to prove the similar assertion in the ∞-category of symmetric monoidal and partially exact functors. Now, we claim that restriction along

N(TorCp)×Sp⊆N(FreeCp)×N(Fin)Sp^⊗_act induces an equivalence of∞-categories

Fun_⊗(N(FreeC_p)×_N(Fin)Sp^⊗_act,Sp)'Fun(N(TorC_p),Funlax(Sp,Sp)),

where Tor_Cp is the category of C_p-torsors. As partially exact functors correspond to Fun(N(Tor_Cp),Fun^Ex_lax(Sp,Sp)) on the right, the result will then follow from [81, Corollary 6.9 (1)].

By [71, Theorem 2.4.3.18], the ∞-category Fun(N(Tor_Cp),Fun_lax(Sp,Sp)) is equiv-alent to the∞-category of∞-operad maps fromN(Tor_Cp)^t×_N(Fin∗)Sp^⊗ to Sp^⊗. Here, N(Tor_Cp)^tis the cocartesian∞-operad associated withN(Tor_Cp); cf. [71,§2.4.3]. Now, as in Proposition III.3.2, the ∞-category of operad maps from any ∞-operad O^⊗ to Sp^⊗ is equivalent to the∞-category of symmetric monoidal functors from the symmet-ric monoidal envelope O_act^⊗ to Sp; cf. [71, Proposition 2.2.4.9]. But the envelope of N(TorC_p)^t×_N(Fin∗)Sp^⊗ is

(N(TorC_p)^t×N(Fin_∗)Sp^⊗)×N(Fin_∗)N(Fin)'N(FreeC_p)×N(Fin)Sp^⊗_act, using the equivalence

N(TorC_p)^t×_N(Fin∗)N(Fin)'N(FreeC_p).

Unraveling, we have proved the desired assertion.

The following corollary is immediate, and finishes our construction of THH as a cyclotomic spectrum.

Corollary III.3.8. The BC_p-equivariant lax symmetric monoidal functor Tep:N(FreeCp)×N(Fin)Sp^⊗_act−!Sp,

(S,(X¯s)_{s∈¯ S})7−! O

s∈S

Xs¯

^tCp

receives an essentially unique BCp-equivariant lax symmetric monoidal transformation from

I:N(Free_Cp)×N(Fin)Sp^⊗_act−!Sp (S,(X_¯s)_¯s∈S)7−!O

¯s∈S

X_s¯.

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