Correspondences and stable homotopy theory

A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra SH$SH$ is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another application recovers stable motivic homotopy theory SH(k)$SH(k)$ from spectral modules over associated spectral categories.


INTRODUCTION
Algebraic Kasparov K-theory is stable homotopy theory of non-unital k-algebras Alg k [9,10].In detail, we start with the category U • (Alg k ) of pointed simplicial functors from Alg k to pointed simplicial sets, where each algebra A ∈ Alg k is regarded as the representable object rA = Hom Alg k (A, −).The category U • (Alg k ) comes equipped with a motivic model structure.Let S 1 be the standard simplicial circle.Stabilization of U • (Alg k ) in the S 1 -direction leads to a stable motivic model category Sp S 1 (Alg k ) of S 1 -spectra in U • (Alg k ).The S 1 -suspension spectrum Σ ∞ S 1 rA of an algebra A is computed as the fibrant spectrum K(A, −) [10], where K(A, B) is the algebraic Kasparov KK-theory spectrum of A, B ∈ Alg k defined in [9].A key non-unital homomorphism involved in the computation is σ A : JA → ΩA, where JA = Ker(TA → A) with TA = A ⊕ A ⊗2 ⊕ • • • the algebraic tensor algebra and ΩA = (x 2 − x)A [x].The morphism r(σ A ) is a motivic equivalence in U • (Alg k ).
The computations of Σ ∞ S 1 rA and Σ ∞ P 1 X + share lots of common properties [11].Inspired by these computations, two categorical constructions are introduced in this paper.The first one produces correspondences associated with two objects P, T ∈ C and ring objects of the category of symmetric sequences C Σ , where C is a symmetric monoidal category with finite colimits and zero object 0. The correspondences are constructed between objects of an arbitrary full subcategory B of C closed under monoidal product.See Theorem 2.4 for details.After Voevodsky, correspondences play a prominent role in motivic homotopy theory.In particular, they are necessary for computing motivic homotopy types as well as for producing triangulated categories of motives.For example, Voevodsky's fundamental graded category of framed correspondences Fr * (k) is recovered from Theorem 2.4 if we take C = M , B = {X + | X ∈ Sm/k}, P = (P 1 , ∞), T = A 1 /(A 1 − {0}), and the commutative ring object (S 0 , T, T 2 , . ..) in M Σ .Next, if σ : P → T is a morphism in C then the second categorical construction produces spectral categories, i.e. categories enriched over symmetric S 1 -spectra Sp Σ S 1 , which are used for applications mentioned below.See Theorem 5.2 for details.Spectral categories are of great utility in classical and equivariant stable homotopy theory (see, e.g., [21,33]) as well as in constructing triangulated categories of K-motives [16,17].
The spectral categories and symmetric spectra constructed in this paper lead to the following applications.We first introduce the stable homotopy category SH k over an arbitrary field k in Section 4. It is defined as the homotopy category of S k -modules, where S k is a commutative symmetric ring spectrum defined over k.Then one reconstructs in Theorem 4.15 the stable homotopy theory of S 1 -spectra SH as SH k if k is algebraically closed (we need to invert the exponential characteristic).Moreover, this reconstruction is given by a functor taking a symmetric S 1 -spectrum N to its symmetric framed motive M Σ f r (N) introduced in this paper (see Definition 4.9).Another application gives yet another genuinely local model of stable motivic homotopy theory SH(k) (in addition to [20]) and, more generally, a local model for the category of E-modules in SH (k), where E is a symmetric Thom ring spectrum.See Theorem 6.7 and Corollary 6.9 for details.For the latter result, we apply Theorem 5.2 to produce a spectral category O E ∆ using data as above: C = M , B = {X + | X ∈ Sm/k}, P = (P 1 , ∞), T = A 1 /(A 1 − {0}).We also use the enriched motivic homotopy theory of motivic spectral categories developed in [16,17].The reader will also find reconstruction theorems for E-modules of SH(k) in terms of ∞-categories of "tangentially framed corrrespondences" in [7,8].The approach presented in Section 6 is combinatorial in the sense that it is based on explicit spectral categories produced by Theorem 5.2 and modules over them defined in terms of original Voevodsky's framed correspondences [35].This approach also produces triangulated categories of E-framed motives out of spectral categories of Theorem 5.2.They are constructed in a similar fashion as the classical Voevodsky category of motives or the category of K-motives in the sense of [16,17].
The author also expects further applications of (spectral) categories of correspondences, constructed in this paper for quite general categories, in classical and algebraic Kasparov K-theory as well as in non-commutative algebraic geometry.This will be the material of subsequent papers.In this paper he has concentrated on applications in classical and motivic stable homotopy theory.
The author thanks the anonymous referee for helpful comments.
Notation.Throughout the paper we employ the following notation.Let (C , ∧, ∨, S) be a symmetric monoidal category with finite coproducts, unit object S and zero object 0. We assume that a canonical morphism In what follows we shall also assume that C has finite colimits.By [22,Section 7] the category of symmetric sequences C Σ is symmetric monoidal with The symmetric sequence (S, 0, 0, . ..) is a monoidal unit of C Σ .This notation needs some explanation (we follow [22,Section 7]).Given a finite set Γ and an object A ∈ C , Γ × A is the coproduct of |Γ| copies of A. If Γ is a group, then Γ × A has an obvious left Γ-action; Γ × A is the free Γ-object on A. Note that a Γ-action on A is then equivalent to a map Γ × A → A satisfying the usual unit and associativity conditions.Also, if Γ admits a right action by a group Γ ′ , and A is a left Γ ′ -object, then we can form Γ × Γ ′ A as the colimit of the Γ ′ -action on Γ × A, where α ∈ Γ ′ takes the copy of A corresponding to β ∈ Γ to the copy of A corresponding to β α −1 by the action of α.
Given two maps f : X → X ′ and g : is the natural map taking the summand (α, X p ∧Y q ) to the summand (α χ q,p ,Y q ∧ X p ) for α ∈ Σ p+q , where χ q,p ∈ Σ p+q is the (q, p)- shuffle given by χ q,p (i) = i + p for 1 i q and χ q,p (i) = i − q for q < i p + q.It is worth noting that the map defined without the shuffle permutation is not a map of symmetric sequences.

Definition (Ring objects).
In what follows we shall refer to monoid objects in C Σ as ring objects.There is a standard description of a ring object E in C Σ which we need later: ⋄ a sequence of objects E n ∈ C for n 0; ⋄ a left action of the symmetric group Σ n on E n for each n 0; 1 Such a category C is also known as a distributive symmetric monoidal category.
This data is subject to the following conditions: (Associativity) The square commutes for all n, m, p 0. (Unit) The two composites are the identity for all n 0. A morphism f : E → E ′ of ring objects consists of Σ n -equivariant maps f n : E n → E ′ n for n 0, which are compatible with the multiplication and unit maps in the sense that f n+m • µ n,m = µ n,m • ( f n ∧ f m ) for all n, m 0, and f 0 • ι 0 = ι 0 .
A ring object E is commutative if the square / / E m+n commutes for all n, m 0.

Definition (Modules).
A right module M over a ring object E ∈ C Σ is defined in a standard way.There is an equivalent definition which we need later: ⋄ a sequence of objects M n ∈ C for n 0 ⋄ a left action of the symmetric group Σ n on M n for each n 0, and The action maps have to be associative and unital in the sense that the following diagrams commute for all n, m, p 0. A morphism f : M → N of right E-modules consists of Σ n -equivariant maps f n : M n → N n for n 0, which are compatible with the action maps in the sense that ) for all n, m 0. We denote the category of right E-modules by Mod E.
The following definition is motivated by the fundamental graded category of Voevodsky's framed correspondences, in which the role of C is played by the category of pointed motivic spaces M , B is given by pointed motivic spaces of the form X + with X ∈ Sm/k, P is the pointed projective line (P 1 , ∞), and E = (pt + , T, T 2 , . ..) with In what follows we shall tacitly use iterated monoidal products and coherence.
2.3.Definition.Suppose B is a full subcategory of C closed under ∧ and P ∈ Ob C .Let E be a ring object of C Σ .We define the set of (E, P)-correspondences of level n between two objects Compositions are defined by pairings ϕ X,Y,Z .
for each n 0 is given by conjugation.In detail, for each f : With this definition each Corr E n (X ,Y ) becomes a pointed Σ n -set.Here Σ n acts on P ∧n by permutations, using the commutativity and associativity isomorphisms.
Since the multiplication maps µ * , * for E are Σ n × Σ m -equivariant and the diagram If there is no likelihood of confusion, we shall simetimes write (X ∧ P ∧n ,Y ∧ E n ) to denote Hom C (X ∧ P ∧n ,Y ∧ E n ).The "associativity square" is commutative for all n, m, p 0 due to the associativity of the multiplication maps µ * , * for E, and so ϕ X,Y,Z is an associative pairing.
The identity morphism is defined by , where ι 0 : S → E 0 is the unit map.We see that B is enriched over Sets Σ * by means of (E, P)correspondences.
Suppose E is a commutative ring object in C Σ .If there is no likelihood of confusion, we shall simetimes write [X ,Y ] to denote the Sets Σ It sends each quadruple and then to the couple The pairing ψ X,Y X ′ ,Y ′ is plainly Σ p × Σ q -equivariant.We have a commutative diagram 3.1.Definition.(1) Following [22,Definition 7.2] the category of symmetric spectra Sp Σ (C , T ) is the category of modules in C Σ over the commutative monoid Sym(T ) in C .That is, a symmetric spectrum X is a sequence of Σ n -objects X n ∈ C and Σ n -equivariant maps X n ∧ T → X n+1 , such that the composite equivariant for all n, p 0. A map of symmetric spectra is a collection of Σ n -equivariant maps X n → Y n compatible with the structure maps of X and Y .
(2) A symmetric ring T -spectrum is a ring spectrum E ∈ C Σ such that there is another unit map ι 1 : T → E 1 subject to the following condition: (Centrality) The diagram (3) A right module M over a symmetric ring T -spectrum E is a symmetric right T -spectrum which is also a right E-module in the sense of Definition 2.2.We denote the category of right E-modules by Mod E (its morphisms are morphisms of symmetric T -spectra satisfying Definition 2.2).
Because Sym(T ) is a commutative monoid, the category Sp Σ (C , T ) is a symmetric monoidal category, with Sym(T ) itself as the unit.We denote the monoidal structure by X Y = X ∧ Sym(T ) Y , where X ∧ Sym(T ) Y is defined similarly to [32, p. 499] as the coequalizer, in C Σ , of the two maps / / X ∧Y induced by the actions of Sym(T ) on X and Y respectively.Given X ∈ C and a pointed set (K, * ), we shall write X ∧ K to denote K\ * X .Let P ∈ C and σ : P → T be a morphism in C .Denote by This notation is inherited from the standard notation for pointed motivic spaces T n and P ∧n associated with pointed motivic spaces T = A 1 /(A 1 −{0}) and (P 1 , ∞), where σ : (P 1 , ∞) → T is the canonical motivic equivalence given by the level 1 framed correspondence In what follows S 1 is the standard simplicial circle with n-simplicies being n + = {0, 1, . . ., n} and Here each E n ∧ S n is performed in every degree to produce a simplicial object in C and a simplicial Hom-set.We also call 1 E the PT -spectrum of E. Each simplicial Hom-set is pointed at the zero morphism.Each structure map coincides termwise with the natural morphisms where coproducts are indexed by non-basepoint elements of S 1 n = n + = {0, 1, . . ., n}.They take an element f : P ∧n → E n ∧ S n of the kth summand to the composition Here ι k is the inclusion into the kth summand.If E = Sym(T ) then 1 E will be denoted by 1 Σ .Though the author was unable to find the following result in the literature, he does not pretend to originality.

3.2.
Theorem.Given a symmetric right T -spectrum E ∈ Sp Σ (C , T ), the following statements are true: (1) the spectrum Proof.(1).We follow [31, §I.1] to verify the relevant conditions for symmetric S 1 -spectra.The left action of the symmetric group Σ n on Hom C (P ∧n , E n ∧ S n ) for each n 0 is given by conjugation.In detail, for each f : P ∧n → E n ∧ S n and each τ ∈ Σ n the morphism τ • f is defined as the composition With this definition each Hom C (P ∧n , E n ∧ S n ) becomes a Σ n -simplicial set.Here Σ n acts on P ∧n and S n by permutations.
(2).Suppose E is a symmetric ring T -spectrum.Define multiplication maps where µ n,m is the multiplication map of The map ν n,m is Σ n × Σ m -equivariant.Indeed, this follows from commutativity of the diagram Clearly, the "associativity square" is commutative for all n, m, p 0 due to associativity of the multiplication maps µ * , * for E. Next, two unit maps are defined as follows.Let ι 0 : S → E 0 and ι 1 : T → E 1 be the unit maps for the symmetric ring T - spectrum E. Then i 0 takes the unbased point of S 0 to ι 0 and i 1 coincides termwise with the natural morphisms taking an unbased simplex j of S 1 ℓ = {0, 1, . . ., ℓ} to the composite map where the right arrow is the inclusion into the jth summand.The two "unit composites" are the identity for all n 0 due to the same properties for E.
Furthermore, the "centrality diagram" commutes for all n 0 due to the same properties for E.Here χ n,m ∈ Σ n+m denotes the shuffle permutation which moves the first n elements past the last m elements, keeping each of the two blocks in order.It follows that 1 E is a symmetric ring S 1 -spectrum.Suppose E is commutative.Then the square commutes for all n, m 0. We also use here commutativity of the diagram for all f : P ∧n → E n ∧ S n and g : Proof.This is straightforward.

RECONSTRUCTING STABLE HOMOTOPY THEORY SH
We refer the reader to [28] for basic facts on compactly generated triangulated categories.Below we will often use the following lemma.4.1.Lemma (see [13]).Let S and T be compactly generated triangulated categories.Suppose there exists a set of compact generators Σ in S and a triangulated functor F : S → T that preserves direct sums such that 1. the collection F(X )|X ∈ Σ is a set of compact generators in T , 2. for any X ,Y in Σ, the induced map is an isomorphism for all n ∈ Z.
Then F is an equivalence of triangulated categories.
Let Sp S 1 ,G (k) denote the category of symmetric (S 1 , G)-bispectra associated with the closed symmetric monoidal category of pointed motivic spaces M (see our notation on p. 3).Here the G-direction corresponds to the pointed motivic space G, which is the mapping cone in M of the map 1 : pt + → (G m ) + .The category Sp S 1 ,G (k) is equipped with a stable motivic model category structure [24].Denote by SH(k) its homotopy category.If E is a bispectrum and p, q are integers, recall that π p,q (E ) is the Nisnevich sheaf of bigraded stable homotopy groups associated to the presheaf A map of bispectra f : E → E ′ is a stable motivic equivalence if and only if π * , * ( f ) is an isomor- phism.The category SH(k) has a closed symmetric monoidal structure with monoidal unit being the motivic sphere bispectrum [24] for details).The following theorem was proven by Levine [25] for algebraically closed fields of characteristic zero, with embedding k ֒→ C and extended by Wilson-Østvaer [37, Corollaries 1.2 and 6.5] to arbitrary algebraically closed fields.4.2.Theorem.Let k be an algebraically closed field of exponential characteristic e and S be the sphere spectrum Σ ∞ S 1 S 0 .For all n 0 the homomorphism Lc : is an isomorphism, where c : SH → SH( k) is the functor induced by the functor c : S • → M sending pointed simplicial sets to constant motivic spaces.
The following statement was proven by Zargar [38,  Recall from [19] that one of the equivalent ways to define Voevodsky's framed correspondences of level n 0 between smooth k-schemes X ,Y ∈ Sm/k is as follows: 4.4.Remark.Due to Voevodsky's lemma [19,Section 3] there is an equivalent description of the sets Fr n (X ,Y ) in terms of explicit geometric data.In detail, each element in that description is the equivalence class of a quadruple (Z,U, ϕ, g), where Z is a closed subset of A n X which is finite over X , U is an etale neighborhood of Z in A n X , ϕ = (ϕ 1 , . . . ,ϕ n ) is a collection of regular functions on U such that ∩ n i=1 {ϕ i = 0} = Z, and g is a morphism from U to Y .The equivalence relation on such quadruples depends on the choice of the neighborhood U .If the base field is C and X = Y = pt, the Hom-sets Hom M (P ∧m , T n ), m, n 0, can also be described in terms of holomorphic framed correspondences -see [12].These are equivalence classes of triples (Z,U, f ), where Z consists of finitely many points in . The latter description essentially follows from the Implicit Function Theorem in Complex Analysis.
Following notation of [19] one sets for any finite pointed set K, There is a distinguished framed correspondence σ : (P 1 , ∞) → T in Fr 1 (pt, pt) associated with the triple ({0}, A 1 ,t).The external smash product by σ gives rise to a map Fr By the Additivity Theorem of [19] the assignment gives rise to a special Γ-space of pointed simplicial sets.Its Segal S 1 -spectrum and is called the framed motive of Y evaluated at X .
The following theorem was proven by Garkusha and Panin [19] for algebraically closed fields of characteristic zero, with embedding k ֒→ C. Proof.The proof literally repeats that of [19,Theorem 11.9] if we use Corollary 4.3.4.6.Definition.Given a field k, denote by 1 k := (Fr 0 (pt, pt), Fr 1 (pt, S 1 ), Fr 2 (pt, S 2 ), . ..), pt = Spec(k), the right S 1 -spectrum of pointed simplicial sets with structure maps defined as The sphere spectrum over a field k is the S 1 -spectrum of pointed simplicial sets

.).
The simplicial set Fr m (pt, S m ) can be regarded as the constant simplicial pointed set S m together with the "coefficients set" Fr m (pt, pt) attached to it.Hence the S 1 -spectrum 1 k can be regarded as a graded "tensor algebra" associated to S 1 with "Fr * (pt, pt)-coefficients".In other words, we add Fr * (pt, pt)-coefficients to the classical sphere spectrum S = (S 0 , S 1 , . ..).

4.7.
Proposition.The spectra 1 k , S k are commutative symmetric ring spectra in Sp Σ S 1 .
Proof.We apply Theorem 3.2: C is replaced by the category of pointed motivic spaces M , P (respectively T ) is replaced by P ∧1 := (P 1 , ∞) (respectively by T = A 1 /(A 1 − {0})), E is replaced by the commutative symmetric sphere T -spectrum S k = (S 0 , T, T 2 , . ..).With this notation 1 k = 1 E of Theorem 3.2, and hence a commutative symmetric ring spectrum.
The commutative symmetric ring structure on S k is defined in a similar fashion if we use the cosimplicial diagonal morphism diag : The natural map of ring objects S → S k induces a pair of adjoint functors where U is the forgetful functor and its left adjoint L is the "extension of framed scalars" functor.
Fr Σ * (N) and C * Fr Σ * (N) are symmetric S 1 -spectra for the same reason as 1 E and S k are.Following notation of [15,19] one sets for any finite pointed set K and any integer k 0, The external smash product by σ gives rise to a map Fr n (X , By the Additivity Theorem of [19] the assignment gives rise to a special Γ-space of pointed simplicial sets.The pointed motivic space [15,19].If Y = pt the latter motivic space is denoted by C * Fr(T k ⊗ K). 4.9.Definition.The symmetric framed motive of a symmetric S 1 -spectrum N ∈ Sp Σ S 1 is the symmetric S 1 -spectrum ), . ..) with structure maps and actions of the symmetric groups defined similarly to C * Fr Σ * (N).The framed motive of the suspension spectrum Σ ∞ S 1 X of a pointed simplicial set X will be denoted by M Σ f r (X ).4.10.Remark.If N is the suspension symmetric spectrum Σ ∞ S 1 X of a pointed simplicial set X , the framed motive in the sense of the preceding definition is a bit different from the framed motive of X evaluated at pt defined as 4.11.Lemma.If k is perfect field, then the canonical map of ordinary S 1 -spectra M f r (X ) → M Σ f r (X ) is a level equivalence in positive degrees for any pointed simplicial set X .
Proof.Repeating the proof of [14, Lemma 4.12] word for word, we have that the canonical map of connected spaces The reader should not confuse π * -isomorphisms (i.e.maps inducing isomorphisms of stable homotopy groups) and stable equivalences of symmetric spectra.The first class is a proper subclass of the second.
Though M f r (X ) is canonically a symmetric S 1 -spectrum, where Σ n acts on each space by permuting S n , the point is that it is not a S k -module in contrast with M Σ f r (X ).
e. a stable equiva- lence of ordinary spectra) whenever the base field k is perfect.( 4) for every N ∈ Sp Σ S 1 the canonical map β : is a π * -isomorphism when- ever the base field k is perfect.
Proof.(1).The desired pairing is defined as follows.Given two morphisms (β : ∆ ℓ + ∧ P ∧q → T q ∧ N q ) ∈ Fr q (∆ ℓ , N q ) and (α : It is straightforward to see that this pairing is Σ q × Σ p -equivariant, satisfies associativity and unit conditions, hence it defines the structure of a right S k -module on C * Fr Σ * (N).For the same reasons, . This is straightforward.
(3).Let Θ ∞ S 1 be the naive stabilisation functor of S 1 -spectra.It has the property that X → Θ ∞ S 1 (X ) is a stable equivalence for every S 1 -spectrum X [22, Proposition 4.7].
Let A n,r := C * (P ∧n , T n ∧ S r ), B m,n,r := C * (P ∧(m+n) , T m+n ∧ S r ), m, n, r 0. We have maps of spaces (P ∧r , T r ∧ S r ) / / • • • (we omit C * here and below for brevity).Each structure map B r → Hom(S 1 , B r+1 ) is defined in a similar fashion.
There is a commutative diagram of spectra The maps a, b, c are defined in a canonical way.By the two-out-of-three property a, b are stable equivalences.Therefore α is a stable equivalence if and only if c is.
But c is the infinite composition Each composition A r → B m,r is isomorphic to the canonical map of spaces Repeating the proof of [14, Lemma 4.12] word for word, the map is a weak equivalence for positive r, and hence c is a weak equivalence in positive degrees.(4).The proof literally repeats that of (3) if we replace S r by N r .

Theorem. If k is an algebraically closed field of exponential characteristic e, then the natural maps of symmetric S
Proof.The map β is a π * -isomorphism by Proposition 4.12(4).Consider a commutative diagram We see that a is a π * -isomorphism.Therefore ν is a π * [e −1 ]-isomorphism if and only if κ N is.We claim that the map of S 1 -spectra κ X : Indeed, if X is a finite pointed set regarded as a constant simplicial pointed set, this follows from Theorem 4.5.If X is any pointed set, then κ X is a directed colimit of maps κ W , where W runs over finite pointed sets of X .Hence κ X is a π * [e −1 ]-isomorphism as directed colimits preserve π * [e −1 ]- isomorphisms.Finally, since the geometric realization of a simplicial π * [e −1 ]-isomorphism is an π * [e −1 ]-isomorphism, then so is κ X for an arbitrary pointed simplicial set X as claimed.
Next, every (ordinary) spectrum N ∈ Sp S 1 equals colim i L i (N), where each spectrum We are now in a position to prove the main result of this section saying that the stable homotopy category of classical symmetric spectra can be recovered from the stable homotopy category of framed spectra over an algebraically closed field (after inverting the exponential characteristic).Moreover, the equivalence L is isomorphic to the functor that takes a symmetric S 1 -spectrum N to its symmetric framed motive M Σ f r (N).
Proof.SH = Ho(Sp Σ S 1 ) is compactly generated by the sphere spectrum S .SH k is compactly generated by the framed sphere spectrum S k .By construction, L(S ) = S k .Therefore our statement that (L,U ) is a Quillen equivalence reduces to showing that the composite map of S 1 -spectra is a e −1 -stable equivalence k is a fibrant replacement of S k in Mod S k (we also use Lemma 4.1 here).We claim that S Proposition 4.12(3).It follows from [19,Theorem 4.1] that M Σ f r (S 0 ) is a positively fibrant symmetric Ω-spectrum, and hence ). Theorem 4.14 implies that ϕ is a π * [e −1 ]-isomorphism.
Next, Theorem 4.14 implies that N → M Σ f r (N) induces a functor ) is an isomorphism of graded Abelian groups, hence M Σ f r is an equivalence of compactly generated triangulated categories by Lemma 4.1.By Theorem 4.14 the canonical map of By the first part of the proof L is a quasi-inverse functor to U , and hence L is isomorphic to the functor M Σ f r , as was to be shown.

SPECTRAL CATEGORIES ASSOCIATED TO SYMMETRIC RING SPECTRA
In what follows by a spectral category we mean a category enriched over the closed symmetric monoidal category of symmetric S 1 -spectra Sp Σ S 1 .Recall from [31,Construction 5.6] that for every pair of symmetric spectra X ,Y a morphism X ∧Y → Z to a symmetric spectrum Z is the same as giving a bimorphism b : (X ,Y ) → Z.We define a bimorphism b : (X ,Y ) → Z as a collection of Σ p × Σ q -equivariant maps of pointed simplicial sets b p,q : X p ∧Y q → Z p+q for p, q 0, such that the "bilinearity diagram" commutes for all p, q 0: X p ∧σ q x x q q q q q q q q q q q X p ∧ S 1 ∧Y q In this section C is the category from Section 2.
5.1.Definition.Suppose B is a full subcategory of C closed under ∧ and σ : Second, repeating the proof of Theorem 3.2(1) word for word the morphism σ induces natural ) is given by the composition 5.2.Theorem.Let E be a symmetric ring T -spectrum in Sp Σ (C , T ) and B is a full subcategory of C closed under monoidal product.Then B can be enriched over the closed symmetric monoidal category of symmetric S 1 -spectra Sp Σ S 1 .Namely, Sp Σ S 1 -objects of morphisms are defined by the symmetric spectra Corr E,σ * (X ,Y ) of (E, σ )-correspondences.Compositions are defined by pairings ϕ X,Y,Z .The resulting Sp Σ S 1 -category is denoted by Corr E,σ * (B).Moreover, the Sp Σ S 1 -category Corr E,σ * (B) is symmetric monoidal with the same monoidal product on objects as in B whenever E is a commutative ring T -spectrum.
Proof.The identity morphism is defined by , where ι 0 : S → E 0 is the unit map.Our proof now literally repeats that of Theorem 2.4.The only thing one has to care about is that the pairings occurring here are bimorphisms of symmetric spectra.This is clearly the case if one chases over the diagram (8).It is also worth noting that The proof of the theorem implies the following result.

Corollary. Under the notation of Theorem 5.2 any morphism of two symmetric ring T -spectra
Day's Theorem [2] together with Theorem 5.2 also imply the following result.

Corollary. Under the notation of Theorem 5.2 if E is a commutative symmetric ring Tspectrum then the category of right
Corr E,σ * (B)-modules is a closed symmetric monoidal category.

Theorem. Suppose E is a commutative symmetric ring T -spectrum in Sp
Under the assumptions of Theorem 5.2 the spectral category Corr E,σ * (B) is also a symmetric monoidal Mod 1 E -category with the same monoidal product on objects as in B, where Mod 1 E is the closed symmetric monoidal category of Corollary 3.3.
r and µ * , * is the multiplication map of E. The proof is like that of Theorem 5.2 if we observe that the diagram x x q q q q q q q q q q q is commutative with

Corollary. Under the notation of Theorem 5.5 any morphism of two symmetric ring T -spectra
Day's Theorem [2] together with Theorem 5.2 also imply the following result.5.7.Corollary.Under the notation of Theorem 5.5 if E is a commutative symmetric ring Tspectrum then the category of Corr E,σ * (B)-modules in the category Mod 1 E is a closed symmetric monoidal category.

ENRICHED MOTIVIC HOMOTOPY THEORY
One of the approaches to Morel-Voevodsky's stable motivic homotopy theory SH(k) over a field k is by means of symmetric T -spectra Sp Σ T (k), where T = A 1 /(A 1 − {0}) (see, e.g., [24]).In detail, we start with motivic spaces M equipped with the flasque motivic model structure in the sense of [23] and then pass to Sp Σ T (k) equipped with the stable model structure.The homotopy category of the latter model category is denoted by SH(k).
A genuinely local approach to SH(k), envisioned by Voevodsky in 2001, is presented in [20].It is based on Voevodsky's framed correspondences and the machinery of framed motives [19].
In this section we suggest yet another (genuinely local) approach to SH(k) and, more generally, a local model for the category of E-modules in SH(k), where E is a symmetric Thom ring spectrum.It is an application of enriched category theory of spectral categories and spectral modules of Section 5.The same approach was used in [16,17] to construct the theory of K-motives.
Following [14], a symmetric T -spectrum E is called a Thom spectrum if each motivic space E n has the form where V n,i → V n,i+1 is a directed sequence of smooth varieties, Z n,i → Z n,i+1 is a directed system of smooth closed subschemes in V n,i .We say that a Thom spectrum E has the bounding constant d if d is the minimal integer such that codimension of Z n,i in V n,i is strictly greater than n − d for all i, n.The T -spectrum E is said to be a spectrum with contractible alternating group action, if for any n and any even permutation τ ∈ Σ n there is an A 1 -homotopy E n → Hom(A 1 , E n ) between the action of τ and the identity map.In other words, E neglects the action of even permutations up to A 1 -homotopy.
Unless it is specified otherwise E is a symmetric Thom ring T -spectrum with the bounding constant d 1 and contractible alternating group action throughout this section.By [14, Lemma 10.2] E ∈ SH eff (k), where SH eff (k) is the full triangulated subcategory of SH(k) of effective T -spectra.It is compactly generated by the suspension T -spectra Σ ∞ T X + , X ∈ Sm/k.For instance, E is the algebraic cobordism T -spectrum MGL or motivic sphere spectrum S k = (S 0 , T, T 2 , . ..).Other examples are commutative symmetric ring T 2 -spectra MSL and MSp [29].The results that use T 2 -spectra are the same with those proven in this section and which use T -spectra.For brevity, we will deal with T -spectra only.
We can apply Theorem 5.5 to the following data: ⋄ C = M ; ⋄ the canonical map σ : P ∧1 → T , where P ∧1 := (P 1 , ∞) ∈ M , given by the framed correspondence ({0}, A 1 ,t) ∈ Fr 1 (pt, pt); Within this notation the symmetric monoidal spectral category Corr E,σ * (B) of Theorem 5.5 will be denoted by O E for brevity and each Corr E,σ n (B)(X ,Y ) = Hom M (X + ∧ P ∧n ,Y + ∧ E n ∧ S n ) will be denoted by Fr E n (X ,Y ⊗ S n ).Recall from [19,Section 3] that each simplicial set Fr E n (X ,Y ⊗ S n ) = Hom M (X + ∧ P ∧n ,Y + ∧ E n ∧ S n ) has an explicit geometric description due to Voevodsky's Lemma.
Similarly to Definition 4.6 we can consider a spectral category O E ∆ which is obtained from O E by applying the Suslin complex to symmetric spectra of morphisms: Let O be a spectral category and let Mod O be the category of O-modules.Recall that the projective stable model structure on Mod O is defined as follows (see [33]).The weak equivalences are the objectwise stable weak equivalences and fibrations are the objectwise stable projective fibrations.The stable projective cofibrations are defined by the left lifting property with respect to all stable projective acyclic fibrations.
Let Q denote the set of elementary distinguished squares in Sm/k (see [27, 3.1.3]) and let O be a spectral category over Sm/k.By Q O denote the set of squares which are obtained from the squares in Q by taking X ∈ Sm/k to O(−, X ).The arrow O(−,U ′ ) → O(−, X ′ ) can be factored as a cofibration O(−,U ′ ) Cyl followed by a simplicial homotopy equivalence Cyl → O(−, X ′ ).There is a canonical morphism 6.1.Definition (see [16,17]).We say that O is Nisnevich excisive if for every elementary distinguished square Q the square OQ (10) is homotopy pushout in the Nisnevich local model structure on Sp Σ S 1 (k) := Sp Σ (M , S 1 ).The Nisnevich local model structure on Mod O is the Bousfield localization of the stable projective model structure with respect to the family of projective cofibrations The homotopy category for the Nisnevich local model structure will be denoted by SH nis S 1 O. Suppose O is symmetric monoidal.By a theorem of Day [2] Mod O is a closed symmetric monoidal category with smash product ∧ and O(−, pt) being the monoidal unit.The smash product is defined as The internal Hom functor, right adjoint to − ∧ O M, is given by By [6,Corollary 2.7] that there is a natural isomorphism k from the definition of C * Fr E (X + ∧ G), one gets motivic spaces Fr E (X + ∧ G). 6.3.Definition.The symmetric E-framed motive of a smooth algebraic variety X ∈ Sm/k is the symmetric S 1 -spectrum ), . ..) with structure maps defined similarly to M Σ f r (N) of Definition 4.9.6.4.Remark.The E-framed motive in the sense of the preceding definition is a bit different from the E-framed motive of X in the sense of [14] defined as Proof.The proof is like that of Lemma 4.11.We also use [14,Section 7] here.
Though M E (X ) is canonically a symmetric S 1 -spectrum in Sp Σ S 1 (k), where Σ n acts on each space by permuting S n , the point is that it is not an O E ∆ -module in contrast with M Σ E (X ).6.6.Proposition.Given a field k and X ∈ Sm/k, the following statements are true: ( ∆ is a sectionwise π * -isomorphism (i.e. a stable equivalence of ordinary spectra) whenever the base field k is perfect.
).The proof is like that of Proposition 4.12.We also use [14,Section 7] and Proposition 6.6 (2).The previous proposition also shows that compact generators can be given by the symmetric E-framed motives Here is regarded as a presheaf of S 1 -spectra.Each structure map is induced by the adjunction unit morphism )).
6.8.Corollary.Let k be a perfect field.There is a triangulated equivalence of compactly generated triangulated categories where " f " refers to level local fibrant replacements of motivic S 1 -spectra.We have a triangulated functor of compactly generated triangulated categories By Lemma 6.5, Proposition 6.6 and Theorem 6.7 Proof.The proof is similar to that of Corollary 6.8.We compare compact generators and Hom-sets between them in both categories.
It is worth mentioning that we do not use any motivic equivalences or the A 1 -relation in any of our definitions above (similarly to constructions of [20]).All constructions here are genuinely local.On the other hand, we can do the usual Voevodsky approach [34] to constructing the triangulated category of motives DM eff (k).We start with the spectral category O E and Cech local model structure on the stable model category of O E -modules ModO E .
For each finite Nisnevich cover {U i → X } we let O E (−, Ǔ * ) be the realization of the simplicial module which in dimension n is ∨ i 0 ,...,i n O E (−,U i 0 ...i n ), with the obvious face and degeneracy maps.Here U i 0 ...i n stands for the smooth scheme U i 0 × X • • • × X U i n .The reader should not confuse O E (−, Ǔ * ) with the realization O E (−, Č(U * )) of the simplicial module which in dimension n is O E (−, ⊔ i 0 ,...,i n U i 0 ...i n ).6.10.Lemma.Each natural map ∨ i 0 ,...,i n O E (−,U i 0 ...i n ) → O E (−, Č(U n )) is a schemewise stable equivalence of ordinary S 1 -spectra.
This completes the proof of the lemma.
The Cech model category ModO E Cech associated with Nisnevich topology is obtained from ModO E by Bousfield localization with respect to all maps η : O E (−, Ǔ * ) → O E (−, X ) running over the set of finite Nisnevich covers.It follows from [36,Corollary 5.10] (see also [5]) that ModO E Cech coincides with the Nisnevich local model category ModO E nis , with stable weak equivalences defined stalkwise.
We say that a spectral category O is Cech excisive if for any finite Nisnevich cover {U i → X } the induced map η : O(−, Ǔ * ) → O(−, X ) is a local stable weak equivalence.Proof.By Lemma 6.10 each map ∨ i 0 ,...,i n O E (−,U i 0 ...i n ) → O E (−, Č(U n )) is a schemewise stable equivalence, and hence the realization is.The proof of [35,Theorem 4.4] shows that the map O E (−, Č(U * )) → O E (−, X ) is a level local equivalence.We see that η is a local stable weak equivalence.The rest is now straightforward.
The homotopy category DO E,eff (k) plays the same role as the derived category D(Shv nis tr (Sm/k)) of cochain complexes of Nisnevich sheaves with transfers.Recall from [34] that Voevodsky's category of motives DM eff (k) is the localisation of D(Shv nis tr (Sm/k)) with respect to the family {Z tr (−, X × A 1 ) → Z tr (−, X ) | X ∈ Sm/k}.If k is perfect, DM eff (k) is equivalent to the full subcategory of D(Shv nis tr (Sm/k)) consisting of chain complexes with homotopy invariant cohomology sheaves [34].Likewise, localize ModO E Cech with respect to the maps {O E (−, X ×A 1 ) → O E (−, X ) | X ∈ Sm/k}.Denote by DO E,eff mot (k) its homotopy category.
6.12.Theorem.Let k be a perfect field.The homotopy category DO E,eff mot (k) is equivalent to the full triangulated subcategory DE eff (k) of DO E,eff (k) consisting of modules with homotopy invariant sheaves of stable homotopy groups.The inclusion DE eff (k) → DO E,eff (k) has a right adjoint C * taking a module M ∈ DO E,eff (k) to its Suslin complex C * (M).
Moreover, there is a triangulated equivalence of compactly generated triangulated categories Proof.The proof of the first part is like that of [17,Theorem 3.5].One also uses here the fact that if k is perfect then by [18] (complemented by [4] in characteristic 2 and by [3, A.27] for finite fields) any A 1 -invariant quasi-stable radditive framed presheaf of Abelian groups F , the associated Nisnevich sheaf F nis is strictly A 1 -invariant.
The equivalence DE eff (k) ≃ SH nis S 1 O E ∆ follows from the fact that both categories are compactly generated by symmetric E-framed motives with the same Hom-sets (as usual we use Lemma 4.1 here).
k and pt a field of exponential characteristic e and Spec(k) Sm/k the category of smooth separated schemes of finite type Fr 0 (k) or Sm/k + the category of framed correspondences of level zero Shv • (Sm/k) the closed symmetric monoidal category of pointed Nisnevich sheaves M = ∆ op Shv • (Sm/k) the category of pointed motivic spaces, a.k.a. the category of pointed simplicial Nisnevich sheaves S • the category of pointed simplicial sets 2. GRADED SYMMETRIC SEQUENCES

4. 5 .
Theorem (Garkusha-Panin[19]). Let k be an algebraically closed field of characteristic 0. Then the framed motive M f r (pt)(pt) of the point pt = Spec( k) evaluated at pt has the stable homotopy type of the classical sphere spectrum S = Σ ∞ S 1 S 0 .If k is an algebraically closed field of positive characteristic e > 0 then M f r (pt)(pt)[1/e], pt = Spec( k), has the stable homotopy type of S [1/e].

4. 12 .
Proposition.Given a field k and N ∈ Sp Σ S 1 , the following statements are true: (1) C * Fr Σ * (N) and M Σ f r (N) are right S k -modules; (2) every map of symmetric S 1 -spectra f : N → N ′ induces morphisms of right S k -modules C * Fr Σ * A n,r σ − → A n+1,r and B m,n,r σ − → B m+1,n,r σ − → B m+1,n+1,r .Let A r := colim n r A n,r , B m,r := colim n B m,n,r , B r := colim m,n B m,n,r .The spaces A r , B r constitute right S 1 -spectra A and B. Each structure map A r → Hom(S 1 , A r+1 ) is the composite map determined by the following commutative diagram A r :

4. 15 .
Theorem.Suppose k = k is an algebraically closed field of exponential characteristic e.The Quillen pair (L,U ) (7) is a Quillen equivalence.In particular, it induces an equivalence of compactly generated triangulated categories L : SH[e −1 ] ⇄ SH k [e −1 ] : U.
is naturally zigzag equivalent to the category of bispectra SH(k) = Ho(Sp S 1 ,G (k)).Let Mod eff SH(k) E be the essential image of Mod eff SH(k) E under this zigzag equivalence.The category Mod eff SH(k) E is compactly generated by the images of {X + ∧ E | X ∈ Sm/k} in Mod eff SH(k) E. By the proof of [14, Theorem 9.13] the latter are isomorphic in Mod eff SH(k) E to motivically fibrant bispectra

6. 11 .
Theorem.Let k be any field.The commutative spectral category O E is Cech excisive.The Cech model structure coincides with Nisnevich local model structure on ModO E .This model structure has all the properties of Theorem 6.2.The homotopy category DO E,eff (k) of ModO E Cech is closed symmetric monoidal compactly generated triangulated with compact generators being the representables {OE (−, X ) | X ∈ Sm/k}.The monoidal product O E (−, X )∧O E (−,Y ) in DO E,eff (k) is isomorphic to O E (−, X ×Y ).
Commutativity of the top square follows from commutativity of the multiplication maps, commutativity of the remaining squares follows from equivariancy of the multiplication maps.It follows that Theorem 1.1] by using the stable étale realisation functor.
4.3.Corollary.Let k be an algebraically closed field of exponential characteristic e.The triangulated functor Lc : SH[1/e] → SH( k)[1/e] is full and faithful.Proof.Using Lemma 4.1, our statement follows from Theorem 4.2 if we note that SH[1/e] (respectively the image of SH[1/e]) is compactly generated by S [1/e] (respectively by S k[1/e]).
[32, Section 4], we define the stable model structure on Mod S k by calling a map f of S k -spectra a stable equivalence or fibration if so is U ( f ).By[32, Theorem 4.1]this model structure is also cofibrantly generated monoidal satisfying the monoid axiom.By construction, (L,U ) is a Quillen pair.
4.8.Definition.The category Mod S k is called the category of framed symmetric S 1 -spectra over a field k.The stable homotopy category over a field k, denoted by SH k , is defined as the homotopy category of Mod S k with respect to the stable model structure.SH k is a closed symmetric monoidal category.
.2. Theorem ([16]).Suppose O is a Nisnevich excisive spectral category.Then the Nisnevich local model structure on Mod O is cellular, proper, spectral and weakly finitely generated.Moreover, a map of O-modules is a weak equivalence in the Nisnevich local model structure if and only if it is a weak equivalence in the Nisnevich local model structure on Sp Σ S 1 (k).If O is a symmetric monoidal spectral category then the model structure on Mod O is symmetric monoidal with respect to the smash product (11) of O-modules.In our setting we regard spectral categories O E , O E ∆ as symmetric monoidal Mod 1 E -categories with the same monoidal product on objects as in Sm/k (see Theorem 5.5), where Mod 1 E is the closed symmetric monoidal category of Corollary 3.3.Denote by ModO E and ModO E ∆ the closed symmetric monoidal categories of O E -and O E ∆ -modules in the category Mod 1 E (see Corollary 5.7).The Nisnevich local model structure on ModO E and ModO E ∆ as well as their homotopy categories SH nis S 1 O E and SH nis S 1 O E ∆ are defined similarly to Definition 6.1.Given X ∈ Sm/k and a motivic space G ∈ M , denote by C * Fr E n here.6.7.Theorem.Let k be a perfect field.The commutative spectral category O E ∆ is Nisnevich excisive and the Nisnevich local model structure on ModO E Nisnevich excisive by [14, Section 9], Lemma 6.5 and Proposition 6.6.The fact that the Nisnevich local model structure on ModO E ∆ has all the properties of Theorem 6.2 follows from the fact that O E ∆ is Nisnevich excisive symmetric monoidal.
follows that LΣ ∞G m takes compact generators to compact generators with isomorphic Hom-sets.It remains to apply Lemma 4.1.
∆ ).Its homotopy category is denoted bySH S 1 ,G m O E ∆ .Given X ∈ SH nis S 1 O E ∆ we write X (1) to denote X ⊠ L G ∧1 m .