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Advances in Mathematics

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Symmetric powers in abstract homotopy categories

S. Gorchinskiya'b, V. Guletskiic'*

a Steklov Mathematical Institute, Gubkina str. 8, 119991, Moscow, Russia b AG Laboratory, HSE, Vavilova str. 7, 117312, Moscow, Russia c Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom

CrossMark

a r t i c l e i n f o

a b s t r a c t

Article history:

Received 21 February 2015

Received in revised form 16

December 2015

Accepted 26 January 2016

Communicated by the Managing

Editors of AIM

MSC: 14F42 18D10 18G55

Keywords:

Model structures

Symmetric monoidal model

categories

Symmetric powers

Generating cofibrations

Localization of model categories

Symmetric spectra

A1-homotopy theory

Motivic spaces and motivic

symmetric spectra

We study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories, in both unstable and stable settings. As an outcome, we prove that symmetric powers preserve the Nisnevich and étale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base. More precisely, if f is a weak equivalence of motivic spaces, or a weak equivalence between positively cofibrant motivic spectra, with respect to the Nisnevich or étale topology, then all symmetric powers Symn (f) are weak equivalences too. This gives left derived symmetric powers in the corresponding motivic homotopy categories of schemes over a base, which aggregate into a categorical A-structures on these categories.

© 2016 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

* Corresponding author.

E-mail addresses: gorchins@mi.ras.ru (S. Gorchinskiy), vladimir.guletskii@liverpool.ac.uk (V. Guletskii).

http://dx.doi.org/10.1016/j.aim.2016.01.011

0001-8708/© 2016 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Contents

1. Introduction...................................................... 708

2. Preliminary results....................................................................................................711

3. Symmetrizable cofibrations........................................................................................713

4. The proof of Theorem 7............................................................................................718

5. Kunneth towers for cofibre sequences..................................... 727

6. Localization of symmetric powers........................................ 731

7. Localization w.r.t. diagonalizable intervals....................................................................736

8. Positive model structures on spectra............................................................................739

9. Symmetric powers in stable categories..........................................................................745

10. Symmetrizable cofibrations in topology........................................................................748

11. Symmetrizable cofibrations in A1-homotopy theory of schemes.................... 749

Acknowledgments....................................................... 752

Appendix A. Categorical v.s. geometrical symmetric powers......................... 752

References............................................................ 754

1. Introduction

In topology, symmetric powers preserve homotopy type of C^-complexes, which is at the heart of the fundamental Dold-Thom theorem connecting the homology of a complex to the homotopy groups of its infinite symmetric power, see [2]. A natural question is to which extent such phenomena could be true in the motivic Ax-homotopy theory of schemes over a base? The first steps in this direction were made in the pioneering work [21]. In [23] Voevodsky developed a motivic theory of symmetric powers, good enough to construct motivic Eilenberg-MacLane spaces needed for the proof of the Bloch-Kato conjecture. His symmetric powers depend on symmetric powers of schemes presenting motivic spaces. The aim of this paper is to develop a purely homotopical theory of symmetric powers in an abstract symmetric monoidal model category, and to give an affirmative answer to the question when symmetric powers preserve weak equivalences in such a category, working out the unstable and stable settings separately.

More technically, working in a closed symmetric monoidal model category C, we address the following two fundamental questions in the paper. Whether left derived symmetric powers exist in the homotopy category Ho(C) and, if they do, whether they aggregate into a (categorical) A-structure on the homotopy category of C? The latter concept means that, given a morphism in Ho(C), there exists a tower connecting the derived symmetric powers of the domain and codomain, whose cones can be computed by the Kunneth rule. A categorical A-structure is then a system of Kunneth towers, func-torial on morphisms in Ho(C). If the categorical A-structure preserves compact objects in C, then it induces a usual A-structure on the ^0-ring of the Waldhausen category of cofibrant compact objects in C.

We develop a general machinery to deal with that kind of questions in C, and in symmetric spectra over C. The methods for the stable and unstable cases are surprisingly different. In the unstable setting, we introduce the notion of symmetrizable cofibrations and study how symmetrizability behaves under cofibrant generation and localization in

the sense of [8]. With this aim, we provide quite a general condition on a left derived functor so that it factors through the localized homotopy category. The main type of localization is the contraction of a diagonalizable interval in C. In the stable setting we construct a positive model structure on the category of symmetric spectra, in which weak equivalences are the usual stable weak equivalences and all cofibrations are isomorphisms on level zero. Our positive model structure is an utmost generalization of the topological positive model structure constructed in [3], and the motivic positive model structure introduced in [9]. Positive model structures are the main tool in the study of symmetric powers of abstract symmetric spectra over C.

Our main destination is, however, the motivic Ax-homotopy theory of schemes, and we anticipate numerous applications of our methods and results in arithmetic and geometry through that theory. For the present, we prove the following two theorems giving positive answers to the questions above in the unstable and stable motivic homotopy theory of schemes over a base:

Theorem A. Symmetric powers preserve the Nisnevich and étale homotopy type of motivic spaces, left derived symmetric powers exist in the unstable motivic homotopy category of schemes over a base and aggregate into a categorical \-structure on it.

Theorem B. Symmetric powers preserve stable weak equivalences between positively cofi-brant motivic symmetric spectra, left derived symmetric powers exist in the motivic stable homotopy category of schemes over a base and aggregate into a categorical X-structure on it. The left derived symmetric powers of motivic spectra coincide with the corresponding homotopy symmetric powers.

In a broader context, homotopical theory of symmetric powers has many potential applications. For example, it can be used to construct a model structure on commutative monoids, and a global model structure for ultra-commutative monoids in a symmetric monoidal model category, see [24] and [20]. The results and technique of the present paper are further developed and extended to the setting of abstract symmetric operads in [18]. In [17] the obtained results are used to prove that the motivic rational homotopy type of symmetric powers of motivic spectra and motivic spectra of geometric symmetric powers coincide. Finally, in [7] we used our theory to discover a new phenomenon in Chow groups of algebraic varieties over a field.

Now we give a road map of the paper. We start by introducing the notion of sym-metrizable (trivial) cofibrations in C. To study left derived symmetric powers, it would be natural to consider (trivial) cofibrations whose symmetric powers are again (trivial) cofibration. However, we need to introduce a stronger property so that it becomes invariant under compositions and pushouts. Loosely speaking, (trivial) cofibrations are symmetrizable (Definition 3) if they are stable under taking colimits of the action of symmetric groups on their pushout products in C. If cofibrations are symmetrizable, then one can associate, to a cofibre triangle in C, a tower of cofibrations connecting symmetric

powers of the vertices of the triangle, and whose cones can be computed by Kunneth's rule. Such Kunneth towers can be viewed as a sort of categorification of A-structures in commutative rings (Definition 24), and give a powerful tool to work out symmetric powers (Theorem 22). If trivial cofibrations between cofibrant objects are symmetriz-able, then symmetric powers preserve weak equivalences between cofibrant objects and so admit their left derived endofunctors on C (Theorem 25). When C is cofibrantly generated by the set of generating cofibrations I and the set of trivial generating cofibrations J, and if the sets I and J are both symmetrizable, then all cofibrations and trivial cofibrations in C are symmetrizable (Theorem 7 and Corollary 9). This is useful in applications to concrete cofibrantly generated monoidal model categories, and will be applied to symmetric spectra in Section 9. If, in addition, symmetric powers of cofibrant replacements of morphisms in a set of morphisms S are S-local equivalences, then trivial cofibrations between cofibrant objects in the left localization CS are symmetrizable (Theorem 33). To show this, we give a condition on a left derived functor (which might not have right adjoint) to factor it through the localized homotopy category (Theorem 29). This result can be applied to a broad range of Bousfield localizations. An important particular case is when S-localization is a contraction of a diagonalizable interval into a point (Theorem 42).

In topology, i.e. when C is the category of simplicial sets, all cofibrations and trivial cofibrations are symmetrizable (Proposition 60). If C is the unstable model category of motivic spaces over a base, i.e. the model category for the unstable A1-homotopy category of schemes, cofibrations come up from the simplicial side, so that they are symmetrizable too. The A1-localization is a crux, and Theorem 42 gives that symmetrizability of trivial cofibrations is stable under A1-localization. In turn, this gives that trivial cofibrations between motivic spaces are symmetrizable, so that the above Theorem 22 and Theorem 25 are applicable in the motivic unstable homotopy theory of schemes over a base. Collecting all these things together we obtain the above Theorem A (Theorem 62 in the text).

In the stable world, the approach is different. In this paper, a stable homotopy category is the homotopy category of the category S of symmetric spectra over a closed symmetric monoidal model category C, stabilizing a smash-with-T functor for a cofibrant object T in C, see Sections 7 and 8 in [11]. This generalizes topological symmetric spectra introduced and studied in [12]. The symmetricity of spectra is essential to have the monoidal structure and its compatibility with the model one, see Theorem 8.11 in [11]. There are two crucial ingredients in working out symmetric powers of symmetric spectra. The first one is the existence and construction of the positive stable model structure for abstract symmetric spectra (Definition 47 and Theorem 50). The second ingredient is that n-th monoidal powers of positively cofibrant spectra are positively level-wise S„-equivariantly cofibrant (Proposition 53). Using these results we prove (Theorem 55) a pretty general version of the theorem due to Elmendorf, Kriz, Mandell and May saying that the n-th symmetric power of a positively cofibrant topological spectrum is stably equivalent to the n-th homotopy symmetric power of that spectrum, see [3], Chapter III, Theorem 5.1,

and [15], Lemma 15.5. Our method, however, is different from the one in [15]. In constructing positive model structures we systematically use Hirschhorn's localization and in proving Theorem 55 we use Theorem 7 on the stability of symmetrizable (trivial) cofi-bration under cofibrant generation. Theorem 55 implies that symmetric powers preserve positive and stable weak equivalences between cofibrant objects in the positive model structure in S (Corollary 56). In one turn, this gives A-structure of left derived symmetric powers in the stable homotopy category Ho(T) (Corollary 57). Notice also that the left derived symmetric powers of symmetric spectra are canonically isomorphic to the corresponding homotopy symmetric powers. Now, applying the above general results for symmetric spectra to motivic symmetric spectra of schemes over a base, we obtain Theorem B (Theorem 64 below).

2. Preliminary results

To get started we recall the notion of a closed symmetric monoidal model category C. Such a category is equipped with three classes of morphisms, weak equivalences, fibra-tions and cofibrations, which have the standard lifting properties and meanings, see Chapter 1, §1 in [19], or Section 1.1 in [10]. The monoidality of C means that we have a functor A : C x C ^ C sending any ordered pair of objects X, Y into their monoidal product X A Y, and that product is symmetric, i.e. there exists a functorial transposition isomorphism X A Y ~ Y A X. Moreover, the product A is also functorially associative, and there exists a unit object 1, such that 1 A X ~ X and X A 1 ~ X for any X in C. The monoidal product could be also denoted by < but we prefer to keep to the "pointed" notation A. Coproducts will be denoted by V.

A substantial thing here is that monoidality has to be compatible with the model structure. Namely, let f : X ^ Y and f' : X' ^ Y' be two morphisms in C and let

(X A Y') Vxax' (Y A X')

be the colimit of the diagram

X A X'-f-Y A X'

X A Y'

A pushout product of f and f' is, by definition, the unique map f Of' : (X A Y') Vxax' (Y A X') Y A Y'

induced by the above colimit. The relation between the model and monoidal structures can be expressed by the following axioms, see Section 4.2 in [10]:

(A1) If f and f' are cofibrations then f □f' is also a cofibration. If, in addition, one of

the maps f and f' is a weak equivalence, then so is f □/'. (A2) If q : Q1 ^ 1 is a cofibrant replacement for the unit object 1, then the maps q A id : Q1 A X ^ 1 A X and id A q : X A Q1 ^ X A 1 are weak equivalences for all cofibrant X.

Here (A1) is called the pushout product axiom, and (A2) is called the unit axiom. The functor X A — has right adjoint functor Hom(X, —). It follows that X A — commutes with colimits.

If C is simplicial, we will require that the simplicial structure is compatible with all the structures above in the sense of Definition 4.2.18 in [10].

Now, for any natural number n let £„ be the symmetric group of permutations of n elements, considered as a category with single object and morphisms being elements of the group. Given an object X in C we have a functor from £„ to C sending the unique object in £„ into XAn, and permuting factors using the commutativity and associativity constrains in C. The n-th symmetric power Symn(X) of X is a colimit of this functor. Clearly, Symn is an endofunctor on C.

Lemma 1. Suppose that C is a closed symmetric monoidal model category. Assume, moreover, that C is a simplicial model category, and the functor K ^ 1 A K from simplicial sets to C is symmetric monoidal. Let f, g : X ^ Y be two morphisms in C which are left homotopic, i.e. there exists a morphism H : X A A[1] ^ Y, such that H0 = f and H1 = g, where A[1] is the simplicial interval in AopSets. Then, for any natural n, the morphism Symn(f) is left homotopic to the morphism Symn(g).

Proof. Let 5n : A[1] ^ A[1]An be the diagonal morphism for the simplicial interval A[1], and let an : XAn A A[1] ^ (X A A[1])An be the composition of the morphism idXA^ASn with the isomorphism between XAnAA[1]An and (XAA[1])An. Then HAnoan : XAn A A[1] ^ YAn is a left homotopy between fAn and gAn. The cylinder functor — A A[1] has right adjoint, so commutes with colimits. Permuting factors does not affect the diagonal, and the functor K ^ 1 A K is symmetric monoidal by the hypothesis. Therefore, the permutation of factors in (X A A[1])An is coherent with the permutation of factors in XAn in the product XAn A A[1]. Taking colimits over £n we obtain a left homotopy between Symn(f) and Symn(g). □

Example 2. The existence of a simplicial structure, and its compatibility with the symmetric monoidal structure on C in Lemma 1 are essential. Indeed, let Kom(Z) be the category of unbounded complexes of abelian groups. The category Kom(Z) inherits the symmetric monoidal structure via total complexes Tot( — <8>—), and has a natural struc-

ture of a model category whose weak equivalences are quasi-isomorphisms and fibrations are termwise epimorphisms, see Section 2.3 in [10]. Then Kom(Z) is a closed symmetric monoidal model category by Proposition 4.2.13 in [10]. It is not known whether Kom(Z) is a simplicial model category, see page 114 in [10]. The following argument, taken together with Lemma 1, shows that there is no a simplicial structure compatible with the monoidal model structure on Kom(Z), such that the functor K — 1 A K would be symmetric monoidal. Let X be the complex ... — 0 — Z — Z — 0 — ..., where Z is concentrated in degrees —1 and 0 respectively. This complex is homotopically trivial. On the other hand, a calculation shows that Sym2(X) is the complex

... — 0 — Z/2 —— Z —— Z — 0 — ... ,

where Z/2 stands in degree —2. Clearly, this Sym2(X) has non-trivial cohomology group in degree —2.

Let now C be as in Lemma 1, and let Ho(C) be the homotopy category of C. A naive way to define symmetric powers in Ho(C) would be through Lemma 1 and the standard treatment of homotopy categories as subcategories of fibrant-cofibrant objects factorized by left homotopies on Hom-sets, see [10, 1.2] or [19]. Indeed, let Ccf be the full subcategory of objects which are fibrant and cofibrant simultaneously. Let, furthermore, ho(C) be the quotient category of Ccf by left homotopic morphisms between fibrant-cofibrant objects in C. As symmetric powers respect left homotopies by Lemma 1, we have now a functor Sym" : ho(C) — Ho(C). The category C, being a model category, is endowed with a fibrant replacement functor R : C — Cf and a cofibrant replacement functor Q : C — Cc. Combining both we obtain mixed replacement functors RQ and QR from C to the full subcategory Ccf of fibrant-cofibrant objects in C, any of which induces a quasi-inverse to the obvious functor from ho(C) to Ho(C). Then one might wish to construct an endofunctor Sym" on Ho(C) as a composition of this quasi-inverse and the above Sym". However, in general this method does not give left derived symmetric powers on Ho(C).

3. Symmetrizable cofibrations

In this section, we introduce the notion of symmetrizable (trivial) cofibrations. The main result, Theorem 7, asserts that this property is stable under pushouts, retracts and transfinite compositions. This gives that, in order to check symmetrizability of (trivial) cofibrations, it is enough to examine it on generating (trivial) cofibrations, see Corollaries 9, 10 and 11.

Let C be a closed symmetric monoidal model category with the monoidal product A : C x C — C. For any two morphisms f : X — Y and f' : X' — Y' in C, let

□ (f,f') = (X A Y') Vxax' (Y A X')

be the coproduct over X A X' (not to be confused with the morphism f df' defined in Section 2 before the axioms (A1) and (A2)). The pushout product □ is commutative and associative in the obvious sense. For example, for any three morphisms f : X — Y, f' : X' — Y' and f" : X" — Y" in C the morphism (fdf')□f" is the same as the morphism fa(f'nf") up to the canonical isomorphism between □(fnf, f") and

□ (f, ff"). Since □ is an associative operation, for any finite collection fi : Xi — Yj, i = 1, ..., l, of morphisms in C we have a well defined morphism

f1D ...aft : a(.h,...,fi) —- Y1 A •••ay .

For simplicity, let X' = X, Y' = Y and f' = f. Then we have the d-squares n1(f) =

□ (f, f) and fn2 = fnf, which can be generalized for higher degrees as follows. Let r be the category with two objects 0 and 1 and one morphism 0 —■ 1, and let rn be the n-fold Cartesian product of r with itself. Objects in rn are ordered n-tuples of 0's and 1's. A functor K : r —■ C is just a morphism f : X — Y in C. It is also natural to write K(f) rather than K, since K is fully determined by the morphism f. For any natural n let Kn be the composition of the n-fold Cartesian product rn ^ Cn and the functor Cn -A C. For any 0 < i < n one has a full subcategory rn in rn generated by n-tuples having not more than i units in them. The restriction of Kn on rn will be denoted by Kn(f), or simply by K" when f is clear. In other words, K" is a subdiagram in Kn having not more than i factors Y in each vertex. Let then

□n(f) = colim Kn(f)

or simply

□n = colim Kn .

Since K$ = Xn and Kn = Kn, we have that = Xn and = Yn, respectively. As Kin 1 is a subdiagram in Kin one has a morphism on colimits

□n-1 —- an

for any 1 < i < n.

Suppose C is cofibrantly generated. Let G be a finite group considered as a one-object category, and let CG be the category of functors from G to C. We shall be using the standard model structure on CG provided by Theorem 11.6.1 in [8]. In particular, given a morphism f in C G, it is a weak equivalence (fibration) in CG if and only if the same f, as a morphism in C, is a weak equivalence (fibration) in C. For any object X in CG, let X/G be the colimit of the action of the group G on X. This is a functor from CG to C preserving cofibrations, see Theorem 11.6.8 in [8].

The group En acts on rn and so on Kn. Each subcategory rn is invariant under the action of En. Then En acts on Kj1 and so on d". Let

□ "(f) = colim s„ n"(f)

for each index i. Obviously, □ "(f) = Sym"(X) and □"(f) = Sym"(Y), and for each index i we have a universal morphism between colimits

□ "_1(f) —— □"(f) . Sometimes we will drop the morphism f from the notation writing

□" = colim s„ □" ,

□" _V □"

□¿-1 > □¿-1 ,

In new notation, the axiom (A1) of a monoidal model category says, in particular, that for any cofibration f : X — Y in C the pushout product

fD2 : □Kf) —— Y A Y

is also a cofibration in C. By associativity, it implies that the morphism

f □" : Q"-1(f) —— YA"

is a cofibration in C for any natural n, not only for n = 2. It doesn't mean, of course, that the £"-equivariant morphism f□" is a cofibration in CEn.

Definition 3. A morphism f : X — Y in C is said to be a symmetrizable (trivial) cofibration if the corresponding morphism

f5" : □ "-1(f) —— Sym"(Y)

is a (trivial) cofibration for any integer n > 1.

A symmetrizable (trivial) cofibration f is itself a (trivial) cofibration because □ ¿(f) — Sym1(Y) is nothing but the original morphism f.

Remark 4. If f : X — Y is a symmetrizable (trivial) cofibration in C, it is not necessarily true that the £"-equivariant morphism fa" is a cofibration in CEn. Theoretically, it would also make sense to say that f is a strongly symmetrizable (trivial) cofibration if f □" is a (trivial) cofibration in CEn. However, such defined strongly symmetrizable cofibrations are not of much use to us because, as the following example shows, they do not occur even in topology.

Example 5. Let C be the model category of simplicial sets AopSets. According to our notation, A in this C stands for the usual Cartesian product of simplicial sets. Let EX"

be the contractible simplicial set with (EX")¿ = X^, and the diagonal action of X". The morphism f : 0 — X is a cofibration for any simplicial set X. Then the morphism f5" from 0 to Sym"(X) is also a cofibration, for any n > 1. Hence, f is a symmetrizable cofibration. Similarly, one can show that all cofibrations in AopSets are symmetrizable. On the other hand, the morphism f □" from 0 to XA" is not a cofibration in CEn. The reason is that the diagonal map from X to XA" is X"-equivariant. This has the effect that there are no X"-morphisms from XA" to EX" A XA", as X" acts term-wise freely on the simplicial set EX" A XA". It follows that the morphism f □" does not have a X"-left lifting property with respect to the trivial fibration EX" A XA" — XA" in CEn and the identity map from XA" to itself. Thus, f is not strongly symmetrizable in the sense of Remark 4.

Symmetrizability of (trivial) cofibrations is not always the case too. Example 2 shows that trivial cofibrations are not symmetrizable in the category Kom(Z). More importantly, cofibrations are not symmetrizable for symmetric spectra over simplicial sets, see Remark 58 below. This is why we shall give one more definition of (strong) sym-metrizability, which will serve all the needs relevant to symmetric powers of symmetric spectra.

Let C be a closed symmetric monoidal model category, let D be a cofibrantly generated model category, and let F : C — D be a functor from C to D. Then F induces a functor from CG to DG, which will be denoted by the same symbol F. A finite collection {n1, ..., ni} of non-negative integers will be called a multidegree.

Definition 6. A class of morphisms M in C will be called a symmetrizable class of (trivial) F-cofibrations in C if for any finite collection {f1, ..., fl} of morphisms in the class M and any multidegree {n1, ..., nl} the morphism

F (ff"1 □ ...□ff"l)

is a (trivial) cofibration in the model category D. The class M will be called a strongly symmetrizable class of (trivial) F-cofibrations in C if for any finite collection {f1, ..., fl} of morphisms in M and any multidegree {n1, ..., nl} the morphism

F (ff"1 □ ...□ff"l) is a (trivial) cofibration in the model category DSn1 x'"xEni.

Notice that if D = C and F is the identity functor, then M is a (strongly) symmetriz-able class of (trivial) Id-cofibrations if and only if M consists of (strongly) symmetrizable (trivial) cofibrations in C. The case l > 1 is essential when F is not monoidal. This will hold in the applications to symmetric spectra in Section 9.

Let now A be an ordinal and let X be a functor from A to a model category C preserving colimits (although A is not necessarily cocomplete). To shorten notation, for

any ordinal a < A let Xa be the object X(a), and for any two ordinals a and 3, such that a < 3 < A, let = X(a < 3). Let also = colim(X) and, for any ordinal a < A, let : Xa — be the canonical morphism into colimit. Since the set of

objects in A has the minimal object 0, we have the canonical morphism : X0 — , which is called a transfinite composition induced by the functor X.

Theorem 7. Let C be a cofibrantly generated closed symmetric monoidal model category, F : C — D a functor from C to a cofibrantly generated model category D commuting with colimits, and let M be a (strongly) symmetrizable class of (trivial) F-cofibrations in C. Let 0 be a morphism of one of the following types:

(A) a pushout of a morphism from M;

(B) a retract of a morphism from M;

(C) a composition g o f, where f and g are two composable morphisms from M;

(D) a transfinite composition : Xo — induced by a functor X : A — C, where A is an ordinal, X commutes with colimits, and for any ordinal a < A, such that a + 1 < A, the morphism fa+1a : Xa — Xa+1 is in M.

Then the class M U{0} is a (strongly) symmetrizable class of (trivial) F-cofibrations in C too.

Remark 8. Item (C) can be considered as a particular case of item (D). The category D is required to be cofibrantly generated merely to have a model structure on the category D.

The proof of Theorem 7 occupies the next section of the paper. Now we discuss its consequences. Suppose C is cofibrantly generated by a set of generating cofibrations I and a set of generating trivial cofibrations J.

Corollary 9. If I is a (strongly) symmetrizable set of F-cofibrations, then the class of all cofibrations in C is a (strongly) symmetrizable class of F-cofibrations. Similarly, if J is a (strongly) symmetrizable set of trivial F-cofibrations, then the class of all trivial cofibrations in C is a (strongly) symmetrizable class of trivial F-cofibrations.

Proof. By Theorem 7, the class of retracts of relative I-cell complexes is a (strongly) symmetrizable class of F-cofibrations. On the other hand, this class coincides with all cofibrations in C. Similar argument applies to trivial cofibrations.

Applying Corollary 9 to a cofibration 0 — X we obtain two more corollaries.

Corollary 10. Suppose all morphisms in I are symmetrizable. Then any symmetric power Symn(X) of a cofibrant object X in C is cofibrant.

Corollary 11. If I is a strongly symmetrizable set of F-cofibrations, then for any cofibrant object X in C we have that F(XAn) is a cofibrant object in D.

For short, by abuse of notation, throughout the text we will say that I is symmetrizable if it consists of symmetrizable cofibrations, and that J is symmetrizable if it consists of symmetrizable trivial cofibrations.

Finally, we compare the pointed v.s. unpointed cases of our setup. Assuming the terminal object * is the monoidal unit and cofibrant in C, the pointed category C* = * I C inherits the monoidal model structure by Proposition 4.2.9 in [10].

Lemma 12. Let f be a morphism in C*, which is a symmetrizable (trivial) cofibration as a morphism in C. Then f is a symmetrizable (trivial) cofibration as a morphism in C*.

Proof. This follows from the fact that fan in C* is a pushout of fan in C. □

4. The proof of Theorem 7

First we collect some technical lemmas needed in proving the theorem. If f : X ^ Y is a morphism in C and there exists a pushout square

then sometimes we will write

f = psht(f') ,

not specifying the horizontal morphisms of the square.

Lemma 13. Let f = psht(f'), e : A ^ B a morphism in C and let

d : n(f ', e) m(f, e)

be the universal morphism between two colimits induced by the pushout square above. Then the commutative square

□(/',e)

f '□ e

Y 'A B

□(/,e)

is pushout, i.e. psht(f')ne = psht(f'ne).

Proof. As A-multiplication is a left adjoint, and so it commutes with colimits, the commutative squares

X' A A-^ X A A X' A B-^ X A B

f 'Aid

Y 'A A

fAid f'Aid

Y A A Y 'A B

are pushout. The morphism e induces a morphism from the left pushout square to the right one. This and the universal property of the colimits □(/', e) and □(/, e) allow to show that the commutative square in question is pushout. □

Lemma 14. Let /1, ..., /n be a collection of morphisms in C. Then we have

psht(/i)n ... Dpsht(/„) = psht(/in ... □/„) .

Proof. Use Lemma 13 and associativity of the Hi-product. □

Let G be a finite group and let H be a subgroup in it. The natural restriction resH : CG A CH has left adjoint functor corH : CH A CG, such that (corH, resH) is a Quillen adjunction, see Theorem 11.9.4 in [8]. Recall that corH(X) ~ (G x X)/H and corH(X)/G ~ X/H.

Lemma 15. Let X A Y A Z be two composable morphisms in C, and let n be a positive integer. Then the morphism (g/)Dn : □ n_i(g/) A ZAn is a composition

gDn o psht(cor|:_! xSi (gD(n-i)□/)) o ••• o psht(cor|»xS„_i (g^/D(n-i))) o psht(/Dn) ,

where S x Sn-i is canonically embedded into Sn for each i, and psht(/Dn) is a pushout of /Dn with respect to the universal morphism i(/) AD n_ i(g/).

Proof. Let J be the category 0 ^ 1 ^ 2. A pair of subsequent morphisms f : X ^ Y and g : Y ^ Z in C can be considered as a functor K(f, g) : J ^ C from J to C. Let Jn and Cn be the Cartesian n-th powers of the categories J and C respectively, and let Kn(f, g) : Jn ^ C be the composition of the n-th Cartesian power of the functor K(f, g) and the n-th monoidal product A : Cn ^ C. In particular, Kn(f, g) is a commutative diagram in C, whose vertices are monoidal products of the three objects X, Y and Z. Notice that the order of the factors is important here.

For short, let Kn = Kn(f, g), and consider a subdiagram L in Kn generated by the vertices containing at least one factor X, and for any index i G {0, 1, ..., n} let Kn be a subdiagram in Kn generated by vertices containing < i factors Z. Let also Li = L U Kn and put L-1 = L. Then we have a filtration

L-i c Lo C Li c L2 C ••• C Ln = Kn and, correspondingly, a chain of morphisms between colimits

colim (L-1) ^ colim (L0) ^ colim (L1) ^ ••• ^ colim (Kn) ,

whose composition is nothing but (gf )Dn.

For any 0 < i < n the object n(gDi, fD(n-i)) is a colimit of a subdiagram in Li-1, so that one has a universal morphism from n(gDi, fD(n-i)) to colim(Li-1). Since ZAi A YA(n-i) is a vertex in the diagram Li, we have a morphism from ZAi A YA(n-i) to colim(Li). Finally, we have a standard morphism gDiU fD(n-i) : [i](gDi, fD(n-i)) ^ ZAi A YA(n-i). Collecting these morphisms together we get a commutative diagram

□(gDi,fD(n-i))

colim (Li-1)

ZAi /\ YA(n-i)

colim (Li)

This is a Si x Sn-i-equivariant commutative diagram, which yields a Sn-equivariant commutative diagram

cor|;x ^(gDi,fD(n-i)))

-5- cor

(ZAi A YA(n-i))

colim (Li-1) -colim (Li)

Straightforward verification shows that this is a pushout square. □

Lemma 16. For any three morphisms X A Y A Z and A A B in C we have that

(g/= (gtt) o k ,

where k : d(g/, e) AD (g, e) is a universal morphism between colimits and the square

□(/, e)-^ □(g/e

Y A V-^ □(g, e)

is pushout, i.e. we have (g/)□ e = (gde) o psht(/de).

Proof. The top horizontal morphism □(/, e) A iu(g/, e) in the above diagram is also a universal morphism between colimits. The proof of the lemma then follows from the appropriate commutative diagrams for the products / e, g/ e and g e involved into the lemma. □

Lemma 17. Let Xi A X2 A •• • A Xn+i and e : A A B be morphisms in C. Then one has

(/n o ••• o /i)tt = (/ntt) o psht(/n_i^e) o ••• o psht(/i^e) .

Proof. Use induction by n. If n = 2 then the lemma is just Lemma 16. For the inductive step,

(/n o ••• o /i)^h = (/n o (/n-i o ••• o /i))^h = (/n^h) o psht((/n-i ,

where the last equality is provided by Lemma 16 too.

Lemma 18. Let G and G be two finite groups, let H be a subgroup in G and H' be a subgroup in G'. Let also / : X A Y and /' : X' A Y' be two morphisms in C. Then

corH(/)rnorH'' (/') = corHXGr' (/□/') .

Proof. The lemma holds true because A-multiplication commutes with colimits and the order of counting colimits is not important.

Lemma 19. Let A be an ordinal. For any two functors X and Y from A to C, not necessarily preserving colimits, one has a canonical isomorphism

colima<\(Xa A Ya) - (colima<\Xa) A (colimp<\Yp) .

Proof. Indeed, as the monoidal product A in C is closed, smashing with an object commutes with colimits. Therefore, we have two canonical isomorphisms

(colima<\Xa) A (colimp<\Yp) - colima<\(Xa A colimp<\Yp) -- colima<Acolimp<\(Xa A Yp) .

Since all arrows in the diagram X A Y are targeted towards the diagonal objects Xa A Ya, the last colimit is canonically isomorphic to the colimit of these objects. □

Let now A be an ordinal and let X be a colimit-preserving functor from the ordinal A to the category C. For any two ordinals a and 3, such that a < ¡3 < A, let i(fa,o) ^ i(/p,o) be the universal morphism from Lemma 15 being applied to the composition fp,o = /p,a ◦ /a,0. Similarly, let □n—1(/a,o) ^ be the universal morphism

from Lemma 15 applied to the composition = o /a,0. It is not hard to verify that the collection of objects 1(/a,0) and morphisms 1(/a,0) ^ □ n_i(/p,o) gives a functor from A to C.

Lemma 20. In the above terms, there are canonical isomorphisms

□n-l(/~) - colim a<\^n—1 (/a,o) , X^n - colim a<xXaAn ,

and the square

n — 1

( /a, o )

is commutative for any a, i.e.

n—1(/^)

->■ X

= colim a<A (/an ) .

Proof. By Lemma 19,

K? (U) - colim a<xKn(/a,o)

for any index i, where the colimit is taken in the category of functors from subcategories in In to C. It implies the following computation:

- colim Knf) - colim (colim a<\Kn(fa, o)) -- colima<A(colimKn(/a,o)) - colima<\^n(/afi) .

In particular,

□n-l(/~) - colim a<\^n-l(/a, o) ,

X" - □ n(/TO) - colima<xnn(/a,o) - colima<xXa ,

and both isomorphisms are connected by the corresponding commutative square. □ Lemma 21. Let E be a model category and let X be an ordinal. Let

U,V : X-T E

be two functors from X to E, both commuting with colimits, and let

^ : U —¥ V

be a natural transformation. For any ordinal a < X, such that a + 1 < X, let

be a pushout square, and let ha be a universal morphism from the colimit Wa to Va+i. Assume that for any a < X, such that a + 1 < X, the morphism ha and the morphism ^o are cofibrations in E. Then the universal morphism

colim : colim (U) — colim (V)

is also a cofibration in E.

Proof. For any ordinal a < A, such that a + 1 < A, let Da be the diagram

colim (U )

Let also D—1 be the diagram

Uo ^ U1 ^ ••• ^ Ua ^ ••• ^ colim (U) , and let D\ be the diagram

Uo -> U1 ->■ ... -> Ua -> Ua+1 -> ... ->■ colim (U)

colim (V)

Let now Sa = colim (Da), S-i = colim (D-1) = colim (U) and Sa = colim (Da) = colim (V ). One has a transfinite filtration of diagrams

D-i С Do С Di С ••• С Da С Da+i С ••• С Da .

Consequently, we obtain a decomposition of the morphism colim (ф) into a transfinite composition

colim (U) = S-i ^ S0 ^ Si ^ ••• ^ Sa ^ Sa+i ^ ••• ^ Sa = colim (V) . For any a < A, such that a + 1 < A, the square

Sa + i

is pushout. Since our input is that all ha and are cofibrations, we get that the morphism colim(^) is a transfinite composition of cofibrations in E. Since a transfinite composition of cofibrations is a cofibration, the lemma is proved. □

Now we are ready to give the proof of Theorem 7. We will only consider the strong symmetrizability case. The symmetrizability assertion then follows by applying in addition the colimit under the action of the symmetric group.

Let f', f2,..., f l be l morphisms in M and let f be a pushout of f'. To prove (A) we need to show that the morphism F(f Dnnf2n"2D .. .nfj3nt ) is a cofibration in the category DEnXEn2 X'"xEni for any multidegree {n, n2,..., nl}.

By Lemma 14, f DnfDn2 □ ... □ fDn' is a pushout of f' DnfDni □ ... □flDni. Since F commutes with colimits, the morphism F (f Dnn fDn2n ...nfDni ) is a pushout of F(f ' Dnnf2DniD .. -Uffni ). Since the latest morphism is a cofibration in DSnXSn2 X'"xSni, the morphism F(f Dnnf2Dn2D .. .nfDn' ) is a cofibration too. So, (A) is done.

To prove (B) we just notice that a retract of a cofibration is a cofibration, and retraction is a categoric property coomuting with colimits. This gives (B).

Let f, g, f2,..., fi be l + 1 morphisms in M, where f and g are composable. To prove (C) we need to show that for any multidegree {n, n2,..., n l} the morphism F ((gf )DnfDn2 □ ...□fDni ) is a cofibration in DE"xS"2 X'-'XE»,.

By Lemma 15 we have that (gf)Dn is a composition of pushouts of the morphisms cor^_.(gDin f D(n-i))fori = 0, 1, ... , n. By and , the morphism

(gf )Dnnf2Dn2D .. .nfDn' is a composition of pushouts of the morphisms

cor|;XE_(gDi□fD(n-i))fDn2□... □fiDni .

By Lemma 18, the latest morphism can be also viewed as the morphism

cnrS"XS"2 X'"XEni (gDin fn(n-i)n fDn2n nfDni ) COrS,XE„_,XEn2 x-XE„^ g □J □j 2 □ -.□h ).

Since any cor is a colimit and the functor F commutes with colimits, the morphism F ((gf )Dnn f2Dn^ .. .dfDni ) is a composition of pushouts of morphisms of type

<Sm (F (gD^ f D(n-i)n fDDn2n .. .VfDni )) .

Since the morphisms f, g, f2,f i are taken from the class M, every morphism F (gniD f n("-i)a f2D"2D .. .nf°n') is a cofibration in the category D X-XS„;.

As corH is a left Quillen functor for any group G and a subgroup H in it, we obtain that F ((gf )D"n f2D"2n ...nf^') is a cofibration in the category DE"xS"2 X"-xEn .

Now we prove (D). For any ordinal A let D(A) be the property (D) in the statement of the theorem being considered for this ordinal A. We need to show D(A) for any ordinal A. To do that we are going to apply the method of transfinite induction. Namely, suppose that for any ordinal a < A the property D(a) is satisfied. We will show that this assumption implies that D(A) holds true.

Consider a finite collection f2,..., fl of morphisms in M. We need to show that for any positive integers n, n2,..., nl the morphism F(f™Df2D"2n • • .OfD"1) is a cofibration in the category DSnXSn2 x'"xSni. If, for short, we denote the morphism f2D"2D .. .afD"1 by e : A ^ B then we need to show that for any positive integer n the morphism F(fD"ne) is a cofibration in DSnXSn2 X ^XS„|.

Our strategy is to apply Lemma 21 to the category E = DEnXE"2 X ^xEni, the functors U = F (df", e)), V = F (XAn A B), and the natural transformation ^ = F (f^de). First we show that colim(^) is nothing but the morphism F(fD"de). This is provided by Lemma 20, which says that fD" = colim (fD"), the commutativity of the functor F with colimits, and the obvious fact that the right m-multiplication is colimit-commutative too:

colim ~ colim F (fafiO e) ~ ~ F (colim (f™D e)) ^ F (colim (f°")n e) ~ F (f°nD e) .

Next, we have that

^o = F (f^D e) = F (idXA^ e) = F (id* a^b )

F (X AnAB)

is a cofibration in DEnXEn2 X ^XS„|. in order to apply Lemma 21 it remains only to show that the universal morphisms ha are cofibrations in DEnXE"2 X' ' XEni. We give an explicit description of ha.

Let ra be a pushout of the morphism fD" with respect to the universal morphism between colimits 1(fa,o) ^ □i(fa +i,o). Applying Lemma 13 to the corresponding pushout square and the morphism e : A ^ B we get a pushout square

a,0, '

XAn A B

□ (r a

Ra A B

Let furthermore sa be the universal morphism from the colimit Ra into the wedge-power XA+i, so that f o = sa ◦ r a • Applying Lemma 16 to this composition and the morphism e : A ^ B, we obtain yet another pushout square

-i,o, '

□ (Sa,<

Composing these two pushout squares we obtain that

/a+ 1,0ne = (saae) o Ka .

This proves that Wa from Lemma 21 equals F(d(sa, e)) and ha equals F(saDe) since F commutes with colimits.

By Lemma 15, the morphism sa is the composition

/a+1,a o psht(cor|:_ixSi (/^D/^)) ◦ • • • O psh^^- (/a+1,/"^ )) . By Lemma 17 the morphism sane is the composition of pushouts of the morphisms

psht(cor|:xS_ /+ i,aD/an,(0n-î)))De, where 0 =1, ..., n — 1. By Lemma 13,

psht(cor|»xE„_i/+i,aü/°,(0n-i)))ne = psht(cor|»xs„_i/+ i,aü/aD,(o"-i))ne) . Since e = /2D"2 □ ...n/"ni, by Lemma 18 we have that

psht(cor|:xS_ /+ i,aD/an,(0n-i)))De =

corE"xE"2 x'"xEni (/ai n/a(n-i)n )

COrEixS„_,xE„2 x-xEnj Ua+1,auJa,0 UJ2 uJl ).

Since F commutes with colimits, it follows that for any ordinal a, such that a + 1 < A, the morphism ha is a composition of pushouts of the morphisms

F(fDi^nfD.lrW"2□... □fD"'),

where i = 0, ..., n — 1. By the inductive hypothesis, any such morphism is a cofibration. Then ha is a cofibration too. As we have shown above, F(fD"ne) = colim(^). By Lemma 21, this morphism is a cofibration in DEnXEn2 X---XSn'. This finishes the proof of Theorem 7.

5. Kunneth towers for cofibre sequences

Here we prove the existence of special towers of cofibrations connecting symmetric powers in cofibre sequences via the Kunneth rule, provided (trivial) cofibrations are symmetrizable, Theorem 22. This suggests to introduce the concept of a categorified A-structure in C and Ho(C). Using the results from Section 3, we prove the existence of the A-structure of left derived symmetric powers provided symmetrizability of generating (trivial) cofibrations in C, Theorem 25 and Corollary 27. An application to categorical finite-dimensionality (with coefficients in Z) is given in Corollary 28.

In a model category D, if X ^ Y is a cofibration, then let Y/X be the colimit of the diagram Y ^ X and if X and Y are cofibrant, then X ^ Y ^ Y/X is a cofibre sequence in D.

Theorem 22. Let C be a closed symmetric monoidal model category, and let X ^ Y ^ Z be a cofibre sequence in C. Then, for any two natural numbers n and i, i < n, there is a cofibration i(f) ^ □"(f) and a X"-equivariant isomorphism

□"/□"-i - corl^xE,(Xa("-i) a Zai)

in C. If f is a symmetrizable cofibration and all symmetric powers Sym*(X) are cofibrant, then the morphism □i(f) ^ □ "(f), obtained by passing to the colimit of the action of the symmetric group X", is a cofibration, and □" /□"_i can be computed by Kunneth's rule,

□ "/□ "-i - Sym"-i(X) A Sym*(Z) .

If f is a symmetrizable trivial cofibration, then all the cofibrations □"_i(f) ^ □ "(f) are trivial cofibrations.

Proof. The proof is similar to the proof of Lemma 15. For any 0 < i < n the diagram XA("-i) a i(f) is a subdiagram in i(f). Since the wedge product commutes with colimits, we obtain a universal morphism from XA("-i) a □ i_i(f) to □i(f). Since XA("-i) a yAi is a vertex in the diagram K"( f), we have a morphism from XA("-i) a YAi to □"(f). Finally, we have a standard morphism XA("-i) a □ ii-i(f) ^ XA("-i) a YAi. Collecting these morphisms together we get a commutative diagram

XA("-i) A ii i(f)

XA("-i) YAi

□"(f)

This is a X"-i x Si-equivariant commutative diagram, which yields a S"-equivariant commutative diagram

cor|" x(XA(n-i) ADU(f)) -- cori" „(XA(n-i) A YAi)

□n-i(f) -- n"(f)

A straightforward verification shows that this is a pushout square.

By the pushout product axiom of a closed symmetric monoidal model category, the morphism n\_ 1 (f) —>■ YAi is a cofibration and we have

YA7oi-i(f) ^ ^Ai.

By the same axiom, the functor XA(n-i) a — commutes with colimits and preserves cofibrations in C as the object X is cofibrant. Also the same is true for the functor cor, because this is a bouquet in the category C. This implies the needed statements about

□n(f).

Now suppose that f is a symmetrizable (trivial) cofibration. Recall that taking a quotient over £„ commutes with colimits being a left adjoint functor. This gives a pushout square

Symn-i(X) A □ i_i(f)

Symn-i(X) A Symi(Y)

□ n(f)

The symmetric power Symn-i(X) is cofibrant by assumption. The morphism □ i_ 1(f)—> Symi(Y) is a (trivial) cofibration by assumption. Therefore the top morphism is a (trivial) cofibration. This finishes the proof.

Corollary 23. Let f be a cofibration between cofibrant objects in C. Suppose that f is a symmetrizable cofibration, and all symmetric powers Symn(X) are cofibrant in C. Then f is a symmetrizable trivial cofibration if and only if Symn(f) is a trivial cofibration for all n > 0.

Proof. Consider the sequence of cofibrations

Symn(X) = □ n(f) — □ n(f) — ••• — □ n(f) — ••• — □ n(f) = Symn (Y)

provided by Theorem 22. The composition of all the cofibrations in that chain is Sym"(f). If f is a symmetrizable trivial cofibration then each cofibration

□"(f) □ "+i(f)

is a trivial cofibration by Theorem 22. Thus, so is Sym"(f). Conversely, suppose Sym"(f) is a trivial cofibration for any n > 0. Let's prove by induction on n that the mor-phism □i(f) ^ Sym"(Y) is a trivial cofibration, i.e. that f is a symmetrizable trivial cofibration. The base of induction, n =1, is obvious. To make the inductive step we observe that in proving Theorem 22 we deduce that □i(f) ^ □ "(f) is a trivial cofibration by only using that □ \_i(f) ^ Symi(Y) is a trivial cofibration for i < n. But the last condition holds by the induction hypothesis. Thus, all morphisms □ "-i (f) ^ □ "(f), 1 < i < n - 1, are trivial cofibrations. Then □ "-i(f) ^ Sym"(Y) is a weak equivalence by 2-out-of-3 property for weak equivalences. Finally, by the assumption of the lemma, □i(f) ^ Sym"(Y) is a cofibration, and so a trivial cofibra-tion.

Definition 24. For any closed symmetric monoidal model category C with monoidal unit 1, a X-structure on C is a sequence A of endofunctors A" : C ^ C, n = 0, 1, 2, ..., such that

(i) A0 = 1, Ai = Id,

(ii) A"(0) = 0 for all n > 1,

(iii) to each cofibre sequence X ^ Y ^ Z in C and any n there is associated a unique sequence of cofibrations between cofibrant objects

A"(X) = L" ^ L" ^----► L" ^----^ L" = A"(Y) ,

called Kunneth tower, such that for each index 0 < i < n one has isomorphisms

L"/L"-i - A"-i(X) A Ai(Z) ,

(iv) such towers are functorial in cofibre sequences in the obvious sense.

In particular, the endofunctors A" preserve cofibrant objects in C. In these terms, Theorem 22 says that if cofibrations in C are symmetrizable, then symmetric powers yield a specific X-structure in C. We will call it the canonical X-structure of symmetric powers in C.

A cofibre sequence in Ho(C) is a sequence of two composable morphisms, which is isomorphic to a sequence coming from a cofibre sequence in C via the functor from C to Ho(C). A similar definition of a X-structure can be then given also in Ho(C). If A* is a X-structure on C such that A" takes trivial cofibrations between cofibrant objects

into weak equivalences, then by Ken Brown's lemma the left derived functors LAn exist, and their collection gives a A-structure in Ho(C). Combining this with Corollary 23, we obtain the following important result.

Theorem 25. Let C be a closed symmetric monoidal model category, such that all cofi-brations are symmetrizable, and all trivial cofibrations between cofibrant objects are symmetrizable in C. Then symmetric powers Symn take weak equivalences between cofibrant objects to weak equivalences, and the canonical A-structure of symmetric powers in C induces the A-structure of left derived symmetric powers LSymn in Ho(C).

Remark 26. Let C be a closed symmetric monoidal model category cofibrantly generated by a set of generating cofibrations I and a set of generating trivial cofibrations J. Suppose I and J are both symmetrizable. Then by Corollaries 9 and 10, the conditions of Theorem 25 are satisfied.

Assume now that C is moreover pointed. According to [10], there is a well-defined S^suspension functor — AL S1 : T — T provided by a Ho(AopSetst)-module structure on the homotopy category T = Ho(C). If it is an autoequivalence on T then T is triangulated, where the translation functor [1] is given by — AL Si and distinguished triangles come from cofibre sequences in C, see Chapter 7 in [10]. Since C is closed symmetric monoidal, so is the triangulated category T, and the functor C — T is monoidal as well, see Section 4.3 in [10]. We will denote the monoidal product in T also by A. A A-structure in T = Ho(C) associates Kunneth towers to distinguished triangles in Ho(C) in the functorial way. Using Theorem 25 we obtain the following result.

Corollary 27. Let T be the homotopy category of a pointed closed symmetric monoidal model category C, so that T is triangulated. Assume, furthermore, that all cofibrations are symmetrizable, and all trivial cofibrations between cofibrant objects are symmetrizable in C. Then T inherits the canonical A-structure of left derived symmetric powers associated to distinguished triangles in T.

As a straightforward consequence of Corollary 27 we also get the following corollary.

Corollary 28. Let T be as above, and let X — Y — Z — X [1] be a distinguished triangle in T. If there exist natural numbers n' and m' such that LSymn(X) = 0 for all n > n' and LSymm(Z) = 0 for all m > m', then there exists N' such that LSymN (Y) = 0 for all N > N'.

6. Localization of symmetric powers

In this section we prove a few results on the Bousfield localization of model categories with regard to monoidal structures and symmetric powers on them, which will

be used later. In particular, Theorem 33 gives a necessary and sufficient condition for trivial cofibrations to remain symmetrizable after Bousfield localization of C by a set of morphisms S. This will be applied to the localization by an abstract interval in Section 7.

Let C be a left proper cellular model category, and denote the model structure in C by M. Recall that left properness means that the pushout of a weak equivalence along a cofibration is a weak equivalence. Cellularity means that C is cofibrantly generated by a set of generating cofibrations I and a set of trivial generating cofibrations J, the domains and codomains of morphisms in I are all compact relative to I, the domains of morphisms in J are all small relative to the cofibrations, and cofibrations are effective monomorphisms. Further details about these notions can be found, for instance, in [10, 11] or [8].

Let S be a set of morphisms in C. Recall that an object Z in C is called S-local if it is fibrant, and for any morphism f : A ^ B in S the morphism between function complexes

map(f, Z) : map(B,Z) ^ map(A, Z)

is a weak equivalence in AopSets, see Definition 3.1.4(1)(a) in [8]. The construction of the function complex bi-functor map(-, —) is given in Sections 17.1-17.4 in [8] (see also Section 5.4 in [10]). A morphism g : X ^ Y in C is said to be an S-local equivalence if the induced morphism

map(g, Z) : map(Y, Z) ^ map(X,Z)

is a weak equivalence in AopSets for any S-local object Z in C, see Definition 3.1.4(1)(b) in [8]. Notice that since map(—, —) is a homotopic invariant, each weak equivalence is an S-local equivalence in C.

By the main result in [8] (see Theorem 4.1.1), under the assumptions above, there exists a new left proper cellular model structure MS on C whose cofibrations remain unchanged and new weak equivalences WS are exactly S-local equivalences in C. The new model structure is cofibrantly generated by the set of generating cofibrations I and a new set of generating trivial cofibrations JS, and it is called a (left) Bousfield localization of M with respect to S. The symbol CS will be used to denote the same category C, endowed with the new model structure MS. Then CS is a (left) Bousfield localization of C with respect to S.

Let F : C ^ D be a left Quillen functor such that F(Q(f)) is a weak equivalence for any f € S, where Q denotes the cofibrant replacement functor in the model structure M. Then F is still left Quillen with respect to the localized model structure MS and has a left derived with respect to MS, see Proposition 3.3.18(1) in [8]. Our main goal is to construct left derived symmetric powers for the localized model category. Since symmetric powers do not admit right adjoints in general, and thus are not left Quillen, we need to strengthen the above result.

Given a functor F : C ^ D to a model category, we say that a morphism g in C is F-acyclic if g is a cofibration between cofibrant objects in C and F(g) is a weak equivalence in D. Obviously, given composable cofibrations between cofibrant objects, their F-acyclicity has 2-out-of-3 property. By an S-local cofibration we mean a cofibration which is an S-local equivalence in C.

Theorem 29. Let F : C ^ D be a functor to a model category such that all trivial cofibra-tions between cofibrant objects in M are F-acyclic and F(Q(f)) is a weak equivalence in D for any f € S. In addition, suppose that F-acyclic morphisms are closed under trans-finite compositions and pushouts with respect to morphisms to cofibrant objects. Then all S-local cofibrations between cofibrant objects are F-acyclic. In particular, by Ken Brown's lemma, the left derived functor LF : Ho(Cs) ^ Ho(D) exists and commutes with the localization functor Ho(C) ^ Ho(Cs).

To prove Theorem 29 we first need to prove an auxiliary result. Fix a left framing on C, see Definition 5.2.7 in [10]. Thus, for each cofibrant object X one has the functorial cofibrant replacement X* of the constant cosimplicial object given by X, with respect to the Reedy model structure on the category of cosimplicial objects in C. The product X A K in C of X and a simplicial set K is then defined as the product X* A K. For any morphism g in C, and a morphism i in AopSets, we have their pushout product gdi. For a non-negative integer m let im : dA[m] ^ A[m] be the embedding of the boundary into the m-th simplex.

Lemma 30. Let F be as in Theorem 29. Then F-acyclic morphisms are closed under taking products with simplicial sets generated by finitely many non-degenerate simplices and pushout products with the embeddings im.

Proof. Let g : X ^ Y be an F-acyclic morphism in C, and let K be a simplicial set. Let m be the maximal dimension of non-degenerated simplices in K, and n be the number of such simplices. We apply induction with respect to the lexicographical order on the set of pairs (m, n). Represent K as a simplicial set obtained by gluing an m-dimensional simplex to another simplicial set K' having one simplex less than in K, i.e. i : K' ^ K is a pushout of im. By Corollary 5.4.4(1) in [10], the functor X A — is left Quillen. It follows that the morphism X = X A A[0] ^ X A A[m] is a trivial cofibration between cofibrant objects, whence it is an F-acyclic morphism by the assumption on F. Since the same is true for Y ^ Y A A[m], we see that the morphism X A A[m] ^ Y A A[m] is also F-acyclic by 2-out-of-3 property for acyclicity. The morphism X A A[m] ^ □ (g, im) is a pushout of g A idgA[m]. Then it is F-acyclic by the pushout property for acyclicity and the induction. Using 2-out-of-3 property once again, we conclude that gnim is F-acyclic. The obvious commutative diagram

o(g,im)

■>■ □(g,i)

Y A A[m]

is a pushout square, and all objects in it are cofibrant. Then g A id^ = psht(gdim) o psht(g Aid^'). The induction and the pushout property finish the proof of the lemma. □

Using a standard transfinite composition argument and Lemma 30 one can also show that F-acyclic morphisms are closed under products with arbitrary simplicial sets and pushout products with arbitrary cofibrations between simplicial sets, though we do not need this. Now we can prove Theorem 29.

Proof. By Ken Brown's lemma and the assumption of the theorem, F sends weak equivalences between cofibrant objects in C to weak equivalences in D. For any morphism f : A — B of S decompose Q(f) into a cofibration f : Q(A) — C and a trivial fibration f" : C — Q(B). Since f" is a weak equivalence between cofibrant objects in C, F(f") is a weak equivalence. Let S' = {f'\f G S}. Then all morphisms in S' are F-acyclic. Since MS = MS', without loss of generality, one may assume that all morphisms in S are F-acyclic.

Next, let g : X — Y be an S-local cofibration between cofibrant objects in C. Let LS(g) : LS(X) — LS(Y) be the fibrant replacement of the morphism g with respect to the localized model structure MS. Then LS(g) is a weak equivalence between cofibrant objects in C, whence F(LS(g)) is a weak equivalence. This gives that the theorem will be proved as soon as we prove that X — LS(X) is F-acyclic.

By Theorem 4.3.1 in [8], the morphism X — LS(X) is a relative A-cell complex, where A consists of morphisms that are either trivial cofibrations between cofibrant objects, or being composed with a weak equivalence between cofibrant objects are equal to fa(dA[n] — A[n]), where f runs S. By Lemma 30 and 2-out-of-3 property, all morphisms in A are F-acyclic and the theorem is proved by the assumptions on F. □

Now we need to investigate when the compatibility between the model and monoidal structures is stable under localization. For that we shall prove Lemma 31 below, following the ideas taken from the proofs of Theorems 6.3 and 8.11 in [11]. The same result is also proven in [25], Theorem 4.5.

Since now we assume that C is a closed symmetric monoidal left proper cellular model category cofibrantly generated by the set of generating cofibrations I and the set of generating trivial cofibrations J, such that the domains and codomains of the cofibrations from I are cofibrant. Let also Q be the cofibrant replacement in C, and so in Cs .

Lemma 31. The model structure Ms is compatible with the monoidal structure in C if and only if for any X G dom(I) U codom(I) and for any f G S the product X A Q(f) is an S-local equivalence.

Proof. If MS is compatible with the monoidal structure in C, then XAQ(f) is an S-local equivalence by the axioms of a monoidal model category. Conversely, let h : X — Y be a cofibration in I and let g : Z — U be an S-local cofibration in C. By Corollary 4.2.5 in [10] all we need to show is that tog is an S-local cofibration in C. By Theorem 2.2 in [11], the functors X A — and Y A — are left Quillen with respect to the localized model structure MS. This is because X A Q(f) is an S-local equivalence for any f from S, and the same for YAQ(f). Since XA— is left Quillen and g : Z — U is an S-local cofibration, the morphism idx Ag : X A Z — X A U is an S-local cofibration. Since trivial cofibrations are stable under pushouts, the pushout Y A Z — □ (h, g) is an S-local cofibration too. The morphism idy A g : Y A Z — Y A U is an S-local cofibration, because Y A — is left Quillen. Since idy A g is the composition Y A Z — □ (h, g) —— Y A U, we obtain that tog is an S-local equivalence. Moreover, tog is a cofibration since C monoidal model. □

Remark 32. Lemma 31 has the following direct generalization. Let C and S be as in the lemma and D be a C-module in the sense of Definition 4.2.18 in [10]. Let R be a set of morphisms in D and assume that D is left proper and cellular. Let I' be a set of generating cofibrations in D. Suppose the condition of Lemma 31 is satisfied, for all X G dom(I) U codom(I) and g G R the product X A Q(g) is R-local, and for all f G S and Y G dom(I') Ucodom(I') the product Q(f) A Y is R-local as well. Then the localized model category Dr is a CS-module.

Theorem 33. Let C and S be such that MS is compatible with the monoidal structure in C, and assume furthermore that all cofibrations are symmetrizable and all trivial cofibrations between cofibrant objects are symmetrizable in C. Assume also that for any f G S and any natural n the morphism Sym"(Q(f)) is an S-local equivalence. Then all S-local cofibrations between cofibrant objects are symmetrizable in Cs. The left derived functors LSym" exist on Ho(Cs), and they commute with the localization functor Ho(C) — Ho(Cs).

Proof. Let F be the composition of Sym" and the localization functor C — CS (this is just the identity functor considered as a functor between two different model structures). Since cofibrations in C are symmetrizable, they are so in CS. By Corollary 23 applied to CS, we see that trivial symmetrizable cofibrations between cofibrant objects in CS are the same as F-acyclic morphisms in C. So, it is enough to show that S-local cofibrations are F-acyclic.

By Theorem 7 applied to the category CS, F-acyclic morphisms are closed under transfinite compositions and under pushouts with respect to morphisms to cofibrant

objects (actually, to treat transfinite compositions it is enough to use Lemma 19 and Theorem 22). We conclude by Theorem 29. □

7. Localization w.r.t. diagonalizable intervals

Let us consider more closely the important particular case of the left Bousfield localization contracting an object A into a point. If A is what we call a diagonalizable interval, then, using the results from Section 6, we prove that trivial cofibrations (between cofibrant objects) remain symmetrizable in the localized category, Theorem 42. As a consequence, we obtain that left derived symmetric powers exist in the homotopy category of the localized category CS, provided we have them in Ho(C), see Corollary 43. This will be applied in Section 11 to the unstable motivic homotopy theory, where A will be the affine line Ai over a base.

Let C be a closed symmetric monoidal left proper cellular model category C cofi-brantly generated by the set of generating cofibrations I and the set of generating trivial cofibrations J, such that the domains and codomains of the cofibrations from I are cofibrant. Let A be a cofibrant object, let n : A ^ 1 be a morphism in C, and let

S = {X A A id—X | X € dom(I) U codom(I)} .

For any morphism f : X ^ Y and any object Z in C the morphism Hom( f, Z) : Hom(Y, Z) ^ Hom(X, Z) in C, as well as the morphism map(f, Z) : map(Y, Z) ^ map(X, Z) in AopSets, will be denoted by f *.

Notice that, if X € dom(I) U codom(I), it is cofibrant, and since A is cofibrant, the monoidal product X A A is cofibrant too.

The following two lemmas and Proposition 36 are well-known to experts. We give complete proofs, as we could not find them in the literature.

Lemma 34. An object Z in C is S-local if and only if Z is fibrant in C and the induced morphism n* : Z — Hom(1, Z) ^ Hom(A, Z) is a weak equivalence in C.

Proof. Let X € dom(I) U codom(I). If n* is a weak equivalence, the morphism map(X, n*) : map(X, Z) ^ map(X, Hom(A,Z)) is a weak equivalence of simplicial sets. If Z is fibrant, then the simplicial sets map(X, Hom(A, Z)) and map(X A A, Z) are canonically weak equivalent, since the objects X and A are cofibrant in C. The composition of the morphism map(X, n*) with this weak equivalence equals to the morphism (idx A n)* : map(X, Z) ^ map(X A A, Z), so that (idx A n)* is a weak equivalence of simplicial sets as well. By definition, it means that Z is S-local. Conversely, if Z is S-local, the morphism (idx A n)* and so map(X, n*) are weak equivalences of simplicial sets. Then Z — Hom(1, Z) ^ Hom(A, Z) is a weak equivalence in C by Proposition 3.2 in [11]. □

Lemma 35. If Y is a cofibrant object in C, the morphism Y A A —-—" Y A 1 ~ Y is an S-local equivalence, i.e. a weak equivalence in Cs.

Proof. For any S-local object Z the morphism n* : Z — Hom(A, Z) is a weak equivalence by Lemma 34 so that map(Y, n* ) is a weak equivalence of simplicial sets. As in the proof of Lemma 34 this implies that (idy A n)* is a weak equivalence of simplicial sets for any S-local Z. This means that the morphism Y A A "———n Y is an S-local equivalence. □

Proposition 36. Let C and S be as above. Then the model structure Ms is compatible with the monoidal structure in C.

Proof. Let X be an object in dom(I) Ucodom(I) and let f be a morphism from the set S. By definition, there exists W G dom(I) U codom(I), such that f = id^ A n : W A A — W. Smashing with X we obtain the morphism id* A f : X A W A A — X A W. Applying Lemma 35 to Y = X A W we obtain that id* A f is a weak equivalence in C. Hence, the category C and the set S satisfy the conditions of Lemma 31. Notice that the cofibrant replacements can be ignored here because X and W are in dom(I) U codom(I), so that they are cofibrant, and A is cofibrant too. □

Notice that the proof of Proposition 36 follows closely the proofs of Theorems 6.3 and 8.11 in [11].

Our aim is now to apply Theorem 33 to CS with S as above. For this we need to impose more conditions on the morphism n. Suppose we are given with two morphisms io, ii : 1 — A, such that n o i0 = n o i1 = id^. If f, g : X ^ Y are two morphisms from X to Y in C, then we say that f and g are A-homotopic if there is a morphism H : X A A — Y, such that H o (id* A i0) = f and H o (id* A i1) = g. If f : X — Y and g : Y — X are two morphisms in opposite directions, such that g o f is A-homotopic to id* and f o g is A-homotopic to idy, then f and g are mutually inverse A-homotopy equivalences in C.

Following [16], we will be saying that n is an interval if there exists a morphism 1 : A A A — A, such that i o (id^ A i0) = i0 o n and ^ o (id^ A i1) = id^ as morphisms from A to itself.

Lemma 37. Let n : A — 1 be an interval in C. Then, for any cofibrant object X in C, the morphism idx A n : X A A — X A 1 ^ X is an A-homotopy equivalence in C.

Proof. From the definition of an interval, it follows that (idx A n) o (idx A i0) = idx. Let H = idx A ¡i, where ¡i is taken from the definition of an interval for A. Then (X A A) A A ~ X A (A A A) X A A is an A-homotopy from (id* A i0) o (id* A n) to id*AA. □

Definition 38. The object A, together with the morphisms i0, i1 : 1 — A, is said to be diagonalizable if A is a symmetric co-algebra (possibly, without a co-unit), i.e. there

exists a morphism S : A — A A A, such that the compositions (id^ A S) o S and (J A id^) o S coincide, t o S = S, where t : A A A — A A A is the transposition in C, and there are two equalities a o i0 = (i0 A i0) o £ and a o ¿1 = (¿1 A ¿1) o £, where £ is the inverse to the obvious isomorphism 1 A 1 — 1.

By co-associativity, we have also the morphisms Sn : A — AAn obtained by iterating S. The following lemma is a straightforward generalization of Lemma 1, where A[1] is being replaced by a diagonalizable object A.

Lemma 39. Let A be diagonalizable. Then, for any two A-homotopic morphisms f, g : X ^ Y, and for any positive integer n, the morphisms Symn (f ) and Symn(g) are A-homotopic in C.

Example 40. Let C be as above and assume furthermore that C is simplicial, and that the structures are compatible with each other. Consider the functor AopSets — C sending a simplicial set K into the object 1 A K, and the same on morphisms. Let n : A — 1 be the image of the morphism A[1] — A[0] under this functor. Then n is a diagonalizable interval in C, where the morphism ^ : A[1] x A[1] — A[1] is induced by the multiplication [1] x [1] — [1].

Example 41. Let B be a Noetherian separated scheme of finite Krull dimension, and let C be the category AopPre(Sm/B) of simplicial presheaves on the category of smooth schemes of finite type over B endowed with the stalk-wise model structure with respect to the Nisnevich or étale topology. By abuse of notation, denote by A1 the simplicial presheaf represented by the affine line AB over B. The monoidal unit 1 is represented by B, as a scheme over itself. The structural morphism n : A1 — 1 is then a diagonalizable interval in C, where ^ : A1 A A1 — A1 is the multiplication induced by the fibre-wise multiplication in AB, see [16].

Now we are ready to prove the main result of this section.

Theorem 42. Let C be a closed symmetric monoidal left proper cellular model category C cofibrantly generated by the set of generating cofibrations I and the set of generating trivial cofibrations J, such that the domains and codomains of the cofibrations from I are cofibrant, and the sets I and J are both symmetrizable. Let A be a cofibrant object and let n : A — 1 be a diagonalizable interval in C. Let also S = {X A A X | X G dom(I) U codom(I)} be the set of morphisms in C. Then all S-local cofibrations between cofibrant objects are symmetrizable.

Proof. By Proposition 36 and Theorem 7, C and S satisfy the first two assumptions of Theorem 33, so that we only need to show that they satisfy the third assumption of it. By Theorem 25, symmetric powers preserve weak equivalences between cofibrant objects

in C. This is why, for any f G S, the morphism Sym"(Q(f)) is an S-local equivalence if and only if the morphism Sym"(f) is an S-local equivalence in C.

Let now f be the morphism id* A n : X A A — X A 1 ^ X in S, where X G dom(I) U codom(I). Then f = id* A n is an A-homotopy equivalence by Lemma 37. By Lemma 39, Sym"(f) is an A-homotopy equivalence too. Since I is symmetrizable, Sym"(X A A) and Sym"(X) are cofibrant by Corollary 10, because X and A are cofibrant.

By Proposition 36, idy A n is an S-local equivalence for any cofibrant Y. This implies that A-homotopic morphisms between cofibrant objects are the same in the homotopy category Ho(CS). Therefore, an A-homotopy between cobibrant objects is an S-local equivalence in C. Summing up, we obtain that Sym"(f) is an S-local equivalence in C.

Corollary 43. If the assumptions of Theorem 42 are satisfied, the left derived functors LSym" exist on Ho(Cs) and commute with Ho(C) — Ho(Cs).

Proof. Follows from Theorem 42 and Theorem 25.

8. Positive model structures on spectra

Now we are going to study symmetric powers in stable categories. In this section we give an outline of the utmost generalization of topological and motivic positive model structures developed, respectively, in [3] and [9]. More details on abstract positive model structures can be found in [4]. Positive model structures will play the key role in Section 9.

Let C be a closed symmetric monoidal model category which is, moreover, left proper and cellular model category. Suppose in addition that all domains of the generating cofibrations in I are cofibrant. Let T be a cofibrant object in C. As it was shown in [11], with the above collection of structures imposed upon C there is a passage from C to a category

S = Spts(C ,T)

of symmetric spectra over C stabilizing the functor

— AT : C —— C .

Let's remind the basics of this construction for reader's sake. Let Ebea disjoint union of symmetric groups £n for all n > 0, where £0 is the permutation of the empty set, so, isomorphic to S1, and all groups are considered as one object categories. Let CE be the category of symmetric sequences over C, i.e. functors from £ to C. Explicitly, a symmetric sequence is a collection (X0, X1, X2,...) of objects in C together with the action of on Xn for each n ^ 1. Since C is closed symmetric monoidal, so is the category CE with the monoidal product given by the formula

(X A Y)n = Vi+j=n£n x^xEj (Xi A Yj) ,

where for any group G and a subgroup H in G the functor G x H — is the functor corH described in Section 4, see [12] or [11]. The restriction to the n-th slice of the symmetry isomorphism X A Y - Y A X is equal to the product of the right translation

£„ — £„ , a — a o j ,

and the symmetry isomorphism Xi A Yj — Yj A Xi in C, where j permutes the first block of j and the second block of i elements, [12, Sect. 2.1].

Let S(T) be the free commutative monoid on the symmetric sequence (0, T, 0, 0, ...), i.e. the symmetric sequence S(T) = (1, T, TA2, TA3, ...), where £„ acts on TAn by permutation of factors (recall that 0 is the initial object in C). Then S is the category of modules over S(T) in CE. In particular, any symmetric spectrum X is a sequence of objects (X0, Xi, X2, ...) in C together with £n-equivariant morphisms

Xn A T -> Xn+i ,

such that for all n, i > 0 the composite

Xn A TAi —- Xn+i A TA(i-i) ^ ••• ^ Xn+i

is £n x Si-equivariant. One has a natural closed symmetric monoidal structure on S given by product of modules over the commutative monoid S(T). For any non-negative n consider the evaluation functor

Evn : S —> C

sending any symmetric spectrum X to its n-slice Xn. Each Evn has a left adjoint

Fn : C -S ,

which can be constructed as follows. First we define a naive functor Fn from C to CE taking any object A in C into the symmetric sequence

(0,...,0, £n x A, 0,0,...) ,

in which £n x A stays on the n-th place. On the second stage we set

Fn (A) = Fn(A) A S(T) ,

see [11, Def. 7.3]. Then, for any non-negative integer m one has

Evm(Fn(A)) = £m xEm_ (A A TA(m-n)) ,

where Sm-n is embedded into £m by permuting the first m — n elements in the set {1, ..^ m}.

The functors Fn have the following monoidal property: there is a canonical isomorphism Fp(A) A Fq(B) ~ Fp+q(A A B). The restriction to the rn-th slice of the symmetry isomorphism Fp(A) A Fq(B) ~ Fq(B) A Fp(A) is the morphism

Sm xEm-p_? (A A B A TA(m-p-q)) A £m xEm-p_? (B A A A TA(m-p-q)) which is equal to the product of the right translation

A j O A O ◦ Tq,p ,

the symmetry isomorphism AAB ~ BAA in C, and the identity morphism on TA(m-p-q).

The model structure on S is constructed in two steps - projective model structure coming from the model structure on C and its subsequent Bousfield localization.

Let IT = Un>0Fn(I) and JT = Un>0Fn(J), where Fn(I) is the set of all morphisms of type Fn(f), f € I, and the same for Fn(J). Let also WT be the set of projective weak equivalences, where a morphism f : X A Y is a projective weak equivalence in S if and only if fn : Xn A Yn is a weak equivalence in C for all n > 0. The projective model structure

M = (It Jt ,Wt )

is generated by the set of generating cofibrations IT and the set of generating weak cofibrations JT. As the model structure in C is left proper and cellular, the projective model structure in S is left proper and cellular too, [11]. Projective fibrations of spectra are level-wise fibrations. The closed monoidal structure on S is compatible with the model structure M.

Remark 44. By Remark 7.4. in [11], each functor Evm has right adjoint. The above formula for Evm(Fn(A)) implies that, given a morphism f in C, the morphism Evm(Fn(f )) is a coproduct of the product of f with a power of T. Since T is cofibrant, Evm(Fn(f)) is a (trivial) cofibration provided f is so. This is why Evm sends generating (trivial) cofibrations, in the sense of the model structure M, to (trivial) cofibrations in the model category C. Applying Lemma 2.1.20 in [10], we see that the functors Evm are left Quillen.

Let now

(A : Fn+i(A A T) a Fn(A)

be the adjoint to the morphism

A A T a Evn+i(Fn(A)) = £n+i x (A A T)

induced by the canonical embedding of S1 into Sn+1. For any set of morphisms U let dom(U) and codom(I) be the set of domains and codomains of morphisms from U, respectively. Let then

s = {(A I A G dom(I) U codom(I) , n > 0} ,

where Q is the cofibrant replacement in the projective model structure. Then a stable model structure

Ms = (It ,Jt,sWt,s)

in S is defined to be the Bousfield localization of the projective model structure with respect to the set S. It is generated by the same set of generating cofibrations IT, and by a new set of generating weak cofibrations JTsS. Here WTsS is the set of stable weak equivalences, i.e. new weak equivalences obtained as a result of the localization. The condition of Lemma 31 is satisfied and the stable model structure is compatible with the monoidal structure on S.

The importance of the stable model structure is that the functor — A T is a Quillen autoequivalence of S with respect to this model structure.

An abstract stable homotopy category, in our understanding, is the homotopy category T of the category of symmetric spectra over a closed symmetric monoidal model category C as above, stabilizing a smash-with-T functor for a cofibrant object T in C, i.e. the homotopy category of S with respect to stable weak equivalences WTsS.

Notice also that by Hovey's result, see [11], the homotopy category T is equivalent to the homotopy category of ordinary T-spectra provided the cyclic permutation on T A T A T is left homotopic to the identity morphism.

Now we introduce positive model structures on S. Let I+ = Un>0Fn(I ), J+ = Un>0Fn(J) and let W+ be the set of morphisms f : X — Y, such that fn : Xn — Yn is a weak equivalence in C for all n > 0. We call such morphisms positive projective weak equivalences.

Proposition 45. There is a cofibrantly generated model structure on S

M+ = (I+,J+,W+) ,

called a positive projective model structure. Positive projective fibrations are level-wise fibrations in positive levels. Positive projective cofibrations are projective cofibrations that are also isomorphisms in the zero level.

Proof. We check that the sets I+, J+ and W+ satisfy the conditions of Theorem 2.1.19 in [10], so that they generate a model structure. Condition 1 is satisfied automatically. Conditions 2 and 3 are immediately implied by the inclusions I+-cell C IT-cell, J+-cell C JT-cell and the fact that M = (IT, JT, WT), whence the sets IT, JT and WT satisfy the conditions 2 and 3.

Obviously, all morphisms in J+-cell are positive level weak equivalences. To check condition 4 it remains only to show that J+-cell C I+-cof. The class I+ -cof is closed under transfinite compositions and pushouts, see the proof of Lemma 2.1.10 on page 31 in [10].

Thus, it is enough to show that JT+ C I+-cof, or, equivalently, that I+-inj C J+-inj. Since the functors (Fn, Evn) are adjoint, we get that

J+-inj = {/ : X — Y in S |Vn > 0 Evn(/) is a fibration in C} ,

i.e. the class J+-inj is the class of positive level fibrations in S. Similarly,

I+-inj = {/ : X — Y in S |Vn > 0 Evn(/) is a trivial fibration in C} .

It follows that I+-inj C J+-inj and condition 4 is done. Also, we obtain that J+-inj f W+ = I+-inj, which gives conditions 5 and 6.

The structure of fibrations and cofibrations in M + can be proved using the definition of I+, J+, left lifting property and the adjunction between Fn and Evn. □

Corollary 46. There is a Quillen adjunction

(Fi(T) A—, Hom(Fi(T), —))

between M and M + and a Quillen adjunction (Id, Id) between M + and M.

Let now

S+ = {(A I A € dom(I) U codom(I) , n> 0} . Definition 47. The localization

M++ = (I+ , J+S+ , W+S+ )

of the positive projective model structure with respect to the set S+ will be called a positive stable model structure on S.

Certainly, we can also localize the positive projective model structure by the set S getting an intermediate model structure M+ = (I+, J+ S, W++ S).

Lemma 48. With respect to the closed monoidal structure on S the model structure M+ is an M-module and the model structure M++ is an Ms-module. In addition, the closed monoidal structure on S defines an adjunction in two variables with respect to both model structures M + and M++ (see Definition 4.2.1 in [10]).

Proof. The proof of the facts that M + is an M-module and that we have an adjunction in two variables with respect to M + is similar to the proof of Theorem 8.3 in [11]. Then we use Lemma 31 and Remark 32. Namely, the domains and codomains of morphisms in IT are of the form Fn(A), n > 0, where A is a domain or a codomain of a mor-phism in I. Morphisms in S have cofibrant domains and codomains. The analogous is

true in the positive setup. Now everything follows from the monoidal properties of the functors Fn. □

Notice that the unit axiom is not satisfied for the model structure M + , thus S is not a closed monoidal model category with respect to M + . Indeed, let S (T)+ denote the spectrum with S(T)+ = 0 and S(T)+ = S(T)n for n > 0. Then the natural morphism S(T)+ —¥ S(T) is a positive cofibrant replacement for the unit in S. However, in general S(T)+ A X — X is not a positive weak equivalence for a positively cofibrant X. For example, if X = Fn(A), n > 0, then a calculation shows that (S(T) + A Fn(A))m = 0 for m < n and (S(T)+ A Fn(A))m = (S(T) A Fn(A))m for m > n. Thus, the morphism in question fails to be a weak equivalence in level n.

Lemma 49. Any positive weak equivalence is a stable weak equivalence.

Proof. Let f : X — Y be a positive weak equivalence. We claim that for any Z in S, there is a canonical bijection

HomHo(M)(Z Al Fi(T),X) = HomHo(M)(Z AL F1(T),Y) .

For this we use Quillen adjunctions from Corollary 46 and the fact that ^Hom(Fi (T), f) is an isomorphism in Ho(M) as f is an isomorphism in Ho(M +).

Let g : Y AL Fi (T) — X be a morphism in Ho(M) that corresponds to the morphism idy Al Co : Y AL F\(T) — Y under the above bijection applied to Z = Y (note that g may be not a class of a morphisms in C, which is the reason to consider homotopy categories). Then we obtain a commutative diagram

The commutativity of the lower triangle is by construction of g, while commutativity of the upper triangle is checked by applying f and using the above bijection for the case Z = X. Since id AL Co is an isomorphism in Ho(MS), we see that f is also an isomorphism in Ho(MS) with the inverse being g o (idy AL Co )-1. n

Theorem 50. In the above terms,

Wt,s = W+S+ = W+s

Proof. Let's apply Theorem 3.3.20(1)(a) from [8] to adjunctions from Corollary 46. Indeed, the domains and codomains of morphisms in S and S+ are cofibrant in the corresponding model structures and we have F1 (T) A S C S+, S+ C S, whence the conditions of the above theorem are satisfied. Therefore, we obtain the corresponding Quillen adjunctions between Bousfield localizations Ms and M++.

We claim that these localized Quillen adjunctions are actually equivalences. More precisely, the functors

Fi(T) Al - : Ho (Ms) — Ho(M++) , LId : Ho(M++) — Ho(Ms)

are quasiinverse. For this it is enough to show that for any (positively) cofibrant X the natural morphism F1(T) A X — X is a (positive) stable weak equivalence. This follows from Lemma 48, because F1(T) — Fo(1) is a stable weak equivalence.

Since cofibrant objects in M++ are the same as in M + , the equivalence LId : Ho(M++) — Ho(MS) sends an object X in S to Q+(X), where Q+ is the cofibrant replacement in M +. Therefore a morphism f : X — Y in S is in W+S+ if and only if Q+(f) is in WTs S. By Lemma 49, the natural morphisms Q+(X) — X and Q+(Y) — Y are in WTsS. Consequently, Q+(f) is in WTsS if and only if f is in WTsS, whence we get W+ s+ = WTsS. This implies that (M++ )S = M++. On the other hand, (M++ )S = M+, because S+ C S.

Corollary 51. The monoidal structure on S is compatible with the model structure M++.

Proof. By Theorem 50, the morphism F1(T) — Fo(1) is a cofibrant replacement in M++. The morphism F1(T) A X — Fo(1) A X = X is a positive stable weak equivalence for any positively cofibrant X by Lemma 48.

Remark 52. For a natural p call a p-level weak equivalence (fibration) a morphism in S which is a level weak equivalence (fibration) for n-slices with n > p. These two classes of morphisms define a model structure M-p on S. Cofibrations in M-p are cofibrations in M which are isomorphisms on n-slices with n < p. By methods similar to those used above one shows that any n-level weak equivalence is a stable weak equivalence. Moreover, stable weak equivalences are obtained by localization of M-p over the set of morphisms | A G dom(/) U codom(/) , n > p}.

9. Symmetric powers in stable categories

Using results from Section 8, we are now going to show that left derived powers exist and coincide with homotopy symmetric powers for abstract symmetric spectra, see Theorem 55 below. This will be applied in Section 11 to the motivic stable homotopy category of schemes over a base.

So, let again C be a closed symmetric monoidal left proper cellular model category, T a cofibrant object in C, and S = SptE(C, T) the category of symmetric spectra.

To obtain results for symmetric spectra, similar to Theorem 25 and Corollary 27, we would require symmetrizability of generating cofibrations in S. However, we can unlikely meet such symmetrizability in applications, see Remark 58 below. Instead, we will be exploring strong Evn-symmetrizability for cofibrations in S. The phenomenon of strong Evn-symmetrizability was first observed in [3] for topological spectra. However, our proof for the case of abstract spectra is different from the one in [3], and heavily relies on Theorem 7.

Proposition 53. Let X be an object in S = Spts(C, T), cofibrant with respect to the positive projective model structure M +. Then, for any two positive integers m and n, the object (XAn)m, as an object of the category CEn, is cofibrant in the canonical model structure in C.

Proof. By Corollary 11, we need only to show that I+ is a strongly symmetrizable set of Evm-cofibrations for all m > 0. Let /i,..., /l be a finite collection of morphisms in I. Recall that I is the set of generating cofibrations in the initial cofibrantly generated category C. Let also pi,...,pl be a collection of l positive integers. We have to show that the morphism

Evm((FPl /i)Dni □ ...a(Fpl /l)Dni)

is a cofibration in CEni x' ' xEni for any multidegree {ni, ..., nl}.

Let r = nipi + ••• + nlpl, / = /°ni □ . ..n/j3nt, and let A and B be the source and target of the morphism /. For any non-negative i the functor Fi commutes with colimits since it is left adjoint. This and the monoidal properties of the functors Fi imply that

(Fpi/i)Dni □ ... □(Fpi/l)Dni = Fmpi+...+nipi (/inni □ ... □/P) = Fr(/) . Applying Evm one has

Evm(Fr(A)) = Sm xEm-r (A A TA(m-r))

Evm(Fr(B)) = Sm xEm_r (B A TA(m-r)) ,

where the group Sni x — ^x Sni acts on A and B naturally, acts identically on TA(m r), and it acts by right translations on Sm being embedded in it as permutations of the blocks in each of the l clusters of blocks, such that the i-th cluster contains ni blocks of pi elements each one, for i = 1, ..., l.

The point here is that this action of the group Sni x ••• x Sni on the set {1, ..., m} induces a free action of the same group on the objects Evm(Fr(A)) and Evm(Fr(B))

because the (right) action of £ni x---x Sni on the right cosets of £m_r in £m is free.1 It follows that the morphism Evm(Fr(f )) in CEni x' ' is isomorphic to a bouquet of several copies of the morphism (Sni x •••x Sni ) x (f ATA(m-r)). Therefore, Evm(F(f)) is a cofibration in CEni x* ' xEni, as required. □

Let now D be a cofibrantly generated model category and let G be a finite group. Then the functor Y — Y/G from DG to D is left Quillen and it has left derived by Theorem 11.6.8 in [8]. Given Y in DG, the homotopy quotient (Y/G)h is the value of this left derived functor at Y. In particular, there is a canonical morphism from (Y/G)h to Y/G, which is a weak equivalence when Y is cofibrant in DG. If D is in addition simplicial, then the homotopy quotient (Y/G)h is weak equivalent to the Borel construction (EG A Y)/G.

Lemma 54. Let Y be an object in SG, such that for any positive integer m the object Ym is cofibrant in the model structure on C G. Then the canonical morphism (Y/G)h — Y/G is a weak equivalence in M +.

Proof. Consider the positive projective model structure M + on the category S and the induced model structure on SG. Let Q+(Y) — Y be the cofibrant replacement in SG. By Remark 44 and Proposition 45, the functors Evm are left Quillen. Lemma 11.6.4 in [8] implies that the functors EvGm : SG — CG are also left Quillen. Therefore, the object Evm(QG(Y)) = QG(Y)m is cofibrant in CG for all m. Combining this with the assumption of the lemma, we see that, for all m > 0, the canonical morphism Q+(Y)m/G — Ym/G is a weak equivalence in C. As colimits in spectra are term-wise, the canonical morphism QG(Y)/G — Y/G is a positive projective weak equivalence. □

Notice that Lemma 54 is also true for the usual projective model structure M, and for more general model structures M-p from Remark 52.

Let now Symn(X)h be the n-th homotopy symmetric power of X, i.e. the homotopy quotient (XAn/£n)h. Combining Proposition 53 and Lemma 54, we obtain the following important result.

Theorem 55. Let X be an object in S = Spt^(C, T), cofibrant with respect to the positive projective model structure M +. Then, for any non-negative integer n the natural morphism

0X,n : Symn(X) — Symn(X)

is a weak equivalence in M +. Hence, it is also a stable weak equivalence by Theorem 50.

1 It is essential that all pi are positive.

Corollary 56. Symmetric powers preserve positive projective and stable weak equivalences between positively cofibrant objects in S.

Proof. The functors Symn, being homotopy quotients, preserve positive projective and stable weak equivalences. Then we apply Theorem 55.

Corollary 57. Let T be the homotopy category of the category of symmetric spectra S. The functors Symn : S — S have left derived functors LSymn : T — T, which are canonically isomorphic to the homotopy symmetric powers Symn. Besides, the left derived functors LSymn give a \-structure in T, which is canonical in the sense of positive stable model structure on symmetric spectra.

Proof. This is a straightforward consequence of Theorem 55, Ken Brown's lemma and the fact that homotopy symmetric powers give rise to Kunneth towers in distinguished triangles.

Remark 58. In contrast to level-wise strong symmetrizability asserted by Proposition 53, (positive) cofibrations in S are not symmetrizable in general. Indeed, if f is a cofi-bration in C, then symmetrizability of Fp(f) in S, for some p > 0, is equivalent to strong symmetrizability of f in C. Then cofibrations are not symmetrizable for spectra of simiplicial sets by Example 5. Furthermore, by a similar argument as in Corollaries 56 and 57, one shows that strong symmetrizability of cofibrations in C implies that left derived symmetric powers exist for C and coincide with the corresponding homotopy symmetric powers. By results from Sections 10 and 11, this gives again that cofibrations are not strongly symmetrizable for (pointed) simplicial sets and, as a consequence, for (pointed) motivic spaces (motivic spaces will be considered in Section 11 below).

10. Symmetrizable cofibrations in topology

Let us illustrate symmetrizability of (trivial) cofibrations in Kelley spaces and simiplicial sets. Recall that the category Top of all topological spaces is not a closed symmetric monoidal category, as it does not have an internal Hom in it. The right category is the category of Kelley spaces K, see Definition 2.4.21(3) in [10]. It is a closed symmetric monoidal model category with regard to the monoidal product defined by means of the right adjoint to the embedding of K into Top, see Theorem 2.4.23 and Proposition 4.2.11 in [10]. The point here is that the realization functor | | from AopSets to Top takes its values in K and, moreover, the it is symmetric monoidal left Quillen, as a functor into K, see Proposition 4.2.17 in [10]. It follows that the category K is simplicial. For any non-negative integer n let A[n] = Hom^(—, [n]) be the n-th simplex. If Is is the set of the canonical inclusions cA[n] — A[n], n > 0, and Js is the set of canonical inclusions Aj[n] — A[n], n > 0, 0 < i < n, then Is and Js are the sets of generating cofibrations and the sets of generating trivial cofibrations for the model structure in AopSets. The sets |Is| = I and |Js| = J cofibrantly generate K.

Lemma 59. If f is a weak equivalence in AopSets then Symn(f) is a weak equivalence in AopSets for any n > 0.

Proof. Let f : X — Y be a weak equivalence in AopSets. Since | | is a left Quillen functor from AopSets to K, all simplicial sets are cofibrant and Kelley spaces are fibrant, |f | is a weak equivalence between fibrant-cofibrant objects in K. Then |f | is a left homotopy equivalence in the simplicial closed symmetric monoidal model category K. Applying Lemma 1, we obtain that Symn(|f |) is a weak equivalence in K for all n > 0. Since | | is monoidal and left adjoint, we have that Symn(|f |) is the same morphism as |Symn (f)|. □

Proposition 60. All (trivial) cofibrations in AopSets, and all (trivial) cofibrations in AopSets* are symmetrizable.

Proof. By Lemma 12, it is enough to prove the proposition in the unpointed case only. For the set of all cofibrations, since the monoidal product and colimits in AopSets are level-wise, it is enough to prove a similar proposition in the category of sets, where cofibrations are injections. This is an easy exercise. For the set of all trivial cofibrations, we apply Lemma 59 together with Corollary 23. □

Proposition 61. All (trivial) cofibrations in K, and all (trivial) cofibrations in K* are symmetrizable.

Proof. Since |Is| = I, | Js | = J, and | | is a symmetric monoidal functor commuting with colimits, we see that by Proposition 60, I and J are symmetrizable. Thus we conclude by Corollary 9.

Since the sets of cofibrations and trivial cofibrations in AopSets, AopSets*, K, and K* are symmetrizable, we can apply Theorem 25 getting A-structures of left derived symmetric powers in the corresponding unstable homotopy categories. In the stable setting, when S = SptE(C, T), the category C is the category AopSets* of pointed simplicial sets and T is the simplicial circle S1 , i.e. the coequalizer of the two boundary morphisms A[0] ^ A[1], then Theorem 55 and Corollary 56 specialize to the results [3], III, 5.1, and [15], 15.5. Corollary 57 yields the A-structure of left derived symmetric powers in the topological stable homotopy category.

11. Symmetrizable cofibrations in A^homotopy theory of schemes

Now we are going to apply the main results of the paper to the Morel-Voevodsky homotopy theory of schemes over a base and prove the existence of A-structures of left derived symmetric powers in both unstable and stable settings of that theory.

Let B be a Noetherian separated scheme of finite Krull dimension, Sm/B the category of smooth schemes of finite type over B, and let Pre(Sm/B) be the category of

presheaves of sets on Sm/B, i.e. contravariant functors from Sm/B to Sets. Let C be the category AopPre(Sm/B) of simplicial presheaves over B. Sometimes it is convenient to think of C as the category Pre(Sm/B x A) of presheaves of sets on the Cartesian product of two categories Sm/B and A. If X is a smooth scheme over the base B, let Ax [n] be a presheaf on Sm/B x A sending any pair (U, [m]) to the Cartesian product of sets Homsm/B(U, X) x HomA([m], [n]). Then we get a fully faithful embedding Sm/B ^ C of Yoneda type, sending X to the presheaf Ax [0] represented by X, and similarly on morphisms. If K is a simplicial set, i.e. a presheaf of sets on the simplicial category A, then it induces another presheaf on Sm/B x A by ignoring schemes and sending a pair (U, m) to the value Km of the functor K on the object [m] in A. This gives a functor AopSets ^ C, which provides a simplicial structure on the category C. The symmetric monoidal structure in C is defined section-wise, i.e. for any two simplicial presheaves X and Y the value of their product on (U, [m]) is the Cartesian product of the values of X and Y on (U, [m]).

Following Jardine, [13], we say that a morphism f : X ^ Y in C is a weak equivalence if f induces weak equivalences on stalks of the presheaves X and Y, where stalks are taken in the sense of Nisnevich (or étale) topology on the category Sm/B. Let W be the class of all weak equivalences in C. Notice that, in spite of that C is a category of simplicial presheaves, the topology is needed to define weak equivalences in C in terms of stalks. Let also I be the set of monomorphisms of type X ^ Au [n] for some simplicial presheaf X, smooth B-scheme U and n > 0. Fix a cardinal ¡3 > 2a, where a is the cardinality of the morphisms in Sm/B. Let J be the set of monomorphisms X ^ Y, which are weak equivalences and such that the cardinal of the set of n-simplices in Y is less than 3 for all n. One can show that the class I-cell consists of all section-wise monomorphisms of simplicial presheaves. Then C together with the above defined weak equivalences and monomorphisms taken as cofibrations is a simplicial left proper and cellular closed symmetric monoidal model category cofibrantly generated by the set of generating cofibrations I and the set of generating trivial cofibrations J. Actually, this is a consequence of a more general result on model structures for simplicial presheaves on a site due to Jardine, see [13]. Such constructed model structure M = (I, J, W) is called the injective model structure in C.

As well as in Example 41, denote by A1 the simplicial motivic space represented by the affine line AB over the base scheme B. Then A1 ^ 1 is a diagonalizable interval, with the multiplication coming from the multiplication in the fibres of the structural morphism from AB to B. The above injective model structure and the set of morphisms S = {XAA1 X | X G dom(I)Ucodom(I)} satisfy the assumptions of the localization theorem in [8]. The corresponding left localized model structure Mai = (I, Jai , Wai ) is one of the motivic model structures on C, and the corresponding localization Cai is again a simplicial left proper cellular closed symmetric monoidal model category cofibrantly generated by the same set of generating cofibrations I and the new localized set of generating trivial cofibrations Jai . The category CAi is called the unstable motivic model

category of schemes over the base B. Its homotopy category Ho(C\i) is nothing but the unstable motivic homotopy category of schemes over B, which we denote by H(B).

The following result is the precise statement of Theorem A mentioned in Introduction.

Theorem 62. Let B be a Noetherian scheme of finite Krull dimension, and let C&i be the unstable motivic model category of schemes over B. Then all symmetric powers Symn preserve weak equivalences in C&i, and the corresponding left derived functors LSymn yield a A-structure in H(B).

Proof. Since cofibrations in C are coming section-wise from cofibrations simplicial sets, all objects are cofibrant in C. By the same reason, and by Proposition 60, we also have that all cofibrations in C are symmetrizable. The class of trivial cofibrations is symmetrizable too. Indeed, let f : X — Y be a trivial cofibration C. Since stalks of presheaves are colimits commuting with symmetric powers, the morphism (Symn(f ))P on stalks at a point P is nothing but the n-th symmetric power Symn(fP) of the morphism fp induced by f at P. So (Symn(f))P is a weak equivalence of simplicial sets by Proposition 60. Since, moreover, A1 — 1 is a diagonalizable interval and all objects are cofibrant in C, we conclude by Theorem 42 and Theorem 25.

Remark 63. Theorem 62 holds true also in the pointed setting by Lemma 12.

Let now T be the motivic (1, 1)-sphere. Recall that T is the A-product of the simplicial circle, i.e. the coequalizer of the two morphisms from A[0] to A[1], and the algebraic group Gm over B in the pointed category C*. The corresponding category of symmetric spectra S = SptE((C\i )*, T), together with the corresponding stable model structure, is the category of motivic symmetric spectra over the base scheme B, and the homotopy category of S, with regard to the stable model structure, is nothing but the Morel-Voevodsky motivic stable homotopy category over B, see [22] and [14]. We will denote it by SH(B).

The category S = Spts((CAi)*, T) of motivic symmetric spectra has a structure of a simplicial closed symmetric monoidal model category by Hovey's result, [11]. Moreover, the simplicial suspension £Si induces an autoequivalence in its homotopy category SH(B), so that it is a triangulated category (use Section 6.5 in [10]). Then we see that the results in Proposition 53, Theorem 55, Corollary 56 and Corollary 57 hold true for symmetric spectra of simplicial sets and for motivic symmetric spectra uniformly. In other words, we have the following result (Theorem B in Introduction).

Theorem 64. Let B be a Noetherian scheme of finite Krull dimension, and let T = S1 A Gm be the motivic sphere. Symmetric powers preserve stable weak equivalences between positively cofibrant objects in the category Spts((CAi)*, T) of motivic symmetric spectra over the base B. The corresponding left derived symmetric powers LSymn exist, they are canonically isomorphic to homotopy symmetric powers and give rise to a A-structure in SH(B).

The category SH(B), being triangulated, can be Q-localized getting the Q-linear triangulated symmetric monoidal category SH(B)q. Hirschhorn's localization allows to make symmetric spectra into a Q-linear stable model category, see Definition 3.2.14 in [1]. One can show that the A-structure from Theorem 64 induces the A-structure of symmetric powers with Q-coefficients defined via idempotents in endomorphism rings, see 3.3.21 in [1]. The latest A-structure coincides with the system of towers constructed in [5]. If now SH(B)Q is the full subcategory of compact objects in SH(B)q, the A-structure of Q-local left derived symmetric powers induces the A-structure in the ^-theory of the triangulated category SH(B)Q considered in [6].

Acknowledgments

The authors are grateful to Peter May for helpful suggestions and stimulating interest to this work. We also thank Bruno Kahn, Dmitry Kaledin, Roman Mikhailov, Oliver Rondigs, Alexander Smirnov, Vladimir Voevodsky and Chuck Weibel for useful discussions along the theme of this paper. We acknowledge the effort of the anonymous referee which helped to improve the text of the paper. The research is done in the framework of the EPSRC grant EP/I034017/1 "Lambda-structures in stable categories". The first named author acknowledges the support of the grants MK-5215.2015.1, RFBR 13-01-12420, 14-01-00178 and the subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program. The second named author is grateful to the Institute for Advanced Study in Princeton for the support and hospitality in 2004-2006, where and when this work began.

Appendix A. Categorical v.s. geometrical symmetric powers

Let k be a field, and let Sch be the category of separated schemes of finite type over k. Let Sm be the full subcategory of smooth schemes in Sch, and let C be the category of simplicial presheaves on Sm. The fully faithful embedding of Sm into C can be extended to Sch, sending a scheme X from Sch to the functor hX = AX [0], and similarly on morphisms. Let E(X) be the motivic symmetric spectrum of the motivic space hX. If any finite subset in X is contained in an affine open subscheme in X, the n-th symmetric power Symn(X) exists as an object in Sch, and the rational homotopy type of the motivic spectrum Symn(E(X)) is the same as the rational homotopy type of the motivic spectrum E(Symn(X)), see [17].

In the unstable motivic category the situation is more complicated, as the rational unstable motivic homotopy theory is not yet in place. Working integrally, the homotopy type of the categorical n-th symmetric power Symn(hX) of the motivic space hX is not the same as the homotopy type of the motivic space hSymn(X). The comparison of these two objects is a question of critical importance, since its understanding would provide the geometrical meaning to our categorical approach to symmetric powers in the

A1-homotopy setup. Below we consider a certain argument, which gives a flavour what the discrepancy between two homotopy types in question might depend on.

First we should look at the category of sets .Sets with the discrete topology on it. Presheaves on .Sets have one stalk only. Therefore, if X is a set and G is a finite group acting on X, it is easy to show that the canonical map from hx/G to hX/G is an isomorphism. So, all is fine in the simplest possible case.

Let now X be a scheme from Sch and let G be a finite group acting on X. Suppose X can be covered by G-invariant affine open subschemes, so that the quotient X/G exists in Sch. The group G acts freely on X if the canonical morphism

n : X —¥ X/G

is étale. Let also

a : hx /G — hx/G

be the obvious canonical morphism in C. In case of symmetric powers, X must be the n-th power of a scheme, and G must be the symmetric group £„ permuting factors in X.

Proposition 65. If G acts freely on X, the canonical morphism a is a weak equivalence in the étale injective model structure on C.

Proof. To prove the proposition it is enough to show that a induces isomorphisms on spectra of strictly Henselian rings. Let R be a strictly Henselian local ring, m be the maximal ideal in it and l = R/m be the corresponding residue field. All we need to show is that the canonical morphism of sets

aR : X(R)/G — (X/G)(R)

is an isomorphism. Let Ar be the category of étale algebras over R and let A be the category of étale algebras over l. As R is Henselian, the residue homomorphism R — l induces an equivalence of categories ^ : Ar — A. Let

f : Spec(R) — X/G

be an element in (X/G)(R). The preimage of f under the morphism n is a set of R-points of the étale R-algebra S, where Spec(S) — X is the pull-back of f with respect to the morphism n. Let

f : Spec(l) — X/G be the precomposition of f with the morphism Spec(l) — Spec(R), and let

Spec(L) — X

be the pull-back of f with regard to n. As ^ is an equivalence of categories,

a—1(f) = a-f ,

where ai is the morphism from X(l)/G to (X/G)(l). In other words, a—1(f ) is in bijection to l-points of the étale l-algebra L. Since R is strictly Henselian, the residue field l is separably closed. This gives that L is isomorphic to the product of n copies of l, where n is the order of G, and G acts freely on Spec(L). Then the quotient of the set of l-points of X by G can be identified with l-points of X/G. Therefore, the quotient of the set of R-points of X by G can be identified with R-points of X/G. Hence, aR is a bijection. □

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