Motivic Tambara Functors

Let k be a field and denote by SH(k) the motivic stable homotopy category. Recall its full subcategory HI_0(k) of effective homotopy modules. Write NAlg(HI_0(k)) for the category of normed motivic spectra with underlying spectrum an effective homotopy module. In this article we provide an explicit description of NAlg(HI_0(k)) as the category of sheaves with generalized transfers and \'etale norms, and explain how this is closely related to the classical notion of Tambara functors.


Introduction
Norms and normed spectra. In [7], we defined for every finiteétale morphism f : S ′ → S of schemes a symmetric monoidal functor of symmetric monoidal ∞-categories f ⊗ : SH(S ′ ) → SH(S). If S ′ = S n and f is the fold map, then f ⊗ : SH(S ′ ) ≃ SH(S) n → SH(S) is the n-fold smash product. These norm maps commute with arbitrary base change and assemble into a functor Here Span(Sch, all, fét) denotes the (2, 1)-category of spans in schemes, where the forward arrows are required to be finiteétale [7,Appendix C]. The category NAlg(SH(S)) is defined as the category of sections of the restriction of SH ⊗ to Span(Sm S , all, fét), cocartesian over backwards arrows [7,Section 7]. In other words, an object E ∈ NAlg(SH(S)) consists of for each X ∈ Sm S a spectrum E X ∈ SH(X), for each morphism p : X → Y ∈ Sm S an equivalence E X ≃ p * E Y , for each finiteétale morphism f : U → V ∈ Sm S a morphism p ⊗ E U → E V , and infinitely many coherences among these data. The forgetful functor U : NAlg(SH(S)) → SH(S), E → E S is monadic [7, Proposition 7.6(2)] and in particular conservative.
Effective homotopy modules. Recall the infinite suspension spectrum functor Σ ∞ + : Sm S → SH(S). Let SH(S) veff ⊂ SH(S) be the full subcategory generated under colimits and extensions by Σ ∞ + Sm S . We call this the category of very effective spectra. Also denote by SH(S) eff ⊂ SH(S) the localizing subcategory generated by Σ ∞ + Sm S ; this is the category of effective spectra. By standard results, SH(S) veff ⊂ SH(S) eff is the non-negative part of a t-structure on SH(S) eff which is called the effective homotopy t-structure. We write SH(S) eff♥ for its heart and τ eff ≤0 : SH(S) veff → SH(S) eff♥ for the truncation functor.
Tambara functors. We define NAlg(SH(S) eff♥ ) ⊂ NAlg(SH(S)) to be the full subcategory on those E ∈ NAlg(SH(S)) such that U E ∈ SH(S) eff♥ . Now suppose that E ∈ NAlg(SH(S) eff♥ ). Then the presheaf π 0 (E) ∈ Ab(Sm S ) acquires norm maps. In other words, if f : X → Y ∈ Sm S is finiteétale, then there is an induced map N f : π 0 (E)(X) → π 0 (E)(Y ). This is a map of sets, not abelian groups; in other words it is not additive. Instead, the maps N f satisfy a generalized distributivity condition related to the transfers tr g [7,Corollary 7.21], making π 0 (E) into a so-called Tambara functor [29] [5,Definition 8]. Let us write T (S) for the category of effective homotopy modules E which are provided with norm maps on π 0 (E) ∈ Ab(Sm S ) in such a way that the distributivity condition is fulfilled; see Definition 1 below for details. Then we have a factorisation π 0 : NAlg(SH(S) eff♥ ) → T (S) → Ab(Sm S ).
Main results. For an additive category C and e ∈ Z >0 , write C[1/e] for the full subcategory on those objects E ∈ C such that E e − → E is an equivalence. For example, Ab(Sm S )[1/e] is the full subcategory of Ab(Sm S ) consisting of those presheaves which are presheaves of Z[1/e]-modules. Write NAlg(SH(S) eff♥ )[1/e] for the full subcategory on those objects E ∈ NAlg(SH(S) eff♥ ) such that U E ∈ SH(S) eff♥ [1/e]. Similarly for T (S) [1/e]. With these preliminaries out of the way, we can state our main results.
Overview of the article. In Section 2 we introduce some standing assumptions and notation, beyond the notation already established in this introduction.
In Section 3 we introduce a first notion of motivic Tambara functors, called motivic Tambara functors of the first kind. These are effective homotopy modules M ∈ HI 0 (k) together with for each finiteétale map f : X → Y ∈ Sm k a norm map N f : M (X) → M (Y ), such that the norms distribute over the finiteétale transfers in a suitable fashion. In the remainder of this section we establish basic structural properties of the category of motivic Tambara functors of the first kind.
In Section 4 we introduce a second notion of motivic Tambara functors, called motivic Tambara functors of the second kind. These are effective homotopy modules M ∈ HI 0 (k) together with for every finiteétale morphism f : X → Y ∈ Sm k and every smooth and quasi-projective morphism p : W → X a norm map N f,W : M (W ) → M (R f (W )), where R f (W ) ∈ Sm Y denotes the Weil restriction of W along f . The norms are again required to distribute over transfers in a suitable fashion. Note that if p = id : X → X, then R f X ≃ X and so we obtain N f,X : M (X) → M (Y ). In other words, any motivic Tambara functor of the second kind naturally induces a motivic Tambara functor of the first kind. The main result of Section 4 is that this is an equivalence of categories.
In Section 5, we introduce a third notion of motivic Tambara functors, called naive motivic Tambara functors. These are just presheaves of sets M on FEt S such that M (X Y ) ∼ = M (X) × M (Y ), together with norm and transfer maps for finiteétale morphisms, satisfying a suitable distributivity condition. Here FEt S denotes the category of finiteétale S-schemes. This definition is closest to Tambara's original definition. Using one of Tambara's original results, we easily show that naive motivic Tambara functors are well-behaved under group completion and localization. This is used at a key point in the proof of the main result.
In Section 6 we study in more detail the category HI 0 (k). Using abstract categorical arguments, we show that if char(k) = 2 then HI 0 (k) ≃ DM eff (k) ♥ . From this we deduce that for X ∈ Sm k , the effective homotopy module EX := π 0 (Σ ∞ + X) 0 ∈ HI 0 (k) is, in a suitable sense, generated under transfers and pullbacks by the maps Y → X for Y ∈ Sm k . In Section 7 we introduce yet another notion of motivic Tambara functors, called normed effective homotopy modules. This is just the category NAlg(SH(k) eff♥ ). We construct it more formally, and establish some of its basic properties.
Finally in Section 8 we put everything together and prove the main theorem. To do so we first note that there is a canonical functor ρ : NAlg(SH(k) eff♥ ) → T 2 (k), where T 2 (k) denotes the category of motivic Tambara functors of the second kind. This just arises from the fact that, by construction, if M ∈ NAlg(SH(k) eff♥ ) then M has certain norm maps, known to distribute over transfers. Next, we observe that both NAlg(SH(k) eff♥ ) and T 2 (k) admit monadic forgetful functors to the category HI 0 (k) of homotopy modules, and that ρ is compatible with these forgetful functors. It is thus enough to prove that the induced morphism of monads is an isomorphism. This reduces to showing that if X ∈ Sm k and M denotes the free normed effective homotopy module on EX, then M is also the free motivic Tambara functor of the second kind on EX. We do this by noting that there is an explicit formula for M as a large colimit, coming from the identification of the free normed spectrum functor [7,Remark 16.25]. From this we can verify the universal property of M as a motivic Tambara functor of the second kind by an essentially elementary (but lengthy) computation.
Use of ∞-categories. Throughout, we freely use the language of ∞-categories as set out in [19,20]. Unless explicitly mentioned otherwise, all categories are ∞-categories, all colimits are homotopy colimits, and so on. That being said, our main categories of interest are actually 1-categories (i.e. equivalent as ∞categories to the nerve of an ordinary category). In a 1-category, the ∞-categorical notions of colimits etc. reduce to their classical counterparts; so in many parts of this article the traditional-sounding language indeed has the traditional meaning.
Acknowledgements. I would like to thank Marc Hoyois for teaching me essentially everything I know about ∞-categories, extensive discussions on normed spectra, and several discussions regarding the results in this article. I would further like to thank Maria Yakerson and an anonymous referee for comments on a draft of this article.

Background and notation
Throughout, k is a perfect field. Recall that the objects Σ ∞ + X ∧ G ∧n m ∈ SH(k), X ∈ Sm k , n ∈ Z generate the non-negative part of a t-structure, known as the homotopy t-structure [23, Section 5.2]. We write HI * (k) ≃ SH(k) ♥ for the category of homotopy modules [23,Theorem 5.2]. The functor i ♥ : SH(k) eff♥ → SH(k) ♥ is fully faithful [2,Propositions 4 and 5]. We write HI 0 (k) ⊂ HI * (k) for its essential image, and call it the category of effective homotopy modules. If X ∈ Sm k then Σ ∞ + X ∈ SH(k) eff ≥0 , and we denote by EX ∈ HI 0 (k) ≃ SH(k) eff♥ the truncation. For M ∈ HI 0 (k) and X ∈ Sm k we abbreviate Hom(EX, M ) =: M (X). The functor HI 0 (k) → Ab(Sm k ), M → (X → M (X)) factors through Ab N is (Sm k ) and the induced functor HI 0 (k) → Ab N is (Sm k ) is conservative and preserves limits and colimits (and also HI 0 (k) has all limits and colimits) [2,Proposition 5(3)]. Moreover, its image consists of unramified sheaves [24,Lemma 6.4.4].
Here Ab(Sm k ) denotes the category of presheaves of abelian groups on Sm k , and Ab N is (Sm k ) the category of Nisnevich sheaves of abelian groups.
Throughout we will be working with full subcategories C ⊂ Sch k which contain Spec(k) and are closed under finiteétale extensions (and so in particular finite coproducts). We also denote this condition by C ⊂ fét Sch k .
Recall that if f : X → Y ∈ Sm k is a finiteétale morphism, then the functor SmQP Y → SmQP X , T → T × Y X has a right adjoint R f called Weil restriction [9,Theorem 7.6.4]. Here SmQP X denotes the category of smooth and quasi-projective X-schemes. In particular if Z → X is finiteétale, then Z → X is smooth and affine, so smooth and quasi-projective, so R f Z exists.
Recall that if f : X → Y ∈ Sm k is a finiteétale morphism and M ∈ HI 0 (k), then there is a canonical transfer morphism tr f : M (X) → M (Y ). These transfer morphisms are natural in M and f [3, Section 4].

Motivic Tambara functors of the first kind
We now come to the most intuitive definition of a motivic Tambara functor as an effective homotopy module with norms.
First recall the notion of an exponential diagram [5, Definition 7]: given finiteétale morphisms A Here e is the X-morphism corresponding by adjunction to the identity R f A → R f A, and p is the canonical projection.
Definition 1. Let C ⊂ fét Sm k . A C-Tambara functor of the first kind consists of an effective homotopy module M ∈ HI 0 (k), together with for each f : X → Y ∈ C finiteétale a map of sets N f : M (X) → M (Y ) such that: (3) Given finiteétale morphisms A q − → X f − → Y in C, the following diagram (induced by the corresponding exponential diagram) commutes A morphism φ : M 1 → M 2 of C-Tambara functors of the first kind is a morphism of the underlying effective homotopy modules such that for every f : X → Y finiteétale, the following diagram commutes We denote the category of C-Tambara functors of the first kind by T 1 C (k), and we write U 1 : T 1 C (k) → HI 0 (k) for the evident forgetful functor.  Here is a basic structural property of the category of C-Tambara functors of the first kind.
Lemma 5. The category T 1 C (k) is presentable and the forgetful functor U 1 : Proof. We first construct auxiliary categories D and D ′ . The objects of both D and D ′ are objects of C. For X, Y ∈ C, the morphisms from X → Y in D ′ are given by equivalence classes spans, i.e. diagrams X f ← − T → Y , where f is required to be finiteétale. In other words D ′ is just the homotopy 1-category of the bicategory Span(C, fét, all).
The morphisms from X → Y in D ′ are given by equivalence classes of bispans, i.e. diagrams X where f and g are required to be finiteétale. We shall identify two bispans if they fit into a commutative diagram X Before explaining composition in D, let us explain what the category D is supposed to do. We will have a functor F : D ′ → D which is the identity on objects and sends X This induces F * : P Sh(D) → P Sh(D ′ ). The objects in P Sh(D) are going to be "presheaves with norm and transfer" in the following sense. Let G : HI 0 (k) → P Sh(D) denote the forgetful functor. Then we have a cartesian square of 1-categories The category HI 0 (k) is presentable, being an accessible localization of the presentable category SH(k) veff . The categories P Sh(D) and P Sh(D ′ ) are of course presentable. The functor F * has a left adjoint, given by left Kan extension. The functor G also has a left adjoint; indeed it is a functor between presentable categories which preserves limits and filtered colimits (see Lemma 11 below), so the claim follows from the adjoint functor theorem [19, Corollary Corollary 5.5.2.9 (2)]. It follows that F * and G are morphisms in P r R . Thus the square is also a pullback in P r R [19,Theorem 5.5.3.18], and in particular T 1 C (k) is presentable and U 1 is a right adjoint.
It remains to finish the construction of D. The composition in D is determined by the following properties: (1) If f, g are finiteétale then τ gf = τ f τ g and ν gf = ν f ν g . (3) The τ and ν morphisms satisfy the basechange law with respect to the ρ morphisms. (4) the distributivity law holds. For a more detailed construction of similar categories, see [28, Section 5, p. 24 and Proposition 6.1].
Remark 6. The cartesian square (1) can be used to elucidate the nature of motivic Tambara functors of the first kind: the category is equivalent to the category of triples (T, M, α) where T is a presheaf on a certain bispan category D, M is an effective homotopy module, and α is an isomorphism between the presheaves with finiteétale transfers underlying T and M . In fact, if char(k) = 0 then one may show that the functor HI 0 (k) → P Sh(D ′ ) is fully faithful (use [8,Corollary 5.17] and [5, paragraph before Proposition 22]), whence so is T 1 C (k) → P Sh(D). We deduce that in this situation the category T 1 C (k) has a particularly simple description: it consists of presheaves on the bispan category D such that the underlying presheaf with finiteétale transfers extends to an effective homotopy module (in particular, is a strictly homotopy invariant sheaf).
We immediately deduce the following.
Corollary 7. The category T 1 C (k) has all (small) limits and colimits. Recall now that if F is a presheaf on a category D, then F extends uniquely to a continuous presheaf on P ro(D), the category of pro-objects. Moreover, consider the subcategory Sm ess k ⊂ Sch k on those schemes which can be obtained as cofiltered limits of smooth k-schemes along diagrams with affine transition morphisms. Then Sm ess k embeds into P ro(Sm k ) [17, Proposition 8.13.5], and consequently for X ∈ Sm ess k the expression F (X) makes unambigious sense, functorially in X. It follows in particular that Definition 1 makes sense more generally for C ⊂ fét Sm ess k . Let C ⊂ Sch. Write C sl for the subcategory of Sch on those schemes obtained as semilocalizations of schemes in C at finitely many points. We write C ⊂ fét,op Sch to mean that C ⊂ fét Sch and C is closed under passage to open subschemes.
A convenient property of the category T 1 C (k) is that, in reasonable cases, it is invariant under replacing C by C sl : In the proof, we shall make use of the unramifiedness property of homotopy modules [24,Lemma 6.4.4]: if X ∈ Sm k is connected and ∅ = U ⊂ X, then M (X) → M (U ) is injective. In particular, if η is the generic point of X, then M (X) ֒→ M (η).
Proof. Let X ∈ C sl and f : Y → X finiteétale. Then X is a cofiltered limit along open immersions, so there exists a cartesian square i an open immersion and f ′ finiteétale [17, Théorèmes 8.8.2(ii) and 8.10.5(x), and Proposition 17.7.8(ii)]. It follows that V ∈ C, and Y is a cofiltered limit (intersection) of open subschemes of V . Since Y → X is finite (so in particular closed and quasi-finite [27, Tags 01WM and 02NU]), Y is semilocal, and so must be a semilocalization of V . This proves the first claim.
Note that T 1 is a morphism of the underlying homotopy modules, compatible with the norms on semilocal schemes, then it is compatible with the norms on generic points, and hence it is compatible with all norms, by Lemma 10 below.
The functor T 1 It remains to show that it is essentially surjective.
Thus let M ∈ T 1 C sl (k). Let p : X → Y ∈ C be finiteétale. We need to construct a norm N p : M (X) → M (Y ). We may assume that Y is connected. Let η be the generic point of Y . We are given a norm map N η p : M (X η ) → M (η). By unramifiedness (and compatibility of norms with base change), there is at most one map N p compatible with N η p . What we need to show is that N η p (M (X)) ⊂ M (Y ). In order to do this, by unramifiedness again, it suffices to prove this for Y replaced by the various localizations of Y at its points of codimension one. But then Y is semilocal, so the result holds by assumption.
It remains to show that these norms turn M into a C-motivic Tambara functor of the first kind. The base change formula (i.e. condition (2) of Definition 1) is satisfied by construction. It implies using unramifiedness of M that it is enough to check conditions (1) and (3) when the base is a field (use that Weil restriction commutes with arbitrary base change [12, Proposition A.5.2(1)]), in which case they hold by assumption.
. This applies for example if C 1 = Sm k and C 2 = SmQP k .
In the course of the proof of Proposition 8, we used the following lemma of independent interest. Let C ⊂ Sch. Write C gen for the full subcategory of Sch consisting of the subschemes of generic points of schemes in C.
and let α ∈ Hom HI0(k) (F, G) be a morphism of the underlying homotopy modules.
where e is the exponential characteristic of k, then the above criterion we may replace C gen by C gen,perf , consisting of the perfect closures of objects in C gen .
Proof. The first claim is proved exactly as in the proof of Lemma 8. Suppose given α with the claimed properties. We need to show that, if p : X → Y ∈ C is finiteétale, then the following diagram commutes The base change formula thus allows us to assume that Y ∈ C gen . It follows that X ∈ C gen , and so the diagram commutes by definition.
For the last claim, we use that if X ∈ C gen has perfect closure X ′ , then G(X) → G(X ′ ) is injective [4, Lemma 17].
Corollary 7 assures us that T 1 C (k) has all limits and colimits. In the final part of this section, we wish to investigate how these limits and colimits are computed. We begin with the case of homotopy modules: Lemma 11. Let X ∈ Sm ess k . The functor ev X : HI 0 (k) → Ab, F → F (X) preserves finite limits and filtered colimits. If X ∈ Sm sl k , then the functor ev X preserves arbitrary colimits as well, whereas if X ∈ Sm k , it preserves arbirary limits.
Proof. By [2, Proposition 5(3)], the functor o : HI 0 (k) → Shv N is (Sm k ) preserves limits and colimits. Taking global sections of sheaves preserves limits, so the claim about preservation of limits when X ∈ Sm k is clear. If X ∈ Sm ess k , say X = lim i X i (the limit being cofiltered), then F (X) = colim i F (X i ). Since this colimit is filtered, and finite limits commute with filtered colimits, the claim about preservation of finite limits follows.
A filtered colimit of Nisnevich sheaves, computed in the category P Sh(Sm k ), is still a Nisnevich sheaf, since Nisnevich sheaves are detected by the distinguished squares and filtered colimits commute with finite limits of sets. It follows that ev X preserves filtered colimits (for any X). Since ev X preserves finite limits and our categories are abelian, ev X preserves finite sums. Now let X be semi-local. It remains to show that ev X preserves cokernels. Let α : F → G ∈ HI 0 (k), let K = ker(α), C = cok(α), I = im(α) and consider the short exact sequences 0 → K → F → I → 0 and 0 → I → G → C → 0. It is enough to show that ev X preserves these exact sequences. Let 0 → F → G → H → 0 ∈ HI 0 (k) be an exact sequence. We can deduce the desired result.
Note that a functor between categories with small colimits (such as ours) preserves sifted colimits if and only if it preserves filtered colimits and geometric realizations [19,Corollary 5.5.8.17], which for 1-categories (such as ours) is the same as preserving filtered colimits and reflexive coequalizers. We will not use this observation.
Proof. By Lemma 8, we may replace C by C sl . Let F : D → T 1 C sl (k) be a sifted diagram, and let C = colim D U 1 F . Note that the forgetful functor Ab → Set preserves sifted colimits. Hence if X ∈ C sl then by Lemma 11 we find that C(X) = colim D F (•)(X), where the colimit is taken in the category of sets. In particular if f : X → Y ∈ C sl is finiteétale, then there is a canonical induced norm N f : C(X) → C(Y ). It is easy to check that C, together with these norms, defines an object of T 1 C sl (k) which is a colimit of F . This concludes the proof.

Motivic Tambara functors of the second kind
We now come to a second, somewhat more technical definition of a category of motivic Tambara functors. We will eventually show that in good cases, it coincides with the first definition.
Remark 13. For the purposes of this article, Tambara functors of the second kind can be viewed just as a technical tool: the proof of our main theorem (that normed effective homotopy modules are the same as Tambara functors of the first kind) is just naturally split into showing both that normed effective homotopy modules are the same as Tambara functors of the second kind, and that Tambara functors of the first and second kind are the same.
Slightly more philosophically, it seems that Tambara functors of the second kind are closer to the "true" nature of normed effective homotopy modules (in cases where Tambara functors of the first and second kind are not the same); see also Remark 46.
Remark 15. The class of smooth quasi-projective morphisms is admissible. Since finiteétale schemes are smooth quasi-projective, and Weil restriction preserves finiteétale schemes (this follows for example from [9, Proposition 7.5.5]), we deduce that the class of finiteétale morphisms is also admissible.
Suppose that the lower square is cartesian, f is finiteétale, X, Y ∈ C and p 1 , p 2 ∈ V. Then the following diagram commutes Here c : namely the one which corresponds by adjunction to a : in Sm k , with f, g finiteétale, X, Y ∈ C and p 1 , p 2 ∈ V. Then the following diagram commutes A morphism φ : M 1 → M 2 of (C, V)-Tambara functors of the second kind is a morphism of the underlying effective homotopy modules such that for each f : X → Y ∈ C finiteétale and V → X ∈ V the following diagram commutes We denote the category of (C, V)-Tambara functors of the second kind by T 2 C (k), and we write U 2 : T 2 C (k) → HI 0 (k) for the evident forgetful functor. Observe that we suppress V from the notation. We give a special name to some of the simplest norm maps on a motivic Tambara functor of the second kind.
as N (1) f = N f,X . Note that this makes sense, since id : X → X ∈ V by assumption.
In some sense, these special norms already determine all the norms: Lemma 19. Let C ⊂ fét Sm k , V ⊂ M or(Sm k ) admissible, and assume that C, V are compatible.
Proof. Apply Definition 17 (2) to the diagram noting that c = id. Note that this makes sense: we have R f V ∈ C by the compatibility assumption, and then X × Y R f V ∈ C since C is closed under finiteétale extension.
We are now ready to prove our main result of this section.
Proposition 20. Let C ⊂ fét Sm k , V ⊂ M or(Sm k ) admissible, and assume that C, V are compatible.
If M ∈ T 2 C (k) then U 2 M ∈ HI 0 (k) together with the norm maps N (1) f of Construction 18 defines a C-motivic Tambara functor of the first kind. Moreover the induced functor T 2 C (k) → T 1 C (k) is an equivalence.
Proof. Write F : T 2 C (k) → T 1 C (k) for this (so far hypothetical) functor. We begin by constructing what will be its inverse G : where f ′ is the projection and a is the counit. Then we put N f,V = N f ′ a * : M (V ) → M (R f (V )). Write GM for M equipped with these norm maps N f,V ; this is enough data to define an object of T 2 C (k) (but we have not shown that the required conditions hold). I claim that (a) if M 2 ∈ T 2 C (k) then F M 2 is indeed a C-motivic Tambara functor of the first kind, and that (b) if M 1 ∈ T 1 C (k) then GM 1 is indeed a (C, V)-motivic Tambara functor of the second kind. Suppose for now that this is true. It is then clear that F, G are functors, i.e. send morphisms to morpisms. It follows from Lemma 19 that GF M 2 = M 2 . Moreover F GM 1 = M 1 by construction. Hence F is an equivalence as claimed. It thus remains to establish (a) and (b).
Proof of (a). Conditions (1) and (2) of Definition 1 follow respectively from Definition 17(1) (with V = X) and (2) (with p 1 = id and p 2 = id). For condition (3), we use that N (1) p e * = N f,A by Lemma 19, and hence the condition follows from Definition 17 (3) Proof of (b). Let M ∈ T 1 C (k). We need to show that Conditions 17(1-3) hold. Proof of (1). The condition about identities is clear. For the composition, let X The maps a are counit maps (use . By definition N gf,V is the composite N p N p a * induced by the left column, whereas N g,R f V N f,V is the composite N p a * N p a * induced by (first row, middle) to (second row, middle) to (third row, middle) to (third row, left) to (fourth row, left). The condition follows from Definition 1(2), because the middle left square is cartesian.
Proof of (2). Consider the diagram Here d, e are counit maps, N means norm along the canonical projections, and g : namely the morphism f ′ * c. We note that the following square is cartesian where the vertical morphisms are the canonical projections. It follows from Definition 1(2) that the right hand square in diagram (2) commutes. Moreover the following square commutes and hence the left hand square in diagram (2) commutes. It follows that the outer rectangle also commutes, which is what we needed to show.
Proof of (3). Consider the following commutative diagram in which all rectangles are cartesian Here r, r ′ are the canonical projections, a is the counit map, and q is induced by the universal property of C from the counit map . ] X denotes the morphisms of X-schemes, and r * : Sch R f V2 → Sch X×Y R f V2 is the base change functor. The first isomorphism is by definition (of Weil restriction), and the second is because the top square in diagram (3) is cartesian. Here we view r * T as a scheme over V 2 via a. We also have where on the right hand side we view T as a scheme over Y via the canonical map R f V 2 → Y . It remains to observe that f * T = r * T , because the lower right hand square in diagram (3) is cartesian. With this preparation out of the way, consider the diagram The unlabelled arrows are restriction along some canonical map, the arrows labelled tr are transfer along some canonical map, and the arrows labelled N are norm along some canonical map. The right hand square commutes by the base change formula, and the left hand rectangle commutes by the distributivity law (i.e. Definition 1 (3)). The top composite is N f,V1 (using that R r C ∼ = R f V 1 , as established above, and r * R r C ∼ = f * R f V 1 , by transitivity of base change) and the bottom composite is N f,V2 , so commutativity is precisely condition (3). This concludes the proof.

Naive Motivic Tambara functors
Throughout this section, k is an arbitrary base scheme. In particular it is not necessarily a field, unless otherwise specified.
Recall the following very naive definition of a motivic Tambara functor [5,Definition 8]. It is closest to Tambara's original definition.
Definition 21. Let k be some base scheme. A naive motivic Tambara functor over k is a presheaf of sets M on FEt k which preserves finite products (when viewed as a functor FEt op k → Set), provided with for every (necessarily finiteétale) morphism f : X → Y ∈ FEt k two further maps of sets tr f , N f : M (X) → M (Y ), such that the following conditions hold: (2) tr f and N f satisfy the base change formula. (For N f this is the condition (2) of Definition 1, with C = FEt k . For tr f the condition is the same, just with tr in place of N everywhere.) (3) The distributivity law holds (in the sense of Definition 1(3)). The morphisms of naive motivic Tambara functors are morphisms of presheaves of sets which commute with the norms and transfers. We write T naive (k) for the category of naive motivic Tambara functors.
The product preservation condition just means that the canonical map M (X Y ) → M (X) × M (Y ) is an isomorphism, and that M (∅) = * ; equivalently M is a sheaf in the Zariski topology. By considering transfer and norm along X X → X, the set M (X) acquires two binary operations + and ×. Considering t : ∅ → X, we have elements 0 = tr t ( * ) and 1 = N t ( * ) which are units for the two binary operations. By condition (3), × distributes over +. Consequently, M is canonically a sheaf of semirings. As before, the conditions imply that f * is a homomorphism of semirings, tr f is a homomorphism of additive monoids, and N f is a homomorphism of multiplicative monoids.
Let us note the following consequence of the axioms.

Lemma 22 (projection formula).
Let M ∈ T naive (k) and g : X → Y ∈ FEt k . Then for a ∈ M (X), b ∈ M (Y ) we have tr g (a · g * b) = tr g (a) · b.
Proof. This follows from the distributivity law applied to the exponential diagram generated by X Y where ∇ is the fold map, using the computation that R f (X Y ) ≃ X × Y Y ≃ X (recall that Weil restriction along a fold map is just the product).
Definition 23. We say that M ∈ T naive (k) is group-complete if the abelian semigroup (M (X), +) is an abelian group for all X ∈ FEt k . We write T naive gc (k) for the full subcategory of group-complete functors.
The inclusion T naive Proof. A naive motivic Tambara functor is essentially the same as a semi-TNR functor in the sense of Tambara [29, Section 2], for the profinite group G = Gal(k). Tambara only treats finite groups, but the extension to profinite groups is immediate. We spell out the details.
Let L/k be a finite Galois extension with group G. Then the category Fin G of finite G-sets is a full subcategory of FEt k , by Grothendieck's Galois theory. The restriction M | FinG defines a semi-TNR functor. It follows from [29, Theorem 6.1 and Proposition 6.2] that M + | FinG has a unique structure of a TNR-functor such that the canonical map (M → M + )| FinG is a morphism of semi-TNR functors, and that M + | FinG is the universal map from M | FinG to a TNR-functor. Now suppose that L ′ /L/k is a bigger Galois extension, with group G ′ . Then (M | Fin G ′ )| FinG = M | FinG and consequently the norms on M + | Fin G ′ obtained by the above universal property, when further restricted to Fin G , coincide with the norms obtained on M + | FinG directly. Now let f : X → Y ∈ FEt k . Then there exists a finite Galois extension L/k with group G such that f is in the image of Fin G → FEt k , and hence we obtain a norm map N f . By the above universal property, extending L does not change this norm, so in particular N f is well-defined independent of the choice of L. This defines the structure of a naive motivic Tambara functor on M + , since all the required conditions can be checked after restriction to Fin G for varying G.
Let A ∈ T naive gc (k) and F : M → A be any morphism of Tambara functors. Then there is a unique morphism of sheaves of additive abelian groups F + : M + → A. It remains to show that F + is a morphism of Tambara functors, i.e. preserves norms. This can be checked on F + | FinG for varying G, where it holds by Tambara's result. This concludes the proof.
Let us also include for the convenience of the reader a proof of the following well-known fact.
Lemma 25. Let A be an abelian group and A 0 ⊂ A an abelian semigroup which generates A as an abelian group. Then the induced map A + 0 → A is an isomorphism. Proof. It suffices to verify the universal property. Thus let B be an abelian group. If f : A → B is a homomorphism and a ∈ A, then there exist a 1 , a 2 ∈ A 0 with a = a 1 − a 2 . Consequently f (a) = f (a 1 ) − f (a 2 ) and Hom(A, B) → Hom(A 0 , B) is injective. To prove that Hom(A, B) → Hom (A 0 , B) is surjective, let f 0 ∈ Hom(A 0 , B). Given a ∈ A, pick a 1 , a 2 ∈ A 0 with a = a 1 − a 2 , and put f (a) = f 0 (a 1 ) − f 0 (a 2 ). I claim that this is independent of the choices.
, which implies the claim. From this it easily follows that f ∈ Hom(A, B). This concludes the proof.
We now investigate the localization of Tambara functors. Proof. Via Grothendieck's Galois theory, we reduce to the analogous statement for TNR-functors for some finite group G. It is known that this category is symmetric monoidal, with initial object the Burnside ring functor A. It is thus enough to prove this result for A, which is done in [7,Lemma 12.9]. Proof. We wish to make M [1/n] into a Tambara functor by defining N (x/n k ) = N (x)/N (n) k , and tr(x/n k ) = tr(x)/n k , and similarly for pullback. Since transfer and pullback are additive, it is clear that they extend as stated; the formula for the norm is well-defined by Lemma 26. In order to check that this is a Tambara functor, the only difficulty is to check that the distributivity law remains valid. Let f : X → Y ∈ FEt k . Note that that for a ∈ M (X), b ∈ M (Y ) we have tr f (a/n k · f * (b/n l )) = tr f (a · f * (b))/n k+l = tr f (a)b/n k+l = tr f (a/n k )b/n l by definition and Lemma 22, i.e. the projection formula still holds for M [1/n]. Now let A q − → X f − → Y ∈ FEt k generate an exponential diagram, and x ∈ M (A). We have N f (tr q (x/n k )) = N f (tr q (x)/n k ) = N f (tr q (x))/N f (n k ). Since M satisfies the distributivity law, this is the same as (tr R f (q) N p e * (x))/N f (n k ). Since M [1/n] satisfies the projection formula (as noted above), it remains to show that N p (n k ) = R f (q) * N f (n k ). This follows from the base change formula.
It is clear that the canonical map M → M [1/n] is a morphism of Tambara functors, which is the initial morphism to an object of T naive gc (k) [1/n]. This concludes the proof.
Our main reason for studying naive motivic Tambara functors is that they can be obtained by restriction from motivic Tambara functors of the first kind. Indeed let k be a field again and M ∈ T 1 C (k). Let X ∈ C. Then FEt X ⊂ C and by restriction we obtain M | FEt X ∈ T naive (X). This observation allows us to reduce the following Corollary to results about naive motivic Tambara functors.
Corollary 28. Let C ⊂ fét Sm k (where k is again a perfect field), and assume that C is closed under passing to summands (i.e. if X Y ∈ C then also X ∈ C). For n > 0 denote by T 1 C (k)[1/n] the full subcategory on those M such that U 1 M ∈ HI 0 (k) [1/n]. Then the inclusion T 1 Proof. As in the proof of Corollary 27, the only difficulty is in extending the norms to M [1/n] and checking the distributivity law. We need to prove that if f : Since M is a sheaf and C is closed under passing to summands, we may assume that Y is connected. In this case the result follows from Lemma 26 applied to M | FEt Y . Now to verify the distributivity law, we may again restrict to Y connected, whence this follows from Corollary 27, again applied to M | FEt Y .

Effective homotopy modules and sheaves with generalized transfers
In this section we provide a more explicit description of the category HI 0 (k) of effective homotopy modules. A similar result (in the non-effective case) was obtained by different means in [1,Theorem 9.11]. We begin with some abstract preparation.
Proof. The t-structure is unique if it exists, since U is conservative. We show existence. Since the t-structure on C is accessible, C ≥0 is presentable, and hence there exists a set of objects P ⊂ C ≥0 generating C ≥0 under colimits. Denote by F : C → A-Mod the left adjoint of U and write A-Mod ≥0 for the full subcategory of A-Mod generated under colimits and extensions by F P . Then A-Mod ≥0 is the non-negative part of an accessible t-structure on A-Mod [20,Proposition 1.4.4.11]. It remains to show that U is t-exact. By construction F is right t-exact, so U is left t-exact. We thus need to show that U (A-Mod ≥0 ) ⊂ C ≥0 . Since U preserves colimits [20,Corollary 4.2.3.7] and extensions, for this it is enough to show that U F P ⊂ C ≥0 . But for X ∈ P we have U F X = X ⊗ A ∈ C ≥0 by assumption. This proves (1).
To prove (2), consider the induced adjunction F ♥ : C ♥ ⇆ A-Mod ♥ : U ♥ . Since U is t-exact and conservative, we need only show that for X ∈ C ♥ the canonical map X → U F ♥ X = (U F X) ≤0 is an equivalence. We have the triangle By assumption, X ⊗ A >0 ∈ C >0 and hence (U F X) ≤0 ≃ (X ⊗ A ≤0 ) ≤0 . But by assumption A ≤0 ≃ ½ ≤0 , and so reversing the steps with ½ in place of A we similarly find that (X ⊗ ½ ≤0 ) ≤0 ≃ (X ⊗ ½) ≤0 ≃ X.
This concludes the proof.
We recall also the following well-known result.
Lemma 30. Let F : C → D be a symmetric monoidal functor of stable, compact-rigidly generated, presentably symmetric monoidal ∞-categories. Assume that F preserves colimits and has dense image. Then F has a lax symmetric monoidal right adjoint U , so U (½ D ) ∈ CAlg(C), and U induces an equivalence D ≃ U (½ D )-Mod.
We note that if F : C → D is a symmetric monoidal functor between stable, presentably symmetric monoidal ∞-categories which preserves colimits and has dense image, and C is compact-rigidly generated, then D is compact-rigidly generated as soon as ½ D is compact.
Proof. The existence of U follows from the adjoint functor theorem, and the factorization D → U (½ D )-Mod is also obtained by abstract nonsense [22,Construction 5.23]. Note that U (½ D )-Mod satisfies the same assumptions as C. In other words we may assume that U ½ ≃ ½. Now apply [4,Lemma 21].
Recall the category of presheaves with generalized transfers [11,Section 4]. We write HI(k) for the category of homotopy invariant Nisnevich sheaves with generalized transfers. Recall also the canonical equivalence HI(k) ≃ DM is an equivalence.
Proof. We first prove that M ♥ : SH(k) ♥ ⇆ DM(k) ♥ : U ♥ is an adjoint equivalence. Since U ♥ is conservative, it suffices to show that for E ∈ SH(k) ♥ the canonical map α : E → U ♥ M ♥ E is an equivalence. Since 2 = e, it suffices to prove that α[1/2] and α[1/e] are equivalences.
The adjunction i : SH(k) eff ⇆ SH(k) : f 0 induces i ♥ : SH(k) eff♥ ⇆ SH(k) ♥ : f ♥ 0 , and i ♥ is fully faithful [2, Proposition 5(2)]. The same argument applies to DM . The diagram commutes (by definition), and hence so does the induced diagram It follows that M ♥ maps the full subcategory SH(k) eff♥ of SH(k) ♥ into the full subcategory DM Remark 32. Note that we do not claim that the inverse of M eff♥ : SH(k) eff♥ → DM eff (k) ♥ is given by U eff♥ . Indeed I do not not know if U ( DM eff (k)) ⊂ SH(k) eff . This is true after inverting the exponential characteristic, by [6, Corollary 5.1 and Lemma 5.3].
Remark 33. I am confident that this result is true even if k is finite. The most natural way to prove this would be to extend the results about DM(k) to finite fields. I am confident that this can be done using the methods of [15,Appendix B], but this would take us too far afield. In the sequel we will treat finite fields by using an alternative description of HI 0 (k) in terms of framed transfers [8,Theorem 5.14].
The following corollary is the main reason we need the above result. It allows us to write down generators for the abelian group (EX)(K).
Corollary 34. Let k be a perfect field of exponential characteristic e = 2.
Let K/k be a field extension and X ∈ Sm k and assume that K is perfect. Then (EX)(K) is generated as an abelian group by expressions of the form tr f g * id X , where id X ∈ (EX)(X) corresponds to the identity morphism, f : Spec(L) → Spec(K) is a finite (hence separable) field extension, and g : Spec(L) → X is any morphism.
We shall give two proofs: one assuming that k is infinite and relying on Theorem 31, and another one that works in general.
Proof assuming k infinite. Since our categories are additive, we may assume that X is connected.
By Theorem 31, it is enough to prove the claim for h 0 ( M X), where M : Sm k → DM eff (k) is the canonical functor. By [13, Corollary 3.2.14] we have M X = L N is Sing * a N isc (X). Consequently h 0 ( M X)(K) is a quotient ofc(X)(K) = Hom Cork (K, X). By [11,Example 4.5], up to non-canonical isomorphism we have where (X K ) (0) is the set of closed points. The class α ∈ Hom Cor k (K, X) corresponding to an element a ∈ GW (K(x)) and f : Spec(K(x)) → X is α = tr K(x)/K (u x · a · f * id X ), where u x ∈ GW (K(x)) is some unit reflecting the non-canonicity of the above isomorphism.
Since char(k) = 2, GW (K(x)) is generated as an abelian group by elements of the form tr L/K(x) (1) with L/K(x) finite separable [5, paragraph before Proposition 22]. It follows from this and the base change formula [3, Proposition 10] that h 0 ( M X) is generated by elements of the form as needed. This concludes the proof.
Proof for general k. We use [8,Theorem 5.14], which tells us that HI 0 (k) is equivalent to the category of homotopy invariant, "stable" sheaves with "equationally framed transfers". The main upshot for us is that (EX)(K) is generated by elements of the form α * (id X ), where α : Spec(K) X is a "framed correspondence". This consists, among other things, of a scheme Z finite over K and a map α ′ : Z → X.
Let Spec(L) = Z red , and write g for the composite Z red ֒→ Z α ′ − → X. Then by [8,Lemma 5.16], we have α * (id X ) = tr L/K (c α g * id X ), where c α ∈ GW (L) is a certain class determined by α.
The rest of the proof proceeds as before.
Remark 35. The only reason above to assume that K is perfect is that then the finite extension f : Spec(L) → Spec(K) is automaticallyétale, and hence we have a transfer morphism as discussed previously. In fact, as long as char(k) = 2, for any finite (but not necessarily separable) field extension, and any homotopy module M , there exist the cohomological transfer morphism tr f : M (L) → M (K) [26,Section 4.3]; it coincides with the previous transfer if f isétale [11,Lemma 2.3]. The above corollary remains true as stated for imperfect L, provided that all finite (not necessarily separable) extensions are considered, and tr f denotes the cohomological transfer.

Normed effective homotopy modules I: construction and basic properties
In this section we construct a final category of motivic Tambara functors, this time as a category of normed spectra.
Proof. The claims about f * and f ⊗ are already implicit in the existence of SH veff⊗ . Since f # preserves colimits (and hence extensions), it is enough to show that f # Σ ∞ + U ∈ SH(X) veff for U ∈ Sm Y , which is clear.
Proof. The statements about f * and f ⊗ are already implicit in the existence of the natural transformation τ eff ≤0 of functors on Span(Sch, all, fét).
The functor f # : SH(X) veff → SH(Y ) veff preserves the subcategory SH(•) eff ≥1 . By adjunction it follows that f * : SH(Y ) veff → SH(X) veff preserves SH(•) eff♥ . From this it is easy to check directly that the composite Definition 38. Let C ⊂ fét Sch S . We denote the full subcategory of NAlg C (SH) consisting of those normed spectra with underlying spectrum in SH(S) eff♥ by NAlg C (SH(S) eff♥ ) and call it the category of C-normed effective homotopy modules. If S = Spec(k) is the spectrum of a field, so SH(S) eff♥ ≃ HI 0 (k), then we also denote NAlg C (SH(S) eff♥ ) by NAlg C (HI 0 (k)).
If C = SmQP S , U has a left adjoint F which satisfies Here the colimit is over the source of the cartesian fibration classified by SmQP op S → S, X → FEt ≃ X . Moreover for (f : X → S, p : Y → X) in the indexing category, the canonical map τ eff where the first map is induced by the unit map E → U F E, the second map is induced by the multiplication p ⊗ (F E) Y → (F E) X , and the third map is a co-unit map.

≤0
preserves colimits and commutes with f # p ⊗ g * by Proposition 37, the result follows.
Remark 40. Under the conditions of Lemma 39, the category NAlg C (SH(S) eff♥ ) is monadic over the 1-category SH(S) eff♥ , and hence a 1-category.

Normed effective homotopy modules II: main theorem
In this section we put everything together: we show that NAlg C (HI 0 (k)) is equivalent T 2 C (k), for an appropriate C. In fact from now on, we set C = SmQP k , V = SmQP and suppress both from the notation. In particular we have T 2 (k) ≃ T 1 (k), by Proposition 20.
Note that the following diagram commutes From now on, let e be the exponential characteristic of k. Write NAlg(HI 0 (k))[1/e] for the full subcategory on those M ∈ NAlg(HI 0 (k)) such that U M ∈ HI 0 (k) [ is an isomorphism.
Proof. Preliminary remarks. By Lemma 39 we have Note that EX = τ eff ≤0 Σ ∞ + X. By Lemma 37, f # , p ⊗ , g * commute with the localization functor τ eff ≤0 . Consequently we find that U F EX = colim and hence From now on, we will suppress [1/e] from the notation; it should be understood that all homotopy modules in sight are in HI 0 (k)[1/e], and if not should be replaced by (?)[1/e]. We will also write X Y for X × k Y , particularly when viewed as a Y -scheme, and similarly for other pairs of schemes. It follows that a morphism of effective homotopy modules α ∈ [U F EX, A] consists of the following data: • For each S ∈ SmQP k and each finiteétale morphism p : Y → S, a class α p ∈ A(R p (X Y )), subject to the following compatibility condition: • For every cartesian square in SmQP k with p 1 (and hence p 2 ) finiteétale, denote by q : R p1 (X Y1 ) → R p2 (X Y2 ) the canonical map induced by k. Then q * (α p2 ) = α p1 ∈ A(R p1 (X Y1 )). We have in particular the class α 1 := α id k ∈ A(X). Injectivity of u. Let us first show that if α : U F EX → A is indeed a morphism in T 2 (k), then the classes α p are all determined by α 1 . In other words, we will show that u is injective. To do this, let for S ∈ SmQP k and p : Y → S finiteétale, a p ∈ (U F EX)(R p (X Y )) denote the class corresponding to the canonical map ER p (X Y ) → U F EX coming from the colimit formula. Then α p = α(a p ) ∈ A(R p (X Y )).
Let r Y : X Y → X denote the canonical projection. It follows from the "moreover" part of Lemma 39 that (5) N p,XY (r * Y (a 1 )) = a p . Thus α p = α(a p ) = α(N p,XY (r * Y (a 1 ))) = N p,XY r * Y α(a 1 ) = N p,XY r * Y α 1 , since α was assumed to be a morphism of Tambara functors. This proves that u is injective.
What remains to be done is to show that this is a morphism of Tambara functors; then u will be surjective. By Lemma 10 (and Proposition 20), it is enough to prove that if K/k is the perfect closure of a finitely generated field extension, and p : Spec(L) → Spec(K) is finiteétale, then the following square commutes We shall do this as follows. Let, for each L ∈ FEt K , (F EX) 0 (L) ⊂ (F EX)(L) denote the subset of those elements obtained by iterated application of norm, restriction and transfer (all along finiteétale morphisms) from elements b ∈ (F EX)(L ′ ), corresponding to maps of schemes b ′ : Spec(L ′ ) → X; in other words b = b ′ * (a 1 ). We shall prove the following: (a) If s 1 ∈ (F EX) 0 (L 1 ) and s 2 ∈ (F EX) 0 (L 2 ), then (s 1 , s 2 ) ∈ (F EX) 0 (L 1 L 2 ) (for any L 1 , L 2 ∈ FEt K ). (b) The Z[1/e]-module (F EX)(L) is generated by (F EX) 0 (L) (for any L ∈ FEt K ). (c) For any p : L → K ∈ FEt K and a ∈ (F EX) 0 (L), we have N p (α(a)) = α(N p (a)).
Let F EX| FEt K ∈ T naive gc (k)[1/e] denote the induced naive motivic Tambara functor. By (a), the subfunctor (F EX) 0 ⊂ F EX| FEtK preserves finite products, and hence (F EX) 0 ∈ T naive (k). By (b), the canonical map (F EX) + 0 [1/e] → F EX| FEt K is an isomorphism (e.g. use Lemma 25). By (c), the composite (F EX) 0 → F EX| FEt K α − → A| FEt K is a morphism of naive motivic Tambara functors. It follows that the unique induced map (F EX) + 0 [1/e] ≃ F EX| FEtK → A| FEtK compatible with the Z[1/e]module structures, is a morphism of naive motivic Tambara functors. Since α|FEt K is compatible with the Z[1/e]-module structures (α being a morphism of homotopy modules), it must be this unique map, and hence a morphism of naive motivic Tambara functors. This proves that square (6) commutes. This concludes the proof, modulo establishing (a)-(c).
Proof of (a). We may assume that s 1 is obtained by a sequence of operations O . This is clear. Proof of (b). The Z[1/e]-module ER p X Y (L) is generated by transfers of pullbacks of a p , by Corollary 34. Consequently F EX(L) is generated as a Z[1/e]-module by transfers and norms of pullbacks of a 1 , by (5). This was to be shown.
Proof of (c). Let us call a section s ∈ (F EX)(L) good if for any span Spec(K ′ ) p ′ ← − Spec(L ′ ) f − → Spec(L) with p ′ finiteétale, we have α(N p ′ f * (s)) = N p ′ f * α(s). We need to show that all sections of (F EX) 0 are good. We shall prove that good sections are closed under norms, transfer and pullback (steps (i)-(iii) below), and that sections of the form b * a 1 are good (step (iv)). This implies the desired result.
Step (i). Suppose s is good and f : Spec(L ′ ) → Spec(L) is arbitrary. Then f * s is good. This follows from transitivity of pullback.
Step (ii). Suppose b : Spec(L ′ ) → Spec(L) is finiteétale and s ∈ (F EX)(L ′ ) is good. Then N b (s) is good. Indeed by the base change formula, we may assume that f = id and p ′ = p. Then N p (N b (s)) = N p•b (s) and so the relevant equality holds by assumption.
Step (iii). Suppose b : Spec(L ′ ) → Spec(L) is finiteétale and s ∈ (F EX)(L ′ ) is good. Then tr b (s) is good. By the base change formula, we may assume that f = id and p ′ = p. Now N p tr b (s) = tr ? N ? e * (s), by the distributivity law. Since N ? e * (s) is good by steps (i) and (ii), and any morphism of homotopy modules commutes with transfers, tr b (s) is indeed good.
Step (iv). Suppose that s corresponds to a map of schemes b : Spec(L) → X. Then s is good. Note that f * (s) corresponds to Spec(L ′ ) → Spec(L) → X, so we may assume that f = id and p ′ = p. Note that s = b * a 1 . Letb : Spec(L) → X L be induced by b, and write b ′ : Spec(K) → R p (X L ) for the Weil restriction ofb along p. We shall use the following observation, proved below: If B ∈ T 2 (k) is arbitrary and t ∈ B(X), then (7) N p b * t = b ′ * N p,XL r * L (t). Thus α(N p b * a 1 ) = α(b ′ * N p,XL r * L a 1 ) = b ′ * α(a p ) = b ′ * α p , using (7) with t = a 1 and (5). On the other hand N p (α(b * a 1 )) = N p b * α 1 = b ′ * N p,XL r * L α 1 = b ′ * α p as well, using (7) with t = α 1 , and (5) again. Hence s is good.
Proof of (6). Apply condition (2) of Definition 17 to the diagram The result follows since R p (b) = b ′ and the composite Lb − → X L rL −→ X is just b.
Proof. The functors U 2 and U are right adjoints that preserve sifted, hence filtered, colimits, and are conservative. See Lemmas 39 and 5, and Corollary 12. It follows that ρ preserves filtered colimits and (small) limits. Consequently ρ is an accessible functor which preserve limits. It follows that it has a left adjoint [19,Corollary 5.5.2.9]. Denote the left adjoint of U 2 by F 2 : HI 0 (k) → T 2 (k), and the left adjoint of ρ by δ : T 2 (k) → NAlg(HI 0 (k)). By the Barr-Beck-Lurie Theorem 3 , both functors U 2 and U are monadic [20,Theorem 4.7.4.5]. Let M 2 = U 2 F 2 and M = U F denote the corresponding monads. We obtain a morphism of monads α : M 2 = U 2 F 2 ⇒ U 2 ρδF 2 ≃ U F = M . If X ∈ SmQP k then F Since α preserves sifted colimits and is an equivalence on the generators, it is an equivalence in general. This concludes the proof.
We have used the following well-known result.
Lemma 44. Let C be an ∞-category with (small) colimits and S a (small) set of objects closed under finite coproducts. The subcategory of C generated by S under sifted colimits coincides with the subcategory of C generated by S under all (small) colimits.
Proof. Let D ⊂ C be the subcategory generated by S under sifted colimits. It suffices to show that D is closed under finite coproducts [19,Lemma 5.5.8.13]. Since ∅ ∈ S ⊂ D, it suffices to consider binary coproducts. For E ∈ D write D E ⊂ D for the subcategory of those D ∈ D with E D ∈ D. Since sifted simplicial sets are contractible, D E is closed under sifted colimits. Let X ∈ S. Then S ⊂ D X , and so D = D X . In other words, for E ∈ D and X ∈ S we have E X ∈ D. Thus for E ∈ D arbitrary we have S ⊂ D E . It follows again that D = D E and so D is closed under binary coproducts, as desired. 3 In light of Remark 40 we are dealing with 1-categories only, so we are really just using the classical monadicity theorem [21,Section VI.7].
Remark 46. Throughout this section we put C = SmQP k and V = SmQP. We cannot reasonably hope to change V. However, C = SmQP k was mainly used as a simplifying assumption. It implies that T 2 C (k) ≃ T 1 C (k). This latter category is somewhat easier to study, and so we were able to deduce somewhat cheaply that T 2 C (k) is presentable, and so on. I contend that all the results about T 2 C remain valid for more general C, such as C = FEt k (except of course that in general T 2 C (k) ≃ T 1 C (k)), and that the same is true for Corollary 43 Remark 47. We were forced to invert the exponential characteristic e essentially because we needed to know that (EX)(K) is generated by transfers along finiteétale morphisms, in some sense. The proof shows that in general, (EX)(K) is generated by transfers along finite, but not necessarilyétale, morphisms (see Remark 35). Inverting e allows us to restrict to K perfect, where these two classes of morphisms coincide.