Rational Enriched Motivic Spaces

Enriched motivic $\mathcal A$-spaces are introduced and studied in this paper, where $\mathcal A$ is an additive category of correspondences. They are linear counterparts of motivic $\Gamma$-spaces. It is shown that rational special enriched motivic $\widetilde{\mathrm{Cor}}$-spaces recover connective motivic bispectra with rational coefficients, where $\widetilde{\mathrm{Cor}}$ is the category of Milnor--Witt correspondences.


Introduction
In his celebrated paper [30] Segal introduced Γ-spaces and showed that they yield infinite loop spaces.In [6] Bousfield and Friedlander defined a model category structure for Γ-spaces and showed that its homotopy category recovers connective S 1 -spectra.They also showed that fibrant objects in this model category are given by very special Γ-spaces.
Garkusha, Panin and Østvaer [23] have recently introduced and studied motivic Γ-spaces.They are M-enriched functors in two variables where M is the category of pointed motivic spaces and Sm k,+ is the M-category of framed correspondences of level 0. Special and very special motivic Γ-spaces are defined in [23] as M-enriched functors X : Γ op ⊠ Sm k,+ → M fr satisfying several axioms, where M fr is the M-category of pointed motivic spaces with framed correspondences.The axioms are a combination of Segal's axioms and axioms reflecting basic properties of framed motives of algebraic varieties in the sense of Garkusha-Panin [21] (see [23] for details).
Inspired by [23] we introduce and study additive versions for motivic Γ-spaces in this paper.We start with a reasonable additive category of correspondences A, such that the exponential characteristic of the base field k is invertible in A, and replace M by the closed symmetric monoidal Grothendieck category ∆ op Shv(A) of simplicial Nisnevich sheaves with A-transfers.The M-category Sm k,+ is replaced here by a ∆ op Shv(A)-category Sm whose objects are those of Sm k .
We define enriched motivic A-spaces as objects of the Grothendieck category of ∆ op Shv(A)enriched functors [Sm, ∆ op Shv(A)].Special enriched motivic A-spaces are defined similarly to special motivic Γ-spaces with slight modifications due to the additive context (see Definition 2.1 for the full list of axioms).In particular, the category Γ op is redundant in this context (see Section 4).
The category [Sm, ∆ op Shv(A)] comes equipped with a local and a motivic model structure.Denote the model categories by [Sm, ∆ op Shv(A)] nis and [Sm, ∆ op Shv(A)] mot respectively (see Section 7).Let D([Sm, ∆ op Shv(A)]) be the homotopy category of [Sm, ∆ op Shv(A)] nis .Define Spc A [Sm] as the full subcategory of D([Sm, ∆ op Shv(A)]) consisting of special enriched motivic A-spaces.It is worth mentioning that D([Sm, ∆ op Shv(A)]) is equivalent to the full subcategory of connective chain complexes in the derived category D([Sm, Shv(A)]) of the Grothendieck category [Sm, Shv(A)].Thus Spc A [Sm] can also be regarded as a full subcategory of D([Sm, Shv(A)]), so that it can be studied by methods of classical homological algebra.
The following result is reminiscent of Bousfield-Friedlander's theorem mentioned above for classical Γ-spaces (see Theorem 7.7).
Theorem.The category Spc A [Sm] is equivalent to the homotopy category of the model category [Sm, ∆ op Shv(A)] mot .The fibrant objects of [Sm, ∆ op Shv(A)] mot are the pointwise locally fibrant special enriched motivic A-spaces.
As applications of the preceding theorem we recover connective motivic bispectra with rational coefficients SH(k) Q, 0 (respectively very effective motivic bispectra with rational coefficients SH veff (k) Q ) from special rational enriched motivic A-spaces Spc A [Sm] (respectively very effective rational enriched motivic A-spaces Spc veff A [Sm]) -see Theorems 9.2 and 9.4.
Here we take A to be the category of finite Milnor-Witt correspondences with rational coefficients Cor ⊗ Q.
Theorem.The (S 1 , G m )-evaluation functor induces equivalences of categories and ev S 1 ,Gm : Spc veff Cor,Q [Sm] → SH veff (k) Q .In particular, the preceding theorem makes SH(k) Q more amenable to methods of homological algebra.It is also worth mentioning that the results of this paper are based on the techniques developed in [5].
The results of the paper were first presented at the Conference on Motivic and Equivariant Topology in May 2023 (Swansea, UK).The author expresses his gratitude to his supervisor Prof. Grigory Garkusha whose patience and keen insight have been indispensable throughout this work.
Notation.Throughout the paper we use the following notation.k field of exponential characteristic p Sm k smooth separated schemes of finite type over k A symmetric monoidal additive V -category of correspondences Psh(A) presheaves of abelian groups on A Shv(A) Nisnevich sheaves of abelian groups on A DM A triangulated category of big motives with A-correspondences SH(k) stable motivic homotopy category over k M A (X) A-motive of X ∈ Sm k M category of motivic spaces f M category of finitely presented motivic spaces Also, we assume that 0 is a natural number.

Preliminaries
Let Sm k be the category of smooth separated schemes of finite type over a filed k.Throughout this paper we work with an additive category of correspondences A that is symmetric monoidal and satisfies the strict V -property and the cancellation property in the sense of [18].Basic examples are given by the categories of finite correspondences Cor or Milnor-Witt correspondences Cor.We also assume that in A the exponential characteristic p of k is invertible.Note that for any additive category of correspondences A we can form an additive category of correspondences A[1/p] in which p is invertible by tensoring all morphism groups of A with Z[1/p].Let Shv(A) be the category of Nisnevich sheaves on A with values in abelian groups.
We shall adhere to the following notations from [18].Let Sp S 1 ,Gm (k) denote the category of symmetric (S 1 , G m )-bispectra, where the G m -direction is associated with the pointed motivic space (G m , 1).It is equipped with a stable motivic model category structure.Denote by SH(k) its homotopy category.The category SH(k) has a closed symmetric monoidal structure with monoidal unit being the motivic sphere spectrum S. Given p > 0, the category Sp S 1 ,Gm (k) has a further model structure whose weak equivalences are the maps of bispectra f : X → Y such that the induced map of bigraded Nisnevich sheaves f * : In what follows we denote its homotopy category by SH(k)[1/p].The category SH(k) Q is defined in a similar fashion.
Following [5], we define a Shv(A)-enriched category Sm, whose objects are those of Sm k , and whose morphism sheaves are defined by In Section 3 we will define a natural local model structure on ∆ op Shv(A).Weak equivalences in this model structure are the local equivalences.
According to [10,Theorem 4.3.12],if G is a Grothendieck category with a generator G, then the category of simplicial objects ∆ op G in G is also Grothendieck and the set {G⊗∆[n] | n 0} is a family of generators for ∆ op G.In particular, a family of generators for the Grothendieck category ∆ op Shv(A) is given by the set Also, the category of enriched functors [Sm, Shv(A)] is Grothendieck by [1].Its family of generators is given by {Sm(X, −)⊗ Shv(A) A(−, Y ) nis | X, Y ∈ Sm k }.Hence ∆ op [Sm, Shv(A)] is Grothendieck by [10].Its family of generators is given by {Sm(X, −)⊗ Note that ∆ op [Sm, Shv(A)] and [Sm, ∆ op Shv(A)] are equivalent, and we will freely pass back and forth between the two.

2.1.
Definition.An enriched motivic A-space is an object of the Grothendieck category ∆ op [Sm, Shv(A)].Similarly to [23,Axioms 1.1], an enriched motivic A-space X is said to be special if it satisfies the following axioms: (1) For all n 0 and U ∈ Sm k the presheaf of homotopy groups ( (4) (Nisnevich excision) For every elementary Nisnevich square in Sm k is homotopy cartesian in the local model structure on ∆ op Shv(A).
For n 0 and every finitely generated field extension K/k, we have the standard algebraic n-simplex ).For every 0 i n we define a closed subscheme v i of ∆ n K by the equations x j = 0 for j = i.We write ∆ n K/k for the semilocalization of the standard algebraic n-simplex ∆ n K with closed points the vertices v 0 , . . ., v n ∈ ∆ n K .2.2.Definition.Similarly to [23, Axioms 1.1], we say that X is very effective or satisfies Suslin's contractibility if for every U ∈ Sm and every finitely generated field extension K/k the diagonal of the bisimplicial abelian group X (G ∧1 m × U )( ∆ • K/k ) is contractible.Since we assume that p is invertible in A the following lemma holds.Proof.If F : A → Ab is additive, then F is an Ab-enriched functor.By the Ab-enriched co-Yoneda lemma we can write F as the following coend: for all U ∈ Sm k we have an isomorphism in Ab, Since the functor −⊗Z[1/p] : Ab → Ab is a left adjoint, it preserves coends, so we can compute For some of our results we will also have to make additional assumptions on the category of correspondences A.
2.4.Definition.Let Fr * (k) be the category of Voevodksy's framed correspondences (see [21,Definition 2.3]).For each (1) We say that the category of correspondences A has framed correspondences if there is a functor Φ : Fr * (k) → A which is the identity on objects and which takes every σ V to the identity of V .(2) We say that A satisfies the ∆-property if for every n > 0 and for every finitely generated field extension Basic examples satisfying both items are given by the categories of finite correspondences Cor or Milnor-Witt correspondences Cor.

The local model structure
In [5,Section 3] we constructed a model structure on Ch(Shv(A)) that is cellular, strongly left proper, weakly finitely generated, monoidal and satisfies the monoid axiom.In this section we construct a model structure on Ch 0 (Shv(A)) that is cellular, strongly left proper, weakly finitely generated, monoidal, satisfies the monoid axiom, and in which weak equivalences are local quasi-isomorphisms.We construct the model structure by taking the right transferred model structure along the inclusion Ch 0 (Shv(A)) → Ch(Shv(A)).We then transfer the model structure along the Dold-Kan correspondence, to get a model structure on ∆ op Shv(A) that is cellular, strongly left proper, weakly finitely generated, monoidal, satisfies the monoid axiom, and in which weak equivalences are stalkwise weak equivalences of simplicial sets.
Proof.Since ιτ naive is a left adjoint functor, it suffices to check it on the set of generating cofibrations So take n ∈ Z, X ∈ Sm k and consider the map 3.2.Definition.Given a model category M and an adjunction L : N ⇄ M : R, we say that the right transferred model structure along the adjunction L ⊣ R exists, if there exists a model structure on N , such that a morphism f is a weak equivalence (respectively cofibration) in N if and only if L(f ) is a weak equivalence (respectively cofibration) in M .Proof.We use [4,Theorem 2.23].All involved categories are locally presentable, and Ch(Shv(A)) is cofibrantly generated, so the theorem is applicable.We now have to show that RLP(ι −1 ({cofibrations})) ⊆ ι −1 ({weak equivalences}).
So take p : X → Y with p ∈ RLP(ι −1 ({cofibrations})).We want to show that ι(p) is a weak equivalence in Ch(Shv(A)).We will show that ι(p) is a trivial fibration, by showing that it has the right lifting property with respect to cofibrations.Let f : A → B be a cofibration in Ch(Shv(A)) and consider a lifting problem By adjunction this diagram has a lift, if and only if the following diagram has a lift Since p ∈ RLP(ι −1 ({cofibrations})), one has to show that τ naive f ∈ ι −1 ({cofibrations}).One has to show that ιτ naive f is a cofibration.As f is a cofibration, this follows from Lemma 3.1.
We now have a model structure on Ch 0 (Shv(A)), in which a morphism f is weak equivalence (respectively cofibration) if and only if ιf is a weak equivalence (respectively cofibration) in Ch(Shv(A)), and a morphism is a fibration in Ch 0 (Shv(A)) if and only if it has the right lifting property with respect to all trivial cofibrations.Furthermore, the adjunction ι : Ch 0 (Shv(A)) ⇄ Ch(Shv(A)) : τ good is a Quillen adjunction.Since weak equivalences in Ch(Shv(A)) are the local quasi-isomorphisms, it follows that also weak equivalences in Ch 0 (Shv(A)) are the local quasi-isomorphisms.
Let us verify the unit axiom.If ½ 0 is the monoidal unit of Ch 0 (Shv(A)), and ½ is the monoidal unit of Ch(Shv(A)), then since ι is strong monoidal we have an isomorphism ι½ 0 ∼ = ½.As ½ = A(−, pt) nis is cofibrant in Ch(Shv(A)) it follows that ½ 0 is cofibrant in Ch 0 (Shv(A)).This implies the unit axiom.
Proof.Let W 0 denote the class of weak equivalences and CW 0 denote the class of trivial cofibrations in Ch 0 (Shv(A)).Let W denote the class of weak equivalences and CW denote the class of trivial cofibrations in Ch(Shv(A)).We need to show that Since ι is a strong monoidal left adjoint functor we have Since ι preserves trivial cofibrations we have ι(CW 0 ) ⊆ CW .Since Ch(Shv(A)) satisfies the monoid axiom (see [5]), it follows that Hence Ch 0 (Shv(A)) satisfies the monoid axiom.
3.6.Lemma.Let I Ch(Shv(A)) be a set of generating cofibrations for Ch(Shv(A)).Then the set τ naive (I Ch(Shv(A)) ) is a set of generating cofibrations of Ch 0 (Shv(A)).In particular, the model category Ch 0 (Shv(A)) has a set of generating cofibrations with finitely presented domains and codomains.
Since the set is a set of generating cofibrations with finitely presented domains and codomains for Ch(Shv(A)), it follows that is a set of generating cofibrations with finitely presented domains and codomains of Ch 0 (Shv(A)).
Next, we want to show that Ch 0 (Shv(A)) is weakly finitely generated.To this end, we need to define a set of weakly generating trivial cofibrations J ′ .For this we need to construct a certain set of morphisms similar to [5, Definition 3.3].

Definition. For every elementary
we have a square Let Q be the set of all elementary Nisnevich squares.Define a set of morphisms be a set of generating cofibrations with finitely presented domains and codomains for Quillen's standard projective model structure on Ch(Ab) 0 .We define sets of morphisms in Ch 0 (Shv(A)) where is the set of all morphisms which are a pushout product of a morphisms from J Q and I Ch(Ab) 0 .
Note that all morphisms from I Ch 0 (Ab) are cofibrations and all morphisms from J proj and J Q are trivial cofibrations.Since Ch 0 (Shv(A)) is a monoidal model category it follows that all morphisms from J ′ are trivial cofibrations.

Lemma. A morphism
has the right lifting property with respect to J proj if and only if for every n 1 the map f n : A n → B n is sectionwise surjective.
Proof.For every n 0, X ∈ Sm k we can solve the lifting problem 0 3.9.Lemma.For an object A in Ch 0 (Shv(A)) the following are equivalent: (3) A → 0 has the right lifting property with respect to (2) =⇒ (3).If A is fibrant in Ch 0 (Shv(A)), then A → 0 has the right lifting property with respect to all trivial cofibrations, hence it has the right lifting property with respect to J Q I Ch 0 (Ab) .
(3) =⇒ (1).Assume that A → 0 has the right lifting property with respect to J Q I Ch 0 (Ab) .We want to show that ι(A) is fibrant in Ch(Shv(A)).By [5,Lemma 3.4] we have to show that A(∅) → 0 is a quasi-isomorphism, and that A sends elementary Nisnevich squares to homotopy pullback squares.Since A is a chain complex of sheaves, we have A(∅) = 0. Let us now show that A sends elementary Nisnevich squares to homotopy pullback squares.Let Q be an elementary Nisnevich square.For X, Y ∈ Ch 0 (Shv(A)) let Hom Ch 0 (Shv(A)) (X, Y ) be the internal hom of Ch 0 (Shv(A)) and let The square A(Q) will be a homotopy pullback square in Ch(Ab) if and only if the map For that we need to show that p * Q has the right lifting property with respect to I Ch(Ab) 0 .Now for every map has a lift in Ch 0 (Ab) if and only if the square 0 has a lift in Ch 0 (Shv(A)).This lift exists, because A → 0 has the right lifting property with respect to In what follows, let Ch(Psh(A)) proj be the model category Ch(Psh(A)) with standard projective model structure.Let Ch(Psh(A)) nis be the model category Ch(Psh(A)) with local projective model structure.See [5, Section 3] for details.Let L nis : Ch(Psh(A)) ⇆ Ch(Shv)A : U nis be the adjunction consisting of the sheafification and the forgetful functors.
3.10.Proposition.Let f : A → B be a morphism in Ch 0 (Shv(A)) such that B is fibrant and f has the right lifting property with respect to J ′ .Then f is a fibration in Ch 0 (Shv(A)).
Proof.Our first claim is that A is fibrant.Since B is fibrant, by Lemma 3.9 B → 0 has the right lifting property with respect to J Q I Ch 0 (Ab) .Since f has the right lifting property with respect to J Q I Ch 0 (Ab) it follows that A → 0 has the right lifting property with resepct to that is B 0 in degree 0 and −1, and which is 0 everywhere else.We claim that D −1 B 0 is fibrant in Ch(Shv(A)).Indeed, the map U nis D −1 B 0 → 0 is a trivial fibration in Ch(Psh(A)) proj , hence it is also a trivial fibration in Ch(Psh(A)) nis .Therefore D −1 B 0 → 0 is a trivial fibration in Ch(Shv(A)), and so In particular, ι(A) ⊕ D −1 B 0 is fibrant in Ch(Shv(A)) and we have that is a map between fibrant objects, and we have a commutative diagram where the horizontal maps are isomorphisms We want to show that f is a fibration in Ch 0 (Shv(A)).Since τ good is a right Quillen functor, we now just need to show that ι(f )+g is a fibration in Ch(Shv(A)).For this it suffices to show that U nis (ι(f ) + g) is a fibration in Ch(Psh(A)) nis .Since U nis ι(A ⊕ D −1 B 0 ) and U nis ι(B) are fibrant in Ch(Psh(A)) nis , it suffices by [24,Proposition 3.3.16]to show that U nis (ι(f ) + g) is a fibration in Ch(Psh(A)) proj .So we have to show that the map ι(f ) + g is sectionwise an epimorphism in Ch(Ab).In degree n 1 the map ι(f ) : ι(A) → ι(B) is sectionwise surjective, because of Lemma 3.8 and the fact that f satisfies the right lifting property with respect to J proj .In degree n −1 the map ι(f )+ g is sectionwise surjective, because ι(B) n = 0. Finally, in degree n = 0 the map ι(f ) + g is sectionwise surjective, because g : 3.11.Corollary.Ch 0 (Shv(A)) is weakly finitely generated and J ′ is a set of weakly generating trivial cofibrations for Ch 0 (Shv(A)).
Proof.By Lemma 3.3 Ch 0 (Shv(A)) is cofibrantly generated, so there exists a set J of generating trivial cofibrations.Since every object in Ch 0 (Shv(A)) is small, the domains and codomains from J are small.By Lemma 3.6 Ch 0 (Shv(A)) has a set of generating cofibrations with finitely presented domains and codomains.All morphisms from J ′ are trivial cofibrations with finitely presented domains and codomains, so Proposition 3.10 implies that J ′ is set of weakly generating trivial cofibrations for Ch 0 (Shv(A)).Proof.If we have a pushout square in Ch 0 (Shv(A)) with f a weak equivalence and g : A → C a cofibration, then the square is a pushout square in Ch(Shv(A)).Since ι(f ) is a weak equivalence, ι(g) is a cofibration, and Ch(Shv(A)) is strongly left proper by [5], it follows that ι(h) is a weak equivalence in Ch(Shv(A)).So h is a weak equivalence in Ch 0 (Shv(A)).
In summary, we have a model category Ch 0 (Shv(A)) that is cellular, weakly finitely generated and where the weak equivalences are the local quasi-isomorphisms.With respect to the usual tensor product of chain complexes ⊗ it is monoidal, strongly left proper and satisfies the monoid axiom.
We can transfer this model structure along the Dold-Kan correspondence So we define a model structure on ∆ op (Shv(A)), where a morphism f is a weak equivalence (respectively fibration, cofibration), if and only if DK −1 (f ) is a weak equivalence (respectively fibration, cofibration) in Ch 0 (Shv(A)).Then weak equivalences in ∆ op Shv(A) are the stalkwise weak equivalences of simplicial sets.Furthermore ∆ op Shv(A) is weakly finitely generated and cellular.From now on, weak equivalences in ∆ op Shv(A) be called local equivalences, fibrations in ∆ op Shv(A) will be called local fibrations, and fibrant objects in ∆ op Shv(A) will be called locally fibrant objects.
Let ⊗ be the degreewise tensor product of ∆ op Shv(A).We want to show that ∆ op Shv(A) is monoidal, strongly left proper and satisfies the monoid axiom with respect to ⊗.
The Dold-Kan correspondence is unfortunately not strongly monoidal with respect to the degreewise tensor product ⊗ on ∆ op Shv(A) and the usual tensor product of chain complexes on Ch 0 (Shv(A)).We define on Ch 0 (Shv(A)) the Dold-Kan twisted tensor product Then the Dold-Kan correspondences is strongly monoidal with respect to the degreewise tensor product ⊗ on ∆ op Shv(A) and the Dold-Kan twisted tensor product ⊗ DK on Ch 0 (Shv(A)).So to show that ∆ op Shv(A) is monoidal, strongly left proper and satisfies the monoid axiom with respect to ⊗, we now just need to show that Ch 0 (Shv(A)) is monoidal, strongly left proper and satisfies the monoid axiom with respect to ⊗ DK .3.14.Lemma.Let f be a cofibration and Z an object in Ch 0 (Shv(A)).Then f So DK(f ) is computed as the morphism This is a direct sum of split monomorphisms.So DK(f ) is a degreewise split monomorphism in ∆ op Shv(A).Hence, if Z is an object in Ch 0 (Shv(A)), then the degreewise tensor product is again a split monomorphism in ∆ op Shv(A).Since DK −1 preserves monomorphisms, this then implies that is a monomorphism in Ch 0 (Shv(A)).satisfies the monoid axiom with respect to ⊗.
Proof.Since Shv(A) is a Grothendieck category, we know that injective quasi-isomorphisms in Ch 0 (Shv(A)) are stable under pushouts and transfinite compositions.So to prove the monoid axiom we just need to show that for every trivial cofibration f : A → B in Ch 0 (Shv(A)) the morphism f ⊗ DK Z is an injective quasi-isomorphism.By Lemma 3.14 we know that it is injective.So we just need to show that it is a weak equivalence.
By [27] we have for all X, Y ∈ Ch 0 (Shv(A)) a natural chain homotopy equivalence between the usual tensor product of chain complexes and the Dold-Kan twisted tensor product.
We then get a commutative diagram where vertical maps are chain homotopy equivalences, and the lower horizontal map is a weak equivalence because Ch 0 (Shv(A)) satisfies the monoid axiom with respect to ⊗.It follows that the upper horizontal map is a weak equivalence.So Ch 0 (Shv(A)) satisfies the monoid axiom with respect to ⊗ is a set of generating cofibrations.All these generating cofibrations are sheafifications of cofibrations from Ch 0 (Psh(A)).So if f and g are generating cofibrations in Ch 0 (Psh(A)), and f g is the pushout-product with respect to ⊗ DK , then we can find cofibrations f ′ and g ′ in Ch 0 (Psh(A)) such that f = L nis (f ′ ) and g = L nis (g ′ ).Then is a monoidal model category with respect to ⊗ DK it follows that f ′ g ′ is a cofibration in Ch 0 (Psh(A)), and therefore f g is a cofibration in Ch 0 (Shv(A)).All we need to show now is that a pushout-product of a cofibration with a trivial cofibration is a weak equivalence in Ch 0 (Shv(A)).So let f : A → B be a cofibration and g : C → D be a trivial cofibration in Ch 0 (Shv(A)).We need to show that the pushout-product f g with respect to ⊗ We document the above lemmas as follows.
3.18.Proposition.The model category ∆ op Shv(A) with the usual degreewise tensor product is cellular, weakly finitely generated, monoidal, strongly left proper and satisfies the monoid axiom.
From now on, weak equivalences in ∆ op Shv(A) be called local equivalences, fibrations in ∆ op Shv(A) will be called local fibrations, and fibrant objects in ∆ op Shv(A) will be called locally fibrant objects.

Relation to Γ-spaces
For every natural number n 0 let n + be the pointed set {0, . . ., n} where 0 is the basepoint.We write Γ op for the full subcategory of the category of pointed sets on the objects n + .Γ op is equivalent to the category of finite pointed sets.We write Γ for the opposite category of Γ op .This category is equivalent to the category called Γ in Segal's original paper [30].
In the additive context we do not need the category Γ as a variable in contrast to framed motivic Γ-spaces in the sense of [23].This section is to justify this fact (see Proposition 4.6).We also associate framed motivic Γ-spaces to enriched motivic A-spaces (see Proposition 4.7).
Let B be an additive model category.By ΓSpc sp (B) we denote the full subcategory of the functor category Fun(Γ op , B) consisting of those functors X : Γ op → B such that for every We have a functor EM : B → ΓSpc sp (B) given by the Eilenberg Maclane construction A. For a function f : m + → n + between pointed finite sets, we define EM(A)(f ) : A → A is the j-th projection morphism.
We have another functor ev 1 : ΓSpc sp (B) → B given by ev 1 (X ) := X (1 + ).Proof.Given a morphism ϕ : X (1 + ) → A in B, we get for every n ∈ N a morphism which together assemble into a morphism Φ(ϕ) : X → EM(A) in ΓSpc sp (B).Conversely, given a morphism ψ : X → EM(A) in ΓSpc sp (B), we can evaluate it at 1 + to get a morphism Ψ(ψ) : It is obvious that for every ϕ : X (1 + ) → A we have Ψ(Φ(ϕ)) = ϕ.Now take a morphism ψ : X → EM(A) in ΓSpc sp (B).We claim that Φ(Ψ(ψ)) = ψ.Take n ∈ N and show that A. By the universal property of the product we need to take i with 0 i n and show that the following diagram commutes But this just follows from the naturality of ψ : X → EM(A).(2) The functors EM : B → ΓSpc sp (B) and ev 1 : ΓSpc sp (B) → B preserve all weak equivalences.
(3) It is a priori not obvious that the hom-sets of the category Ho(ΓSpc sp (B)) are small.However, Proposition 4.6 below implies that they are in fact small.4.4.Lemma.A morphism ϕ : ev 1 (X ) → A is a weak equivalence in B if and only if its adjoint morphism Φ(ϕ) : X → EM(A) is a weak equivalence in ΓSpc sp (B).
Proof.Let ϕ : ev 1 (X ) → A be a weak equivalence.Take n ∈ N. Then Φ(ϕ) evaluated at n + is defined as the composite The first map is a weak equivalence, because X is a special Γ-space.The second map is a weak equivalence, because ϕ : X (1 + ) → A is a weak equivalence.Therefore Φ(ϕ) : X → EM(A) is a weak equivalence.
The following lemma is folklore.For every X ∈ ΓSpc sp (B) the identity morphism ev 1 (X ) → ev 1 (X ) is a weak equivalence, so by Lemma 4.4 applied to A = ev 1 (X ), the adjunction unit map η X : X → EM(ev 1 (X )) is a weak equivalence.This implies that the natural transformation Ho(η) is in fact a natural isomorphism of functors.
Let Fr * (k) be the category of framed correspondences.For each V ∈ Sm k let σ V : V → V be the level 1 explicit framed correspondence ({0} × V, A 1 × V, pr A 1 , pr V ).For the next result, assume that A has framed correspondences in the sense of Definition 2.4.So there is a functor Φ : Fr * (k) → A which takes every σ V to the identity on V .Let M f r be the category of pointed simplicial Nisnevich sheaves on Fr * (k): We have a monoidal adjunction L M : M ⇄ ∆ op Shv(A) : U M , where the right adjoint U M is the forgetful functor.For X, Y ∈ ∆ op Shv(A) we have a canonical map Let Sm/k + be the category of framed correspondences of level 0 as defined in [23,Example 2.4].Its morphism objects are defined by For every enriched motivic A-space X we can now define a M-enriched functor

It acts on morphism sets via the composite
With this enriched functor we can then also define a framed motivic Γ-space EM f r (X ) in the sense of [23,Definition 3.5] by defining
Axioms 2)-5) for motivic Γ-spaces follow directly from axioms 1)-4) of special enriched motivic A-spaces, except that for Axiom 2) we need to explain why the presheaf of stable homotopy groups V → π s n EM f r (X )(S, U )(V ) is radditive and σ-stable.The σ-stability follows from the fact that Φ : Fr * (k) → A sends σ V to the identity.Let us now check that it is radditive.For every U ∈ Sm k , we have that X (U ) is a sheaf of simplicial abelian groups.This implies that EM f r (X )(S, U ) is a sheaf of S 1 -spectra.So we have isomorphisms of S 1 -spectra EM f r (X )(S, U )(∅) = 0 and Since stable homotopy groups π s n preserve products and zero objects, it follows that is radditive.Axiom 7) follows from the fact that X lands in sheaves of abelian groups.
4.8.Lemma.Suppose that A has framed correspondences in the sense of Definition 2.4.Let X be an enriched motivic A-space and let EM f r (X ) be its associated framed motivic Γ-space from Proposition 4.7.Then X is very effective in the sense of Definition 2.2 if and only if EM f r (X ) is very effective in the sense of [23, Axioms Proof.This follows directly from the definitions of effectiveness for X and EM f r (X ).

Enriched functors of chain complexes
In this paper we freely use the canonical isomorphism of categories Ch([Sm, Shv(A)]) ∼ = [Sm, Ch(Shv(A))] constructed in [19].Likewise, there is a canonical isomorphism of categories In what follows we shall freely use this isomorphism.
In the previous section we associated framed motivic Γ-spaces to enriched motivic A-spaces.In this section we associate Ch(Shv(A))-enriched functors in [Sm, Ch(Shv(A))] to enriched motivic A-spaces.
5.1.Definition.Let X be an special enriched motivic A-space and let be the normalized Moore complex functor from the Dold-Kan correspondence.Denote by Λ the composite functor where the latter category is defined in [5, Section 4].
Proof.Four axioms defining special enriched motivic A-spaces correspond to four properties of functors in DM A [Sm].More precisely, the following four properties are true.
(1) X satisfies axiom (1) of special enriched motivic A-spaces if and only if for every U ∈ Sm k the complex of sheaves Λ(X )(U ) has A 1 -invariant cohomology sheaves.
(2) X satisfies the cancellation axiom (2) if and only if Λ(X ) satisfies cancellation in the sense of [5,Definition 4.6].

The Röndigs-Østvaer Theorem
Throughout this section X is a pointwise locally fibrant special enriched motivic A-space.
6.1.Definition.We can extend X to an enriched functor EM(X ) : We can take the (S 1 , G m )-evaluation of EM(X ) to get a motivic bispectrum ev S 1 ,Gm (EM(X )) ∈ SH(k).We define the bispectrum associated to X to be this bispectrum ev S 1 ,Gm (X ) := ev S 1 ,Gm (EM(X )).If A has framed correspondences, then ev S 1 ,Gm (X ) is also the evaluation of the framed motivic Γ-space EM f r (X ) from Proposition 4.7.Then by [23,Section 2.7] the bispectrum ev S 1 ,Gm (X ) = ev S 1 ,Gm (EM f r (X )) is a framed bispectrum in the sense of [22,Definition 2.1].In this case we say that ev S 1 ,Gm (X ) is the framed bispectrum associated to X .
In this section we prove the following theorem extending Röndigs-Østvaer's Theorem [29].6.2.Theorem.For every U ∈ Sm k we have a natural isomorphism where p is the exponential characteristic of k.
To prove it we will need a few lemmas.For a finite pointed set n + = {0, . . ., n} and U ∈ Sm k let n + ⊗ U be the n-fold coproduct n i=1 U .Let f M be the category of finitely presented motivic spaces in the sense of [15].Given an enriched motivic A-space X we can define an extended functor X : where A c is a cofibrant replacement of A in f M. We have for all U ∈ Sm that X (U ) ∼ = X (U ) in ∆ op Shv(A).
Let ev S 1 ,Gm ( X ) be the (S 1 , G m )-evaluation bispectrum of the extended functor X : f M → ∆ op Shv(A).
6.3.Lemma.We have a canonical isomorphism of motivic (S 1 , G m )-bispectra ev S 1 ,Gm ( X ) ∼ = ev S 1 ,Gm (X ) between the (S 1 , G m )-evaluation of the extended functor X , and the bispectrum associated with X in the sense of Definition 6.1.
Proof.By [5,Lemma 6.2] we have for all U, V ∈ Sm k an isomorphism X (U V ) ∼ = X (U ) ⊕ X (V ) in ∆ op Shv(A).This implies that we have for all U ∈ Sm k , n 0 an isomorphism So X naturally extends EM(X ) from Γ op × Sm/k + to f M.This then implies that as required.
Using [5,Section 7] we can associate to Λ(X ) an M-enriched functor Λ(X ) M : f M → Sp S 1 (M).We can take the 0-th level of this functor to get a motivic functor Λ(X ) M 0 : f M → M. By [5,Lemma 7.7] the motivic functor Λ(X ) M 0 preserves motivic equivalences between cofibrant objects.By [26, Lemma 7.2] it follows that we have an isomorphism To prove the theorem, we now just need to show that there is a natural isomorphism ev S 1 ,Gm (Λ(X ) M 0 ) → ev S 1 ,Gm (X ) in SH(k).For this we need some intermediate steps.Firstly, by Lemma 6.3 we have an isomorphism ev S 1 ,Gm ( X ) → ev S 1 ,Gm (X ).
Using [5, Section 6, Equation ( 1)] we can extend Λ(X ) to a functor Now for every A ∈ f M we have a natural quasi-isomorphism in Ch(Shv(A)) for the following reason: X (A) is the diagonal of the bisimplicial sheaf By [9, page 37, equation 24], or [12, Theorem 2.9], for every bisimplicial object S ∈ ∆ op ∆ op Shv(A) there is a quasi-isomorphism DK −1 (diag(S)) → Tot(DK −1 double (S)) in Ch 0 (Shv(A)).So for every A ∈ f M there is a quasi-isomorphism By construction Λ(X ) lands in Ch 0 (Shv(A)), so we can take the functor and form the naive (S 1 , G m )-evaluation bispectrum The above quasi-isomorphism, then induces an isomorphism in SH(k).So to prove the theorem we now just need an isomorphism By [5,Lemma 7.5], for every A ∈ f M with cofibrant replacement A c we have an isomorphism in Sp S 1 (M), where Û : Ch(Shv(A)) → Sp S 1 (M) is the canonical functor defined in [5, Section 7].Let ev 0 : Sp S 1 (M) → M be the functor taking the 0-th level of a S 1 -spectrum.So Λ(X ) M 0 = ev 0 • Λ(X ) M .By the proof of [5,Lemma 7.4], the functor ev 0 • Û is isomorphic to the composite Ch(Shv(A)) where τ 0 is the good truncation functor and U is the forgetful functor.Since Λ(X ) lands in Ch 0 (Shv(A)), it does not get changed by truncation.So we get that Since S 1 and G m are cofibrant in f M, we get an isomorphism Putting it all together, we get a commutative diagram in which all the vertical maps and the top horizontal map are isomorphisms in SH(k) [1/p].It follows that the bottom horizontal map is also an isomorphism in SH(k) [1/p].This completes the proof.

A motivic model structure for enriched motivic A-spaces
In Section 3 we showed that ∆ op Shv(A) with the degreewise tensor product ⊗ has a model structure that is cellular, weakly finitely generated, monoidal, strongly left proper and satisfies the monoid axiom (see Proposition 3.18).We can apply [14,Theorem 4.2]  In this section we define another model structure on [Sm, ∆ op Shv(A)] such that the fibrant objects are the pointwise locally fibrant special enriched motivic A-spaces.
, which induces a map on homotopy fibers We let N is be the family of morphisms consisting of p Q for every elementary Nisnevich square Q.
Finally, we let ∼ denote the union of all these four classes of morphisms.
be the simplicial sheaf of morphisms from X to Y .It is defined by taking the internal hom Hom [Sm,∆ op Shv(A)] (X, Y ) and evaluating it at the point pt ∈ Sm.
For U ∈ Sm k and n 0 we have Similarly to [5,Definition 4.3], given a class of morphisms S in [Sm, ∆ op Shv(A)] and an object X ∈ [Sm, ∆ op Shv(A)] with pointwise locally fibrant replacement X f we say that X is strictly S-local if for every s : A → B with s ∈ S the morphism  Proof.Let F f be a pointwise locally fibrant replacement of F .For every s : A → B, s ∈ S let s c : A c → B c be a cofibrant replacement of s.This means we have a commutative square such that the vertical maps are trivial fibrations, A c and B c are cofibrant and s c is a cofibration.Note that for every s ∈ S the domain A and codomain B are already cofibrant, but s is not neccessarily a cofibration.
For X, Y ∈ [Sm, ∆ op Shv(A)] let map ∆ op Set (X, Y ) ∈ ∆ op Set denote the non-derived simplicial mapping space.It can be defined by map ∆ op Sets (X, Y ) := Hom [Sm,∆ op Shv(A)] (X, Y )(pt)(pt).Now F f is S-local in the usual model category theoretic sense if and only if for every s ∈ S the map s c, * : map Since the functor map ∆ op Sets (−, F f ) sends trivial cofibrations to trivial fibrations, it follows by Ken Brown's lemma [25,Lemma 1.1.12],that map ∆ op Sets (−, F f ) sends weak equivalences between cofibrant objects to weak equivalences.Since the maps A c → A and B c → B are weak equivalences between cofibrant objects, it follows that the vertical maps in the above commutative diagram are weak equivalences.Therefore F f is S-local if and only if for every s ∈ S the map s * : map is a weak equivalence.Every s ∈ S is of the form t ⊗ Z for some Z ∈ Sm k and t : C → D with t ∈ S. We have a commutative diagram in which the vertical maps are isomorphisms:  The preceding theorem is also reminiscent of Bousfield-Friedlander's theorem [6] stating that fibrant objects in the model category of classical Γ-spaces are given by very special Γ-spaces.

Reconstructing DM
) n 0 , where M A (X) := C * A(−, X) nis is the A-motive of X.We call M Gm A (U ) the big A-motive of U .So we just need to show for X ∈ Spc A [Sm] that X ∈ Spc veff A [Sm] if and only if ev Gm (X ) ∈ DM eff A, 0 .By Proposition 8.4 we know that X ∈ Spc veff A [Sm] if and only if ev S 1 ,Gm (X ) ∈ SH f r nis (k) is effective.By [22,Theorem 3.6] this is the case if and only if ev S 1 ,Gm (X ) lies in SH eff (k).By Lemma 8.10 we have a canonical isomorphism ev S 1 ,Gm (X ) ∼ = U (ev Gm (X )) in SH(k).So ev S 1 ,Gm (X ) ∈ SH eff (k) if and only if U (ev Gm (X )) ∈ SH eff (k) and by Lemma 8.9 this is the case if and only if ev Gm (X ) ∈ DM eff A , which proves the theorem.9. Reconstructing SH veff (k) Q In this section we apply the techniques and results from the previous sections to give new models for the stable motivic homotopy category of effective and very effective motivic bispectra with rational coefficients.It also requires the reconstruction theorem by [18] and the theory Milnor-Witt correspondences [3,7,8,11,16,17].
Let Cor be the category of finite Milnor-Witt correspondences in the sense of [8].Then Cor is a strict V -category of correspondences satisfying the cancellation property (See [16] for details).Furthermore it has framed correspondences by [11].It also satisfies the ∆-property by [3].
Denote by SH(k) Q the category of motivic bispectra E whose sheaves of stable motivic homotopy groups π A 1 * , * (E) are sheaves of rational vector spaces.The category SH(k) Q is also called the rational stable motivic homotopy category.It is the homotopy category of a stable model structure in which weak equivalences are those morphisms of bispectra f : E → E ′ for which π A 1 * , * (f )⊗Q is an isomorphism.Let SH(k) Q, 0 be the full subcategory of SH(k) Q on the connective objects.Here a bispectrum object X ∈ SH(k) Q with rational stable A 1 -homotopy groups π A 1 p,q (X) ⊗ Q is called connective, if π A 1 p,q (X) ⊗ Q ∼ = 0 for all p < q.
Throughout this section we assume the base field k to be perfect of characteristic different from 2. The assumption on the characteristic is typical when working with finite Milnor-Witt correspondences.A theorem of Garkusha [18,Theorem 5.5] states that the forgetful functor U : DM Cor,Q → SH(k) Q is an equivalence of categories.This theorem was actually proven under the assumption that k is also infinite.The latter assumption is redundant due to [13, A.27] saying that the main result of [20] about strict invariance for Nisnevich sheaves with framed transfers is also true for finite fields.
For this we just need to show that a special enriched motivic A-space X is very effective if and only if ev S 1 ,Gm (X ) is very effective in SH(k).
According to Proposition 8.4 the special enriched motivic A-space X is very effective if and only if the framed bispectrum ev S 1 ,Gm (X ) is effective in SH(k) f r nis .By [22,Theorem 3.6] this is the case if and only if ev S 1 ,Gm (X ) is effective in SH(k).This concludes the proof of the theorem.

DK . 3 . 16 . 3 . 17 .
Lemma.Ch 0 (Shv(A)) is strongly left proper with respect to ⊗ DK .So ∆ op Shv(A) is strongly left proper with respect to ⊗. Proof.Since Shv(A) is a Grothendieck category, quasi-isomorphisms in Ch 0 (Shv(A)) are stable under pushouts along monorphisms.For any cofibration f the map f ⊗ DK Z is a monomorphism by Lemma 3.14.So Ch 0 (Shv(A)) is strongly left proper with respect to ⊗ DK .Lemma.Ch 0 (Shv(A)) is a monoidal model category with respect to ⊗ DK .So ∆ op Shv(A) is a monoidal model category with respect to ⊗. Proof.The unit for ⊗ DK is the chain complex Z concentrated in degree 0. That is a cofibrant object, so Ch 0 (Shv(A)) satisfies the unit axiom.Let us now show the pushout-product axiom.The category of simplicial abelian groups ∆ op Ab is monoidal and satisfies the monoid axiom with respect to the degreewise tensor product of chain complexes ⊗.If we define a Dold-Kan twisted tensor product ⊗ DK on chain complexes of abelian groups Ch 0 (Ab) by X ⊗ DK Y = DK −1 (DK(X) ⊗ DK(Y )) then Ch 0 (Ab) with the standard projective model structure and tensor product ⊗ DK is a monoidal model category satisfying the monoid axiom.Similarly, we can also define a Dold-Kan twisted tensor product ⊗ DK on chain complexes of presheaves Ch 0 (Psh(A)), and it coincides with the Day convolution product induced by the Dold-Kan twisted tensor product on Ch 0 (Ab) and the monoidal structure of A. By [19, Theorem 5.5] it follows that Ch 0 (Psh(A)) with standard projective model structure and the Dold-Kan twisted tensor product ⊗ DK is a monoidal model category.For Ch 0 (Shv(A)) the set

D
The morphism h is a base change of A ⊗ DK g.Since g is a trivial cofibration and Ch 0 (Shv(A)) satisfies the monoid axiom with respect to ⊗ DK , this means that h is a weak equivalence in Ch 0 (Shv(A)).Similarly B ⊗ DK g is a weak equivalence in Ch 0 (Shv(A)).So by 2-of-3 it follows that f g is a weak equivalence in Ch 0 (Shv(A)).So Ch 0 (Shv(A)) is a monoidal model category.

4. 2 .
Definition. (1) Let B be an additive model category.A morphism f : X → Y in ΓSpc sp (B) is called a weak equivalence if and only if for every n ∈ N the map f (n + ) : X (n + ) → Y(n + ) is a weak equivalence in the model category B. We write W for the class of weak equivalences in ΓSpc sp (B).(2) We write Ho(ΓSpc sp (B)) for the localization of ΓSpc sp (B) with respect to the class of weak equivalences W : Ho(ΓSpc sp (B)) := ΓSpc sp (B)[W −1 ]. 4.3.Remark.(1) All isomorphisms in ΓSpc sp (B) are weak equivalences.Weak equivalences in ΓSpc sp (B) satisfy the 2-out-of-3 property.

4. 5 .
Lemma.Let C, D be categories, each equipped with a class of morphisms, called the weak equivalences, satisfying the 2-out-of-3-property.Let Ho(C), Ho(D) be the homotopy categories of C, D, i.e. the categories obtained by inverting the weak equivalences.Let ℓ C : C → Ho(C) be the localization functor of C, and ℓ D : D → Ho(D) be the localization functor of D. Let F, G : C → D be functors sending weak equivalences in C to weak equivalences in D. Let τ : F → G be a natural transformation.Then the functors F, G induce functors Ho(F ), Ho(G) : Ho(C) → Ho(D) satisfying Ho(F ) • ℓ C = ℓ D • F , Ho(G) • ℓ C = ℓ D • G, and τ : F → G induces a natural transformation Ho(τ ) : Ho(F ) → Ho(G) such that for every A ∈ C, the component of Ho(τ ) at A is given by Ho(τ ) A = ℓ D (τ A ).The following statement informally says that Γ-spaces in an additive category B are entirely recovered by B itself (up to homotopy).4.6.Proposition.The adjunction ev 1 ⊣ EM induces an equivalence of categories Ho(ev 1 ) : Ho(ΓSpc sp (B)) ∼ ⇆ Ho(B) : Ho(EM).Proof.Since ev 1 and EM preserve weak equivalences, they induce two functors Ho(ev 1 ) : Ho(ΓSpc sp (B)) → Ho(B) and Ho(EM) : Ho(B) → Ho(ΓSpc sp (B)) on the homotopy categories.For the adjunction ev 1 ⊣ EM there is a unit η : Id ΓSpc sp (B) → EM • ev 1 .By Lemma 4.5, applied to F = Id ΓSpc sp (B) , G = EM • ev 1 and τ = η , it induces a natural transformation Ho(η) : Id Ho(ΓSpc sp (B)) → Ho(EM) • Ho(ev 1 ).
is a very special framed motivic Γ-space in the sense of[23, Axioms 1.1].
to this model structure to get a weakly finitely generated model structure on the category of enriched functors [Sm, ∆ op Shv(A)] in which the weak equivalences, respectively fibrations, are the Sm-pointwise local equivalences, respectively Sm-pointwise local fibrations.We call this the local model structure on [Sm, ∆ op Shv(A)].By [14, Theorem 4.4] the local model structure on [Sm, ∆ op Shv(A)] is monoidal with the usual Day convolution product.By [14, Corollary 4.8] the local model structure on [Sm, ∆ op Shv(A)] is left proper.Since [Sm, ∆ op Shv(A)] is weakly finitely generated, and all cofibrations in ∆ op Shv(A) are monomorphisms, it follows that [Sm, ∆ op Shv(A)] is cellular.Note that for every U ∈ Sm k the representable functor Sm(U, −) ∼ = Sm(U, −) ⊗ pt is cofibrant in [Sm, ∆ op Shv(A)].

( 1 )
We let A 1 1 be the family of morphisms consisting ofSm(U, −) ⊗ A 1 → Sm(U, −) for every U ∈ Sm k .(2)We let τ be the family of morphisms consisting of the evaluation mapSm(G ∧n+1 m × U, −) ⊗ G ∧1 m → Sm(G ∧n m × U, −)for every n 0 and U ∈ Sm k .(3) We let A 1 2 be the family of morphisms consisting ofSm(U, −) → Sm(U × A 1 , −)for every U ∈ Sm k .(4) We let N is be the following family of morphisms: For every elementary Nisnevich square Q of the form

7. 4 .
Definition.Given a class of morphisms S in [Sm, ∆ op Shv(A)], we write S for the class of morphisms S := {s ⊗ Z | s ∈ S, Z ∈ Sm k }.We define the enriched motivic model structure on [Sm, ∆ op Shv(A)] to be the left Bousfield localization of the local model structure on [Sm, ∆ op Shv(A)] with respect to the class of morphisms ∼.This model category will be denoted by [Sm, ∆ op Shv(A)] mot .7.5.Lemma.Let S be a class of morphisms in [Sm, ∆ op Shv(A)] with cofibrant domains and codomains.Then an object F ∈ [Sm, ∆ op Shv(A)] is strictly S-local if and only if its local fibrant replacement F f is S-local in the usual model category theoretic sense of [24, Definition 3.1.4].

7. 6 .
Definition.Let D([Sm, ∆ op Shv(A)]) be the homotopy category of [Sm, ∆ op Shv(A)] with respect to the pointwise local model structure.Define Spc A [Sm] as the full subcategory of D([Sm, ∆ op Shv(A)]) consisting of special enriched motivic A-spaces.We document above lemmas as follows.7.7.Theorem.The category Spc A [Sm] is equivalent to the homotopy category of the model category [Sm, ∆ op Shv(A)] mot .The fibrant objects of [Sm, ∆ op Shv(A)] mot are the pointwise locally fibrant special enriched motivic A-spaces.
Let U : DM A → SH(k) be the canonical forgetful functor, and let L : SH(k) → DM A be its left adjoint.Let DM eff A be the full triangulated subcategory of DM A compactly generated by the set {M Gm A (U ) | U ∈ Sm k }.See 8.1 for the definition of M Gm A (U ).Recall that SH eff (k) is the full subcategory of SH(k) generated by the suspension bispectra Σ ∞ S 1 ,Gm U + for U ∈ Sm k .8.6.Lemma.Let C and D be triangulated categories, and let F : C → D be a triangulated functor.Assume that F preserves small coproducts.Let S C be a full triangulated subcategory of C compactly generated by a set Σ C .Let S D be a full triangulated subcategory of D closed under small coproducts.Assume that for every A ∈ Σ C we have F (A) ∈ S D .Then for every A ∈ S C we have F (A) ∈ S D .In particular F restricts to a triangulated functor F : S C → S D .Proof.Consider the full subcategory F −1 (S D ) in C consisting of all those objects A ∈ C for which F (A) ∈ S D .We need to show that S C ⊆ F −1 (S D ).Since Σ C ⊆ F −1 (S D ), it suffices due to[28, Theorem 2.1]  to show that the subcategory F −1 (S D ) is a triangulated subcategory closed under triangles and small coproducts in C.If we have a triangleX → Y → Z → ΣX in C with X, Y ∈ F −1 (S D ), then F (X) → F (Y ) → F (Z) → ΣF (X)is a triangle in D with F (X), F (Y ) ∈ S D .Since S D is closed under triangles it follows that F (Z) ∈ S D , so Z ∈ F −1 (S D ), so F −1 (S D ) is closed under triangles.Since F preserves small coproducts and S D is closed under small coproducts, it follows that F −1 (S D ) is closed under small coproducts.Therefore F −1 (S D ) is closed under triangles and small coproducts.We get that S C ⊆ F −1 (S D ), which proves the lemma.8.7.Lemma.If X ∈ SH eff (k), then L(X) ∈ DM eff A .So the functor L : SH(k) → DM A restricts to a functor L eff : SH eff (k) → DM eff A .Proof.By Lemma 8.2 we have L(Σ ∞ S 1 ,Gm U + ) ∼ = M Gm A (U ) ∈ DM eff A .Since the Σ ∞ S 1 ,Gm U + compactly generate SH eff (k) the result now follows from Lemma 8.6.8.8.Lemma.The triangulated functor U : DM A → SH(k) preserves small coproducts.Proof.Let I be a set, and {A i | i ∈ I} a family of objects.We want to show that the canonical morphismi∈I U (A i ) → U ( i∈I A i ) is an isomorphism in SH(k).The triangulated category SH(k) is compactly generated by the set Σ SH(k) := {Σ ∞ S 1 ,Gm U + ∧ G ∧n m | U ∈ Sm k , n ∈ Z}.Proof.By Theorem 8.3 we have an equivalence ev Gm : Spc A [Sm] → DM A, 0 .
[14,.Lemma.The model category Ch 0 (Shv(A)) is cellular.Proof.Due to Corollary 3.11 we just need to show that cofibrations in Ch 0 (Shv(A)) are effective monomorphisms.If f is a cofibration in Ch 0 (Shv(A)), then ι(f ) is a cofibration in Ch(Shv(A)).Then f is a monomorphism in Ch(Shv(A)) and in Ch 0 (Shv(A)).Since Ch 0 (Shv(A)) is an abelian category, every monomorphism is effective.Hence f is an effective monomorphism.3.13.Lemma.The model category Ch 0 (Shv(A)) is strongly left proper in the sense of[14,  Definition 4.6] of sheaves.7.3.Lemma.A enriched motivic A-space X : Sm → Shv(A) is special if and only if it is strictly ∼-local.Proof.By Lemma 5.2 we know that X is special if and only if Λ(X ) lies in DM A [Sm].By [5, Proposition 4.13] this is the case if and only if Λ(X ) is strictly ∼-local in the sense of [5, Definition 4.3], and this is the case if and only if X is strictly ∼-local in the sense of Definition 7.2.