Symmetric powers in abstract homotopy categories

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 etale 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 etale topology on schemes, then all symmetric powers Sym^n(f) are weak equivalences too. This gives left derived symmetric powers which aggregate into a categorical lambda-structures on the corresponding motivic homotopy categories of schemes over a base.


Introduction
On page 596 in [28] Voevodsky pointed out the importance of the correct analog of symmetric powers in A 1 -homotopy theory of schemes. In [29] he has developed a motivic theory of symmetric powers on the ground of symmetric powers of schemes, valid enough to construct motivic Eilenberg-MacLane spaces needed for the proof of the Bloch-Kato conjecture. However, the symmetric powers in loc.cit. seem to be not universal enough as their construction depends greatly on the underlying category of schemes over a field. The aim of this paper is to make an attempt to develop a purely homotopical theory of symmetric powers and to understand to which extent such a theory is possible at all. Our approach is to construct symmetric powers in general symmetric monoidal and simplicial model categories first, and then to find precise axioms which would guarantee an existence of left derived symmetric powers on the homotopy level.
Let C be a closed symmetric monoidal model category cofibrantly generated by a class of cofibrations I and a class of trivial cofibrations J. We introduce a new property of I and J to be symmetrizable, see Subsection 3.2. We claim that left derived symmetric powers exist in the homotopy category of C if both classes I and J are symmetrizable (Theorem 25).
On the other hand, symmetric powers should always come together with some naturally arising towers whose cones might be computed by the Künneth rule, bringing an efficient tool to compute symmetric powers in various cellular situations. Being in a model category we should work with towers of cofibrations, while dealing with distinguished triangles in the triangulated setting. In order to develop the machinery of such Künneth towers we introduce a sort of -calculus of colimits, where comes from the notion of a push-out product in a monoidal model category. Here the idea is the same: if the generating classes I and J are symmetrizable then the above -calculus gives Künneth towers (Theorem 26).
Working with spectra we need to stabilize symmetric powers. Stabilization of spectra is a particular case of localization of a model category with respect to a localizing class of morphisms in it. We prove a general result saying that if symmetric powers carry morphisms from the localizing class into local equivalences, then symmetric powers factor through localization (Theorem 28). One expect that the localization needed to stabilize multiplication by a cofibrant object through symmetric spectra does satisfy the above axiom, provided extra symmetricity assumptions for generating (trivial) cofibrations. Hopefully, all necessary axioms hold in the topological setting, and in the present version of this manuscript we give an outline how that can be proved. This should give a suitable construction of symmetric powers in the topological stable homotopy category T .
Notice that Künneth towers guarantee a nice theory of zeta-functions with coefficients in the Grothendieck group of compact objects in T . Recall that to any strongly dualizable object in T one can assign its Euler characteristic in the ring of endomorphisms of the unit object in T . Euler characteristics of symmetric powers of a spectrum X give rise to a zeta-function ζ X (t) with coefficients in End(½). The existence of Künneth towers implies that rationality of zeta-functions has 2-out-of-3 property for vertices in distinguished triangles in T . This is an integral version of the result obtained in [6]. Working with simplicial sets or CW -complexes, we show that the above theory leads to that any finite spectrum has a rational zeta-function with coefficients in Z, which is a stable homotopy version of MacDonald's theorem for C ∞ manifolds, [18].
But most important prospective applications should lie in the area of motivic symmetric spectra, of course. Unfortunately, we don't know whether all the necessary axioms hold true in the category of motivic spaces over a Noetherian base, making our theory applicable in the motivic setting. The crucial point here is as follows. Let SH(k) be the motivic stable homotopy category over a field k. For any scheme X over k let E(X) be the motivic symmetric spectrum of X in SH(k). How to connect symmetric powers Sym n E(X) of the spectrum E(X) with the spectra E(Sym n X) of symmetric powers of X constructed in the category of schemes? A suitable answer to that question would provide geometrical meaning to our construction, unifying our homotopical approach with Voevodsky's geometrical one. We hope that there exist certain appropriate towers in SH connecting Sym n E(X) and E(Sym n X) in a nice way and collect some naive reasoning for that in the end of the paper.
When T is Morel-Voevodsky's motivic stable category over a field k then the situation with zeta-functions is much more interesting because the ring End(½) is now isomorphic to the Grothendieck-Witt group GW (k) of quadratic forms over the ground field k, see [22], and the Euler characteristic is now highly non-trivial. It seems to be interesting to approach rationality of zeta-functions with coefficients in GW (k) for certain types of algebraic varieties, such as varieties whose motivic spectra admit cellular decomposition to Voevodsky's spheres, see [4].
Acknowledgements. The authors are grateful to Roman Mikhailov, Alexander Smirnov and Vladimir Voevodsky for useful discussions along the theme of that paper. We both wish to thank Bruno Kahn, Dmitri Kaledin, Oliver Röndigs and Chuck Weibel for stimulating remarks. The first named author was partially supported by the grants MK-297.2009. 1, NSh-19871, NSh- .2008.1 and RFBR 08-01-00095. The second named author was partially supported by RDF 6677, and he is also grateful to the Institute for Advanced Study (Princeton) for the support and hospitality in 2004 -2006, where he had got the inspiration originated this paper.

Symmetric powers in homotopy categories
2.1. Simplicial monoidal model categories: recollections. As it was mentioned in Introduction, we will work not in general triangulated categories but with some nice ones, arising as homotopy categories of some suitable model categories.
Assume we are given with a category C which has three distinct but pair-wise compatible structures: it is a model category, it is a closed symmetric monoidal category, and it is a simplicial category. So, first of all, C is equipped with three classes of morphisms, weak equivalences (W ), fibrations (F ) and cofibrations (C) which have the standard lifting properties and meanings, see [25] (or [9] for modern reference).
The monoidality of C means that we have a functor sending any ordered pair of objects X, Y into their monoidal product X ∧Y , and that product is symmetric, i.e. there exists a functorial transposition isomorphism X ∧ Y ∼ = Y ∧ X .
• If q : Q½ → ½ is a cofibrant replacement for the unit object ½, then the maps q ∧ id : Q½ ∧ X → ½ ∧ X and id ∧ q : X ∧ Q½ → X ∧ ½ are weak equivalences for all cofibrant X.
The simpliciality of the category C means that it is enriched by the category of simplicial sets △ op Sets, i.e for any two objects X and Y in C there exists a simplicial set map(X, Y ), for any three X, Y, Z ∈ Ob(C ) there exists a functorial composition where ⊗ is the product of simplicial sets, and for any two X and Y in C one has map(X, Y ) 0 = Hom C (X, Y ) . Compatibility of the simplicial structure and the model one is provided as follows. For any X, Y ∈ Ob(C ) and K ∈ Ob(△ op Sets), there exist two objects X ∧ K and Y K in C and simplicial adjunction isomorphisms map(X ∧ K, Y ) ∼ = Hom △ op Sets (K, map(X, Y )) ∼ = map(X, Y K ) functorial in X, Y and K. For any fibration p : X → Y and cofibration j : X ′ → Y ′ the morphism j * × p * : map(Y ′ , X) → map(X ′ , X) × map(X ′ ,Y ) map(Y ′ , Y ) must be a fibration, and a weak equivalence, if either j or p are so.
Equivalently, C is a △ op Sets-model category in the following sense. Assume we are given with a monoidal model category S and a monoidal model category C . Then C is called a (right) S -model category if it is a (left) S -module, the action is a Quillen bifunctor, [9], p. 107, and for a cofibrant X ∈ Ob(C ) and for cofibrant replacement q : Q½ → ½ in the category S the map id ∧ q : X ∧ Q½ −→ X ∧ ½ is a weak equivalence, loc.cit, p. 114. It is important for what follows that, given a simplicial model category C , we can multiply objects in C by simplicial sets. In particular, we have a cylinder functor X → Cyl X = X ∧ I , where I is the simplicial interval, a suspension functor as well as a path object functor X → X I , which is right adjoint to X ∧ I.
Notice that we should require that the monoidal functor K → K ∧ ½ from simplicial sets to C is symmetric, i.e. both symmetric structures on C and △ op Sets are compatible.
Notice, that simplicial monoidal model categories naturally arise from monoidal model categories C endowed with a cosimplicial object in it, i.e. a functor from the category of finite ordered sets to C allowing to enrich Hom-sets in C by simplicial sets.
In most applications we suppose to deal with pointed model categories, i.e. model categories whose initial and terminal objects are isomorphic. Let △ op Sets * be the model category of pointed simplicial sets. A pointed simplicial model category is a pointed model category which is also simplicial. A pointed model category C is simplicial, i.e. a △ op Sets-model category, if and only if C is a △ op Sets * -model category. Indeed, let C be a pointed model category which is also an △ op Sets * -model category. Then the structure of a △ op Sets-model category on C , with the corresponding product denoted by ⊗, can be defined as follows. For any two objects X and Y in C , and a simplicial set K, we put X ⊗ K = X ∧ (K + ) , the simplicial set map(X, Y ) does not change, and a path object X K is C over △ op Sets is the path object X K + in C over △ op Sets * , where Conversely, let C be a pointed model category which is also an △ op Setsmodel category. Define the structure of a △ op Sets * -model category on C by setting, for objects X, Y in C and a pointed simplicial set * p −→ K, a product X ∧ K to be the colimit of the diagram The simplicial set map(X, Y ) does not change as it is canonically pointed, a path object X K is the limit of the diagram where the vertical arrow is an evaluation at the point p morphism in C as a △ op Sets-category.

2.2.
Symmetric powers with coefficients in Z. For any natural number n let Σ n be the symmetric group of permutations of n elements, say {1, 2, . . . , n}. We look at a group as a category with single object and morphisms being elements of the group. Let C be a closed symmetric monoidal model category. Given an object X in C we have a functor F X,n : Σ n −→ C , sending the unique object in the symmetric group to X ∧n and permuting factors using the commutativity and associativity constrains in C . Definition 1. Given an object X in C , its n-th symmetric power Sym n X as an honest colimit colim (F X,n ).
Clearly, Sym n is an endofunctor in C because, given a morphism f : X → Y , it gives a morphism on n-th powers inducing a universal morphism on colimits Sym n f : Sym n X −→ Sym n Y .
The following simple lemma shows that symmetric powers respect homotopy equivalence provided the model category is simplicial: Lemma 2. Suppose C is a simplicial closed symmetric monoidal model category. Let f, g : X ⇉ Y be two morphisms in C which are left homotopic, i.e. there exists a morphism H : X ∧ I → Y , such that H 0 = f and H 1 = g. Then, for any natural n, the morphism Sym n f is left homotopic to the morphism Sym n g.
Proof. Consider a smash product of the homotopy H with itself, and let a : (X ∧ I) ∧n −→ X ∧n ∧ I ∧n .
be an isomorphism permuting factors in the desired order, and let b = H ∧n • a −1 , so that the diagram is commutative. The specificity of a simplicial monoidal model category is that we always have the diagonal morphism which does not exists in a general model category. Precomposing b with the morphism id X ∧n ∧ ∆ we obtain a new homotopy The second essential feature of any simplicial model category is that the cylinder functor has right adjoint (see Section 2.1), so commutes with colimits. As permutation of factors in (X ∧ I) ∧n is coherent with permutation of factors in X ∧n in the product X ∧n ∧ I, because the permutation does not effect the diagonal, we obtain a left homotopy between Sym n f and Sym n g.
The following example shows that the simpliciality of C is essential here.
Example 3. Let Com − (Z) be the category of bounded on the right cohomological type complexes whose terms are finitely generated abelian groups. The category Com − (Z) inherits the monoidal structure via total complexes Tot(− ⊗ −) and has a natural structure of a model category whose cofibrations are term-wise monomorphisms with torsion-free cokernels, fibrations are term-wise epimorphisms and weak equivalences are quasi-isomorphisms. The category is model symmetric monoidal but not simplicial. Let X be the complex where Z is concentrated in degrees −1 and 0 respectively. This complex is homotopically trivial. On the other hand, its naive symmetric square Sym 2 X is computed via the diagram where Z/2 stands in degree −2. Clearly, this Sym 2 X has non-trivial cohomology group in degree −2.
A naive way to define integral symmetric powers in T would be through Lemma 2 and the standard treatment of homotopy categories as subcategories of fibrant-cofibrant objects factorized by left homotopies, see [9, 1.2]. Indeed, let C cf be the full subcategory of objects which are fibrant and cofibrant simultaneously. Let, furthermore, ho(C ) be the quotient category of C cf by left homotopic morphisms between fibrant-cofibrant objects in C . In other words, objects in ho(C ) are the same as in C cf , and for any two fibrant-cofibrant objects X and Y we define Hom ho(C ) (X, Y ) to be the classes of left homotopic morphisms from X to Y . Recall that for fibrant-cofibrant objects left homotopies coincide with right ones, and two morphisms f and g from X to Y are left homotopic if and only if the morphism (f g) : X X → Y factors through the object X ∧ I, see [7, p. 94]. As integral symmetric powers respect homotopies by Lemma 2, we have now a functor Sym n : ho(C ) −→ Ho(C ) .
The category C , being a model category, is endowed with a fibrant replacement functor R : C → C f and a cofibrant replacement functor Q : C → C c . Combining both we obtain mixed replacement functors RQ and QR from C to the full subcategory C cf of fibrant-cofibrant objects in C , any of which induces a quasiinverse to the obvious functor ho(C ) → Ho(C ). Then one might wish to construct a functor Sym n on Ho(C ) as a composition However, this way is not too much natural as we still have no derived symmetric powers in Ho(C ). A desired way of defining symmetric powers in T is through their left derived, as it requires cofibrant replacements only. Let C be a closed symmetric monoidal model category (not necessarily simplicial). Recall that if F : C → C is an endofunctor on C which takes trivial cofibrations between cofibrant objects to weak equivalences then F preserves all weak equivalences between cofibrant objects by Ken Brown's lemma. In this situation there exists a left derived endofunctor LF on Ho(C ). Our aim then is to figure out when the endofunctors Sym n take trivial cofibrations between cofibrant objects to weak equivalences in C . If this takes place then we define symmetric powers in Ho(C ) as left derived functors So, our strategy now is to determine under which assumptions the symmetric powers in C carry trivial cofibrations between cofibrant objects to weak equivalences.

Symmetrizable cofibrations
3.1. Cubes in C . In order to show the existence of Künneth tower we need to develop some easy calculus of diagrams in monoidal model categories. So, let C be a closed symmetric monoidal model category with symmetric product ∧ : C × C → C . Let I be the category with two objects 0 and 1 and one morphism 0 → 1 between them, and let I n be the n-th fold Cartesian product of I with itself. Objects in I n then are n-tuples of zeros and units, where the order is important. A functor For any morphism f : X → Y and any natural n let be the composition of the n-th fold Cartesian product of the functor I K → C induced by f and the functor C n ∧ → C (under freely chosen but fixed order of parenthesis in computing of the product in n arguments). For example, For any 0 ≤ i ≤ n one has a full subcategory I n i in I n generated by n-tuples having not more than i units in them. The restriction of K n (f ) on I n i will be denoted by K n i (f ), or simply by K n i when f is clear. For example, Obviously, K n 0 = X n and K n n = K n . Respectively, n 0 = X n and n n = Y n . As K n i−1 is a subdiagram in K n i one has a morphism on colimits The n-th symmetric group Σ n acts on I n and so on K n . Since permutations do not change numbers of unites in objects of I n , each subcategory I n i is invariant under the action of Σ n . Then Σ n acts on K n i and so on n i . Let˜ n i (f ) = colim Σn n i (f ) for each index i. Obviously,˜ n 0 (f ) = Sym n X and˜ n n (f ) = Sym n Y , and for each index i we have a universal morphism between colimits n i−1 (f ) −→˜ n i−1 (f ) . Sometimes we will drop the morphism f from the notation when confusion is unlikely, i.e.˜ 3.2. Symmetrizable morphisms. The first axiom of a monoidal model category implies that, for any cofibration f : X → Y in C the pushoutproduct 2 1 (f ) −→ Y ∧ Y is a cofibration. As we will see later, it implies that the morphism n n−1 (f ) −→ Y n is a cofibration for any natural n, not only for n = 2. It turns out that, in order to develop an interesting theory of symmetric powers in T = Ho(C ), we need to have that for any (trivial) cofibration f all morphisms From now on we will be using the following terminology. A morphism f : X → Y in C is said to be a symmetrizable (trivial) cofibration if the corresponding morphism˜ n n−1 (f ) −→ Sym n Y is a (trivial) cofibration for any integer n ≥ 1. Notice that a symmetrizable (trivial) cofibration f is itself a (trivial) cofibration because˜ 1 0 (f ) → Sym 1 Y is nothing but the original f . Suppose C is generated by a class of generating cofibrations I and a class of generating trivial cofibrations J. If all the morphisms in I are symmetrizable cofibrations then we say that the class I is symmetrizable. Similarly, all morphism from J is a symmetrizable trivial cofibration then we say that J is symmetrizable. Now our aim is to prove that if C is cofibrantly generated by a set of generating cofibrations I and a set of trivial generating cofibrations J then all cofibrations and all trivial cofibrations are symmetrizable as soon as the classes I and J are symmetrizable. To show this we will need to prove that symmetrizibility for both cofibrations and trivial cofibrations is stable under push-outs, transfinite compositions and retracts in C .

Proposition 5. Composition of two subsequent symmetrizable (trivial) cofibrations is a symmetrizable (trivial) cofibration.
The next propositions involve some terminology and notation coming from the theory of ordinals. It will be recalled in Section 3.5 below.

Proposition 7. Retract of a symmetrizable (trivial) cofibraion is a symmetrizable (trivial) cofibration.
In order to prove each of the above propositions we will first prove small and easy lemmas, and then aggregate them into proofs of the propositions.
3.3. Push-outs. We need some more notation. If F : I n → C is a functor, let F i be the restriction of F on the subcategory I n i . Let n and m be two positive integers, and let F : I m → C and G : I n → C be two functors. Let F ∧ G be a composition For any natural n let 1 n be the n-th tuple consisting of n units, considered as an object in I n .
Lemma 8. For any two functors F : I m → C and G : I n → C the following commutative diagram is push-out: Proof. Change the order of computation of the colimit of the diagram (F ∧ G) m+n−1 in an appropriate way.

Lemma 9. If the morphism
is an isomorphism, so is the morphism Proof. Since colim F m−1 → F (1 n ) is an isomorphism, so is the left vertical arrow in the commutative square in Lemma 8. Since it is a push-out commutative square, its right vertical arrow is an isomorphism too. The composition coincides with the morphism (colim F m−1 → F (1 n )) ∧ G(1 n ) which is an isomorphism. Then the desired morphism is also an isomorphism.
For any diagram D and any object X in C let D → X be a cone over D in the categorical sense. To be more precise, if we look at D as a functor from a certain index category J to C and X is being considered as a constant functor from J to C sending everything to X and all morphisms in J to the identity, then D → X is a morphism of functors from J to C .

Lemma 10. For any push-out
Proof. For each i, 0 ≤ i ≤ n − 1, consider an n + 1-dimensional cube, i.e. a functor from I n+1 to C It is enough to prove by decreasing induction on i, 0 ≤ i ≤ n − 1, that this diagram is a push-out, because for i = 0 this is actually the needed statement. With this aim we show that the diagram is a push-out for any i, 1 ≤ i ≤ n. Let's denote this commutative diagram by B i . Notice that B i is a functor from I n+1 to C , so that the notation (B i ) n does make sense.
It is not hard to see that We are going to apply Lemma 9 twice. Let first Applying Lemma 9 again we get: Lemma 11. For any push-out Proof. Use Lemma 10 and choose an appropriate order of colimit calculation in subdiagrams.
be a push-out in C . Assume a group G acts on A i for any index i = 1, 2, 3, 4 such that the morphisms in the above diagram commute with the action of G. Let for each i. Then the induced commutative square is push-out.
Proof. Just obvious.
The proof of Proposition 4. Assume we are given with a push-out in C . By Lemma 12 and 11 the commutative squarẽ n n−1 (f ) is push-out as well. As f is symmetrizable (trivial) cofibration, the upper horizontal arrow is a (trivial) cofibration. Then the lower horizontal arrow is a (trivial) cofibration. Hence, the push-out f ′ is a symmetrizable (trivial) cofibration.

3.4.
Compositions. Now we will prove Proposition 5. Let f : X → Y and g : Y → Z be two symmetrizable (trivial) cofibrations in C . We need to show that their composition gf is a symmetrizable (trivial) cofibration too. Consider the following commutative trianglẽ n n−1 (gf ) 6 6 r r r r r r r r r r r r r r r r r r r r Since g is a symmetrizable (trivial) cofibration,˜ n n−1 (g) → Sym n Z is a (trivial) cofibration. Therefore, as soon as we show that the morphism n n−1 (gf ) →˜ n n−1 (g) is a (trivial) cofibration, it would follow immediately that the resulting morphism˜ n n−1 (gf ) → Sym n Z is a (trivial) cofibration, i.e. the composition gf is a symmetrizable (trivial) cofibration. Thus, in order to prove Proposition 5 all we need is to show that the morphism where E → B is a (fibration) trivial fibration. Again we need to introduce some more convenient notation. Let J be a small category and let F : J → C be a functor considered as a diagram in C . Recall that for any object X in C the notation F −→ X stands for a categorical cone over the diagram F with a vertex at X, i.e. a natural transformation of the functor F to the constant functor X from J to C . If L is a subcategory in J, let F | L be a restriction of F on L. If a solid commutative diagram with a trivial fibration E → B admits the dotted arrow making the diagram commute, then we will say that A pair of subsequent morphisms in C is just a functor K : J −→ C from the category J to C . As K is fully determined by the pair f, g it is natural to write: Let now J n be the Cartesian n-th power of the category J, and let be the composition of the n-th Cartesian power of the functor K(f, g) and the n-th monoidal product ∧ : C n → C . Notice that objects in J n are precisely finite ternary codes consisting of n numbers, i.e. ordered strings of 0, 1 and 2 of length n. Given such a string s ∈ J n its "weight" is just the number of "2" in it. Let then J n i be a full subcategory in J n generated by ternary strings of weight ≤ i, and let J n i be a full subcategory in J n i generated by strings containing at least one zero. Let also We see that K n n−1 (gf ) is a subdiagram inK n n−1 (f, g) and K n n−1 (g) is a subdiagram in K n n−1 (f, g). Also, the group Σ n acts on the diagrams K n i (f, g) and K n i (f, g). Notice that a priori n n−1 (gf ) = colim K n n−1 (gf ) and n n−1 (g) = colim K n n−1 (g). On the other hand, changing the order of colimit calculation, we get n n−1 (gf ) = colimK n n−1 (f, g) and n n−1 (g) = colim K n n−1 (f, g) . Let's say again that to prove that˜ n n−1 (gf ) →˜ n n−1 (g) is a (trivial) cofibration we need only to show that for any commutative solid diagram n n−1 (gf ) with (fibration) trivial fibration E → B the dotted arrow does exist leaving the diagram to be commutative. This is equivalent to construct a dotted Σ n -invariant lifting n n−1 (gf ) Finally, this is the same as to construct a dotted Σ n -invariant cone-liftinĝ where both α and β are Σ n -invariant cones over diagrams.
To prove existence of such γ we will also need the following fairly general construction. Let G be a finite group and let H be a subgroup in it. Remind that we look at a group as a category with a single object and morphisms being elements of the group. So, let C G be a category of functors from G to C , and the same stands for C H . The natural restriction res G H : C G −→ C H has left adjoint functor cor G H : C H −→ C G , which can be handcrafted as follows. For any object X in C H let G × X be a coproduct of X g over g ∈ G, where X g = X for each element g. Symbolically, the group H acts on G × X by the rule To be pedantic, for any h ∈ H the corresponding automorphism of G × X consists of automorphisms Let cor G H (X) be the colimit of the above action of H on G × X. The obvious left action of G on G × X induces a left action of G on the colimit cor G H (X), so that cor G H (X) is a object in C G .
Then, for any two objects X in C H and Y in C G one has a bijection of sets natural in both arguments.
Proof. Use induction by i. The existence of γ 0 is equivalent to symmetrizibility of f . Assume γ i ′ exists for all i ′ < i and let's then deduce the existence of γ i . By induction hypothesis we have a commutative diagram Choose and fix any vertex in the complement First we take the canonical morphisms and consider their pushout-product diagram (see (♯)). By induction hypothesis, we have a Σ n -invariant cone-lifting Then the commutative diagram (♭) extends to a commutative diagram in which the dotted arrow exists because the square is commutative (because (♯) commutes), and it is unique as it is a universal morphism for the push-out square in the diagram (♭ ′ ). Now we consider the commutative square Hence, the group Σ n−i × Σ i acts also on the object V appearing in the diagram (♭). Actually, all the morphisms in the diagram (♭ ′ ), except for the morphism E → B, are Σ n−i × Σ i -invariant. It follows that all the morphisms in the diagram (♮), except for E → B, are Σ n−i × Σ i -invariant as well. Our nearest aim now is to build up a As C is monoidal model, the morphismṽ is a (trivial) cofibration too (see the definition of a monoidal model category in [9]). This gives a lifting ξ which is The next step would be spreading out ξ onto all the vertices in the complement L. Notice that coproduct of all objects from L, and this equality respects the action of the group Σ n . Let ξ ′ be a morphism matching the morphism ξ under the adjunction bijection . We identify ξ ′ with the corresponding cone based on objects from L (no arrows in the base). Now we observe that the objects from L are exactly the ends of the "crown" K n i (f, g), and the union of diagrams ) contains all the vertices of K n i (f, g) for which we have an arrow to Y n−i ∧Z i indecomposable into compositions of other arrows from K n i (f, g). It follows that precompositions of ξ ′ with such arrows give morphisms included into the cones coming from the previous step of iteration, because the morphism The proof of Proposition 5. As we have already seen above, in order to prove Proposition 5 all we need is to prove that the morphism n n−1 (gf ) → n n−1 (g) is a (trivial) cofibration. And to prove this we need only to construct a dotted cone-liftinĝ where both α and β are Σ n -invariant, and E → B is a (fibration) trivial fibration. Obviously, this follows from Lemma 13.

3.5.
Ordinals. Before to go any further we should recall some basics about ordinals, transfinite inductions and some other important things. Let X be an ordered set. This means that we have a relation ≤ on X and for any two elements x, y ∈ X either x ≤ y or y ≤ x. The set X is called to be well-ordered if for any non-empty subset Y ⊂ X there exists a minimal element in Y , i.e. there exists a ∈ Y such that for any y ∈ Y one has a ≤ y.
An isomorphism of two well-ordered sets is just a bijection between them respecting the orders. Then an ordinal is just an isomorphism class of wellordered sets. If α is an ordinal and X is a well-ordered set representing α, then we will write X ∈ α.
Notice that if X is a non-empty well-ordered set then X is a non-empty subset in itself, so that it has a minimal element which is normally denoted by 0. If a ∈ X then let where b < a means that b ≤ a and b = a. The set [0, a) is called a semiinterval in X determined by the element a. In the set theory it is proven the following result: Proposition 14. For any two well-ordered sets X and Y exactly one from the following three alternatives holds true: Proof. See [30].
Proposition 14 allows to make an order on the class of all ordinals Ord . Namely, given two ordinals α and β, we say that α < β if there exist a representative X ∈ α and an element b ∈ Y , such that X ∼ = [0, b). If there exist representatives X ∈ α and Y ∈ β, such that X ∼ = Y , we say that α = β. The ordinal represented by the empty set ∅ is the minimal element 0 in Ord .
Another important result coming from set theory is the following one.
Proposition 15. Let α be an ordinal. Then the set of all ordinals β < α forms a well-ordered set with regards to the above order in Ord representing α.
Notice that any ordinal α is the ordinal of all ordinals strictly smaller than α.
Any ordinal α is obviously a category which will be denoted by the same letter α. Its objects are all ordinals β < α, and for any two objects β and β ′ the Hom-set from β to β ′ consists of ≤ if β ≤ β ′ , and it is an empty-set otherwise.
Let α be an ordinal. Its successor, denoted by α + 1, is the ordinal of the disjoint union of any representative X ∈ α and a separate single element * which is bigger than any element in X. Then α + 1 is the minimal element in the set of all ordinals β > α.
As Ord contains 0, then it also contains 0 + 1 = 1, 1 + 1 = 2, and so on. Thus, Ord contains the set of natural numbers N.
An ordinal α is a limit ordinal if it is neither zero nor a successor ordinal. This is equivalent to say that α = colim β<α β in the category Ord .
Let us recall also the mechanism of transfinite induction which will be used below for several times. Let P (α) be a property of ordinals α. Assume P (β) holds true for all β < α. Then P (α) is true as well.
Notice that formally the base of induction P (0) is always true as anything for the empty-set is true. But the substantial thing is the following one. In practice the inductive step has to be proved for two cases separately -when α is a successor for some smaller ordinal β, i.e. α = β + 1, and when α is a limit ordinal, i.e. α = colim β<α β. Clearly, these two cases are alternative. In particular, checking the things in the successor case we have to check P (1) as 1 = 0 + 1. So, the real base of induction is to prove P (1), not P (0).
Let λ be an ordinal and let be a functor from λ to C preserving colimits (although λ is not necessarily cocomplete). For any ordinal α < λ let X α be the object X(α). Since the set of objects in λ has the minimal object 0, we have the canonical morphism which is called a transfinite compositions induced by the functor X. If I is a set of morphisms in the category C and X(α ≤ α + 1) ∈ I for any ordinal α such that α + 1 is in λ, then we say that X 0 → colim X is a transfinite composition of morphisms from I. For example, if I consists of symmetrizable morphisms (say, symmetrizable cofibrations or trivial cofibrations) then it makes sense to speak about a transfinite composition of symmetrizable morphisms in C . Now all the terminology appearing in Proposition 6 has been explained and we are going to prove it in the next section.

Lemma 16. For any two functors
Proof. Indeed, as the monoidal product ∧ in C is closed, smashing with an object commutes with colimits. This is why Since all arrows in the diagram A ∧ B are targeted to the diagonal objects A α ∧ B α , the last colimit is the colimit of the diagonal objects Lemma 17. Let λ be an ordinal, X : λ → C a colimit-preserving functor, and let X 0 → Y = colim X be the corresponding transfinite composition. For any ordinal α < λ let g α = X(0 ≤ α) be a morphism from X 0 to X α . Then: Respectively,˜ n n−1 (f ) = colim α<λ˜ n n−1 (g α ) , Sym n Y = colim α<λ Sym n X α , and the square˜ n n−1 (g α ) Proof. By Lemma 16, where the colimit is taken in the category of functors from subcategories in I n to C . It implies the following computation: Y n = n n (f ) = colim α<λ n n (g α ) = colim α<λ X n α , and both isomorphisms are connected by the corresponding commutative square. Passing to the coequalizers of the actions of the symmetric group Σ n completes the proof of the lemma.
To prove that f : X 0 → Y is symmetrizable we need to construct a dotted lifting˜ for a (fibration) trivial fibration E → B. Lemma 17 says that to construct γ it is enough to prove the following proposition.

Lemma 18. For any ordinal α < λ there exists a dotted lifting
Let α < λ be an ordinal, and consider the following two assertions: Q(α) : for any ordinal β < α there exists a lifting n n−1 (g β ) commutes. Later on we will show (applying transfinite induction once again) that Lemma 18 follows actually from Lemma 19. For any ordinal α < λ, the assertion Q(α) implies the assertion R(α).
Proof. If the ordinal α is a limit ordinal, i.e. α = colim β<α β, then Sym n X α = Sym n (colim β<α X β ), because X is a colimit-preserving functor. By Lemma 17, colim β<α Sym n X β , so we set Let now α be a successor ordinal, i.e. α = α ′ + 1 for an ordinal α ′ . To go any further we need to recall some episodes from the proof of Proposition 5 which says that for any subsequent symmetrizable morphisms X there exists a dotted lifting γ. But in fact the proof gives more: the lifting γ is compatible with the lifting As α ′ is the biggest ordinal amongst all ordinals β < α, one has Since X is a colimit-preserving functor, Then X(0 < α ′ ) : X 0 → X α ′ is a transfinite composition of the restriction of X on the ordinal α. As α < λ, by the transfinite induction hypothesis the morphism X(0 < α ′ ) is a symmetrizable (trivial) cofibration. The morphism X(α ′ < α ′ + 1) is a symmetrizable (trivial) cofibration by assumption. Then the resulting composition X(0 < α) : X 0 → X α is a symmetrizable (trivial) cofibration by Proposition 5. If γ α ′ is a lifting for X(0 < α ′ ) : X 0 → X α ′ , then there exists a lifting γ α for X(0 < α) : X 0 → X α such that γ α is compatible with γ α ′ in the above sense. But α ′ is the biggest ordinal among all ordinals β < α. It follows that γ α is compatible with all liftings γ β for all β < α.
Thus, in order to complete the proof of Proposition 6 it remains to show that Lemma 19 implies Lemma 18. This will be done immediately by applying transfinite induction once again. Actually, all we will do right now is just an adaptation of what is written on the pages 70 -72 of the book [30] to our setting. For any ordinal α < λ a system of liftings over all β < α (respectively, β ≤ α) will be called compatible if for any two The overall collection of all such fixed liftings gives a rule F assigning to any pair (α, {γ β } β<α ) the corresponding fixed lifting which can be now denoted by . In order to make things parallel to the reasoning in [30] for any ordinal α < λ a compatible system of liftings {γ β } β≤α will be called F -correct if for any ordinal β ≤ α we have that Now, for any ordinal α < λ we introduce the following property We are going to prove S(α) for all α < λ by transfinite induction. The assertion S(1) (when the ordinal is 1 = 0 + 1) is just a choice of a lifting Assume S(β) holds true for all β < α. Let us show that it follows S(α).
It is important to emphasize that for any β ′ < β < α the restriction of an F -correct system of liftings Applying this argument we can do now the step of transfinite induction. Indeed, by induction hypothesis, for any ordinal β < α there exists a unique F -correct system of liftings {γ β ′ } β ′ ≤β . Since all liftings in the rule F were chosen and fixed, once and forever, we obtain that there exists a unique lifting γ α such that γ α = F (α, {γ β } β<α ). Then the system {γ β } β≤α is Fcorrect and unique. Thus, the inductive proof of S(α) is completed. Now, unifying all F -correct systems of liftings and using the uniqueness of F -correct systems, we obtain a compatible system of liftings {γ α } α≤λ , i.e. Lemma 18. This completes the proof of Proposition 6.

3.7.
Retracts. The proof of Proposition 7 is fairly easy. Morphisms˜ n n−1 (f ) → Sym n Y are functorial in f . It follows that if f : X → Y is a retract of f ′ : X ′ → Y ′ then˜ n n−1 (f ) → Sym n Y is a retract of˜ n n−1 (f ′ ) → Sym n Y ′ . As retract of a (trivial) cofibration is a (trivial) cofibration, all is done.
3.8. Cells. Recall that for any set X its cardinality |X| is the smallest ordinal α for which there is a bijection between X and a representative in α. A cardinal is an ordinal α, such that α = |X| for some representative X ∈ α. Speaking less sophisticated: an ordinal α is a cardinal if it is the smallest among all ordinals which are "in bijection" with α.
Let κ be a cardinal. An ordinal α is said to be κ-filtered if α is a limit ordinal and for every subset A ⊂ α with |A| ≤ κ, we have sup A < α. Here we consider α as a set of all smaller ordinals, thus A is a set of some ordinals smaller than α, and sup A is the minimal ordinal among all ordinals β such that β ≥ γ for any γ ∈ A. In other words, a limit ordinal α is κ-filtered if α can not be represented as a colimit of a κ-sequence of smaller ordinals.
Let I be a set of morphisms in C , where C is a cocomplete category. Let κ be a cardinal, and let X be an object in C . Then X is said to be κsmall relative to I if for any κ-filtered ordinal λ and any colimit-preserving functor X : λ → C , such that X(α < α + 1) ∈ I for any ordinal α with α + 1 < λ the natural morphism of sets is a bijection. We say that X is small relative to I if there exists at least one cardinal κ, such that X is κ-small relative to I. Finally, X is small if X is small relative to the class of all morphisms in the category C . Now let C be a model category, and let again I be a class of morphisms in C . Recall that I-inj is a class of all I-injective morphisms in C , i.e. morphisms having the right lifting property with respect to every morphisms from I. Dually, I-proj is a class of all I-projective morphisms in C , i.e. morphisms having the left lifting property with respect to every morphisms from I. Let also I-cof be the class (I-inj)-proj in C . For example, if I is the class of all cofibrations in C then I-inj is the class of trivial fibrations and I-cof is I.
A relative I-cell complex is a morphism f : U → V in C which is a transfinite compositions of push-outs of morphisms taken from I. This means that there exists an ordinal and colimit-preserving functor X : λ → C , such that the corresponding transfinite composition X 0 → colim X is f : U → V , and for any ordinal α such that α + 1 < λ the morphism X(α < α + 1) is a push-out of a morphism from I. Let I-cell be the set of all relative I-cell complexes. Always I-cell ⊂ I-cof.
An object X in C is called an I-cell complex if the morphism ∅ → X is a relative I-cell complex, where ∅ denotes the initial object in C . Now we are ready to prove the following important technical result: Proof. By the assumption of the proposition, and by Proposition 4, any push-out of a morphism from I is symmetrizable. Then any I-cell complex is symmetrizable by Proposition 6. Finally, by Proposition 7, any retract of a relative I-cell complex is symmetrizable. The proof for symmetrizibility of relative J-cell complexes is similar.
Now assume that C is a model category, I and J are two classes of morphisms in C . The category C is said to be cofibrantly generated if the following four conditions hold true: In a cofibrantly generated model category C the class of cofibrations coincides with the class I-cof, and every cofibration is a retract of a relative I-cell complex, see Proposition 2.1.18 in [9]. Proof. Combine Proposition 20 with the fact that any cofibration in C is a retract of a relative I-cell complex and, similarly, any trivial cofibration in C is a retract of a relative J-cell complex in C (see [9, 2.1.18]).

Corollary 22.
Let C be a closed symmetric monoidal model category cofibrantly generated by a class of generating cofibrations I and a class of trivial cofibrations J. Suppose I is symmetrizable. Then any symmetric power Sym n X of a cofibrant object X in C is cofibrant.

3.9.
Pointed v.s. unpointed. Let C be a monoidal category with terminal object * , which we also refer as a point. Recall that C is pointed if the unique morphism ∅ → * is an isomorphism. Let C * = * ↓ C be the category under the point * . Then C is obviously pointed, and we have the usual functor C −→ C * sending an object X to and the same on morphisms. That functor is, of course, left adjoint to the forgetful functor from C * to C . If C is pointed itself then C → C * is an honest isomorphism of categories. Let now C be a closed symmetric monoidal model category with the monoidal product ⊗ : C × C → C and the unite ½. The model structure on C inherits a model structure on C * . Moreover, if the point * is cofibrant in C , there is a standard way to inherit the monoidal structure described in [9,Prop. 4.2.9]. Namely, if * x −→ X and * y −→ Y are two objects in C * then the "pointed" monoidal product X ∧ Y can be defined as the colimit of the diagram On morphisms the product ∧ can be defined in the same way. The monoidal structure ∧ : C * × C * −→ C * is closed symmetric if so is the product ⊗ on C . If the original category C is pointed, then ∧ = ⊗. It is proved in loc.cit. that C * is also a closed symmetric monoidal model category. Moreover, if C is a simplicial category, i.e. a (left) module over the category of simplicial sets △ op Sets, then C * is also simplicial being a (left) module over the category of pointed simplicial sets △ op Sets * .
Since the monoidal structures in the categories C and C * are different, symmetrizibility of morphism also is different. However, one has the following result.
Lemma 23. Let C be a closed symmetric monoidal model category and let f : X → Y be a morphism in the category C * . Suppose f is a symmetrizable (trivial) cofibration as a morphism in C . Then f is a symmetrizable (trivial) cofibration as a morphism in C * .
Proof. Let K n i (f ) ⊗ , (Sym n X) ⊗ , . . . be the diagrams and objects in C defined via the monoidal product ⊗, and let K n i (f ) ∧ , (Sym n X) ∧ , . . . be the corresponding diagrams and objects in C * defined via the monoidal product ⊗. We need to show that for any n ≥ 0, the morphism be an n-dimensional cube, where * is considered as a constant functor from I to C sending objects from I to the terminal object * . Let G be an n-dimensional cube defined by the formula Since the objects X and Y are pointed, there is a canonical morphism of diagrams * → K(f ) ∧ in C * , hence there is a canonical morphism of diagrams G → K n (f ) ⊗ in C . It follows from the definition of the wedge product in C * that the diagram is a push-out. Denote by G n−1 the restriction of the functor G to the full subcategory I n n−1 in I n . It is easy to see that Therefore in the diagram the left square and the whole square are push-outs. Hence the right square is a push-out. Passing to colimits over the action of Σ n , we obtain a pushout square˜ Since f is a symmetrizable (trivial) cofibration as a morphism in the category C , the top horizontal morphism is a (trivial) cofibration. Then the bottom horizontal morphism is a (trivial) cofibration.
The moral is that if we need to verify symmetrizibility of (trivial) cofibrations in C * then it is enough to do that for C .

Symmetric powers in homotopy categories
In this section we will prove the central result in the paper -the existence of a Künneth tower for symmetric powers of vertices in cofiber sequences in C . Throughout C stands for a pointed closed symmetric monoidal model category, and we assume that C is cofibrantly generated with symmetrizable generating cofibrations. Notice that the below Theorem 24 and Theorem 33 have analogs for unpointed categories, but for simplicity we formulate them only in the pointed case.
with (fibration) trivial fibration E → B the dotted arrow does exist leaving the diagram to be commutative. This is equivalent to construct a dotted Σ n -invariant lifting For one's turn, this is the same as to construct a dotted Σ n -invariant conelifting Choose and fix any vertex from the complement Respectively, the restriction of K n (f ) on (00 . . . 0 This is the largest subdiagram in K n (f ) generated by all morphisms to the object X n−i ∧ Y i . Consider the morphism Passing to colimit we get a new morphism which we denote by the same letter Note that θ commutes with the action of the group Σ n−i × Σ i embedded into Σ n in the canonical way. Passing to colimit with regards to this action of Σ n−i × Σ i we obtain a morphism As X is cofibrant, its symmetric power Sym n−i X is cofibrant by Corollary 22. The morphism˜ is a (trivial) cofibration by assumption. Thenθ is a (trivial) cofibration since ∧ : C × C → C is a Quillen bifunctor by axioms of a monoidal model category, see [9, §4.2]. Hence, we have a dotted lifting ) . Then ξ ′ is a Σ n -invariant lifting. If L is the collection of the ends of the crown K n i (f ) then ξ ′ induces a morphism ξ ′ : L −→ E .
Precomposing ξ ′ with the morphisms from the diagram Applying cor to ξ ′′ and using Σ n -invariance of the given morphism K n i−1 (f ) → E we obtain a Σ n -invariant lifting γ making the diagram The formulas for the cones are now obvious.

4.2.
Left derived symmetric powers. Let C be a closed symmetric monoidal model category and let Ho(C ) be its homotopy category. We are now ready to give sufficient conditions when left derived symmetric powers exist in Ho(C ). Proof. Let f : X → Y be a trivial cofibration between cofibrant objects in C . The composition Sym n X =˜ n 0 (f ) →˜ n 1 (f ) → · · · →˜ n i (f ) → · · · →˜ n n (f ) = Sym n Y is nothing but the morphism Sym n f : Sym n X −→ Sym n Y , and it is a trivial cofibration by Theorem 24 applied together with Corollary 21. Thus, Sym n takes trivial cofibrations between cofibrant objects to weak equivalences. By Ken Brown's lemma, Sym n takes all weak equivalences between cofibrant objects to weak equivalences. Then left derived functors LSym n exist, see [9, 1.3.2].

4.3.
Künneth towers in triangulated categories. Let C be a pointed simplicial closed symmetric monoidal model category and let Σ S 1 be the S 1 -suspension functor Σ S 1 : C −→ C sending any object X to the product X ∧ S 1 , and the same on morphisms. It has a right adjoint loop-functor be the homotopy category of the category C . As C is a pointed simplicial monoidal model category the category T has a structure of a pretriangulated category, in the sense of [9, 6.5], with respect to the above suspension and desuspension functors Σ S 1 and Ω S 1 . The suspension Σ S 1 induces an endofunctor [1] : If it is an autoequivalence on T then T is triangulated in Hovey's sense, and so it is triangulated also in the classical sense, where distinguished triangles come from cofiber sequences in C , see [9, §7]. Since C is closed symmetric monoidal, so is the triangulated category T , and the functor C → T is monoidal as well, see [9, §4.3]. As T is additive, we will denote coproducts in it by ⊕ and the monoidal product in it by ⊗.
Then we can restated Theorem 24 in triangulated terms:

Theorem 26. Let T be a closed symmetric monoidal triangulated category, which is the homotopy category of a pointed simplicial closed symmetric monoidal model category C . Assume that C is cofibrantly generated by a symmetrizable class of generating cofibrations I and a symmetrizable class of generating trivial cofibrations J. Then for any distinguished triangle
in T and any natural number n we have two towers LSym n X =˜ n 0 (f ) →˜ n 1 (f ) → · · · →˜ n i (f ) → · · · →˜ n n (f ) = LSym n Y in T whose cones can be computed by Künneth's rules as , and, respectively, Proof. Using cofibrant replacement one can assume, without loss of generality, that f is a cofibration between cofibrant objects, and that Z is the cofibrant quotient Y /X, see [20], Lemma 5.3. By Theorem 25 the left derived functors of Sym n exist, and to compute LSym n we need only to use cofibrant replacements. As all three objects X, Y and Z are cofibrant, to get the theorem we apply Theorems 24 and 25, and rewrite everything in the notation of the triangulated category T .

Localization of symmetric powers
The above results give a suitable theory of symmetric powers in the ordinary homotopy category of topological spaces or simplicial sets. In order to extend it to spectra we need to stabilize symmetric powers. Stabilization is a particular case of the general notion of a localization of a model category with regard to a given set (class) of morphisms in it. Here we establish necessary and sufficient conditions for localization of symmetric powers, working in as much as possible general setup.

Localization theorem.
Recall the main result in [8]. Let C be a left proper cellular model category cofibrantly generated by a class of generating cofibrations I and a class of generating trivial cofibrations J. We will also assume that C is a simplicial closed symmetric monoidal model category with a unit ½. Let S be a set of morphisms in C . An object W in C is is a weak equivalence in △ op Sets for any S-local object W in C . Notice that a morphism f : X → Y is a weak equivalence in C if and only if map(f, W ), for each fibrant W , is a weak equivalence in △ op Sets. Therefore, each weak equivalence is an S-local equivalence in C . Assuming all the above, there exists a new left proper cellular model structure on C whose cofibrations remain unchanged and new weak equivalences are exactly S-local equivalences in C . The new model structure is cofibrantly generated by two classes I S and J S , where I S is the same as I, and J S consists of inclusions of I-cell subcomplexes which are S-local equivalences, see [8], Chapter 4. The new model category is then denoted by L S C . Notice that, actually, L S C is the same category, but the model structure is now new.
Moreover, as C is symmetric monoidal and simplicial, so is the category L S C . The model structure in L S C is compatible with the existing simplicial structure, [8], Theorem 4.1.1 (4). We need to impose an assumption on S to get a compatibility of the new model structure in L S C with the existing monoidal structures in it: ( ) for any cofibrant X in C and any morphism f in S the product is an S-local equivalence in L S C , where Q is the cofibrant replacement in C . Proof. Actually, the proof of the compatibility of the monoidal structure with the new model one is similar to the proof of Theorem 8.11 in [10]. All we need is to make the reasoning slightly more abstract and replace the main verification by the condition ( ), which is satisfied by assumption.
Since now we will assume that C satisfies ( ), so that L S C is again a simplicial model monoidal category (left proper, cellular, if necessary). In particular, we have functors Our main result in this chapter is the following theorem.

5.2.
Reductions. The proof of the above theorem is a bit long, so that we divide it into a number of reduction steps, and then aggregate all them together.
Step 1 Without loss of generality one may think that all morphisms f in S are cofibrations between cofibrant objects. Indeed, having f : A → B in S we can take its cofibrant replacement Qf : QA → QB and decompose Qf into cofibration QA → C and trivial fibration C → QB. Then we have to take into account that an adding of weak equivalences to S does not make any impact upon the localization, and that the morphism Sym n (C → QB) is a weak equivalence since J is symmetrizable, so that Sym n carries all weak equivalences between cofibrant objects into weak equivalences by Corollary 21 and the lemma of Ken Brown.
Step 2 Now we wish to reduce the assertion of the theorem to the case of an S-fibrant replacement R S X of a cofibrant object X in C . Let g : X −→ Y be an S-local trivial cofibration between cofibrant objects in C . Our aim is to show that Sym n (g) is an S-local equivalence. Let be the commutative square produced by the S-local fibrant replacement in C , i.e. the fibrant replacement in L S C . The vertical arrows are S-local equivalences and cofibrations between cofibrant objects in L S C .
It is important to observe that, since X and Y are cofibrant, R S X and R S Y are fibrant-cofibrant objects in L S C . Moreover, R S g is a weak equivalence in L S C . Since a weak equivalence between fibrant-cofibrant objects is the same as a homotopy equivalence, we see that Sym n (R S g) is a homotopy equivalence, and so a weak equivalence, by Lemma 2.
Therefore, the theorem will be proved as soon as we prove that Sym n (X → R S X) is an S-local equivalence for any cofibrant X in C .

Step 3
We also need yet another general lemma: Lemma 29. Let D be a closed symmetric monoidal model category, and let g be a trivial cofibration between cofibrant objects which is symmetrizable as a cofibration in D. Then g is a symmetrizable trivial cofibration if and only if Sym n (g) is a trivial cofibration for all n ≥ 0.

Proof. Consider the sequence of cofibrations
Sym n X =˜ n 0 (g) →˜ n 1 (g) → · · · →˜ n i (g) → · · · →˜ n n (g) = Sym n Y provided by Theorem 24. The composition of all the cofibrations in that chain is Sym n (g). If g is a symmetrizable trivial cofibration then each cofibration˜ n i (g) −→˜ n i+1 (g) is a trivial cofibration by Theorem 24. Then so is Sym n (g).
Conversely, suppose Sym n (g) is a trivial cofibration for any n ≥ 0. Let's prove by induction on n that the morphism˜ n n−1 (g) → Sym n Y is a trivial cofibration, i.e. that g is a symmetrizable trivial cofibration. The base of induction, n = 1, is obvious. To make the inductive step we observe that in proving Theorem 24 we deduce that˜ n i−1 (g) →˜ n i (g) is a trivial cofibration by only using that˜ i i−1 (g) → Sym i Y is a trivial cofibration for i < n. But the last condition holds by the induction hypothesis. Thus, all morphisms n i−1 (g) →˜ n i (g) are trivial cofibrations. Then˜ n n−1 (g) → Sym n Y is a weak equivalence by 2-out-of-3 property for weak equivalences. Finally, by the assumption of the lemma,˜ n n−1 (g) → Sym n Y is a cofibration, and so a trivial cofibration.
Corollary 30. Let D be as in Lemma 29. Let g be a symmetrizable cofibration between cofibrant objects in D, and let h be a push-out of g. Then, if Sym n (g) is a trivial cofibration in D for any n ≥ 0, the morphism Sym n (h) is also a trivial cofibration in D for any n ≥ 0.
Proof. Applying Lemma 29 to the morphism g in the category D we obtain that g is a symmetrizable cofibration because Sym n (g) is a trivial cofibration in D. As symmetrizability is stable under push-outs by Proposition 4, the morphism h is a symmetrizable cofibration in D. Applying Lemma 29 in the reverse direction we get that Sym n (h) is a trivial cofibration in D for all n.
Step 4 Now we need to construct a fibrant approximation X → R S X for any cofibrant X in C . For the sake of simplicity of notation we will assume that S consists of only one morphism The general case follows from the one-morphism localization case by taking coproducts of all morphisms in a given localizing set S. For this one uses that the category C is pointed. As we have already seen on Step 1, without loss of generality one may also think that f is a cofibration, and the objects A and B are cofibrant.
We will follow the reasoning and the notation of Chapter 1 in [3]. Namely, for any cofibrant object X in C there exist an ordinal λ and a functor Z : λ −→ C , satisfying the following properties. If β + 1 < λ then where L f is a certain functor, constructed below, equipped with a natural transformation j : Id −→ L f .
Passing to colimit we obtain the desired fibrant approximation This approximation will be functorial as soon as L f will be functorial, so that it will be a fibrant replacement in L S C . Notice that R S X is S-local because for sufficient large λ in a cellular model category every cofibrant object is small relatively to all cofibrations in it, see Theorem 12.4.3 in [8].
Since any weak equivalence is stable under transfinite compositions, and taking into account the reduction in Step 2, we see that in order to prove Theorem 28 we need only to prove that the morphism is an S-local equivalence.

5.3.
The functor L f . Thus, our aim now is to construct L f following Farjoun in [3].
Consider the following commutative diagram: where C is a push-out and, respectively, θ is a push-out product Let ev A,X : A ∧ map(A, X) −→ X be the evaluation morphisms coming from the adjunction in two variables between C and △ op Sets, and the same for B. Then let ev : C −→ X be a push-out product of ev A,X and ev B,X , i.e. ev is defined by the obvious commutative diagram In order to represent this homotopy colimit as an honest colimit we need to decompose the morphism θ into a cofibration η and a trivial fibration ρ in C , As η is a cofibration, so is its push-out X → E. As E → RE is a cofibration as well, the composition j is a cofibration.
One more general lemma: Lemma 31. Let D be a simplicial closed symmetric monoidal model category, and let g : X → Y be a cofibration between cofibrant objects in D such that Sym n (g) is a trivial cofibration for all n ≥ 0. Let K be an arbitrary simplicial set. Then Sym n (g ∧ K) is a weak equivalence in D for all n ≥ 0.
Proof. For simplicity, let's assume that K is a finitely generated simplicial set, i.e. there is a finite number of non-degenerate simplices in each dimension. The general case can be reduced to the finite one via transfinite induction.
We will do a double induction by the number of simplices and the dimension of K, i.e. the maximal dimension of non-degenerated simplices. First we represent K as a simplicial set obtained by gluing a simplex to another simplicial set K ′ having one simplex less than in K, i.e. there is a push-out square Smashing it with X and Y we get two push-out squares which can be connected by the morphism g smashed with ∂∆ m , ∆ m , K ′ and K. Consider also the following two push-out squares It is not hard to verify that the obvious commutative diagram is a push-out square. Notice that the morphisms Sym n (∂∆ m ∧ g) and Sym n (K ′ ∧ g) are weak equivalences by induction, and Sym n (∆ m ∧ g) is a weak equivalence by Lemma 2 because X and ∆ m ∧ X are homotopic. As Sym n (∂∆ m ∧ g) is a weak equivalence, then Sym n (∆ m ∧X → D) is a weak equivalence by Corollary 30. Besides, ∆ m ∧X and D are cofibrant. Since Sym n (∆ m ∧X → D) is a weak equivalence and Sym n (∆ m ∧ g) is a weak equivalence, so is the morphism Sym n (D → ∆ m ∧ Y ) by 2-out-of-3 property for weak equivalences. Then, again, Sym n (H → K ∧ Y ) is a weak equivalence by Corollary 30. Similarly, Sym n (K ∧Y → H) is a weak equivalence. Composing morphisms we see that Sym n (K ∧ X → K ∧ Y ) is a weak equivalence, as required. Now we are coming back to the diagram defining θ in the beginning of Subsection 5.3. By Lemma 31, applied in L S C , for any n the morphism Sym n (f ∧ id) is an S-local equivalence. By assumption of Theorem 28 this morphism is also a cofibration in C , so in L S C . By Corollary 30, is an S-local equivalence. Lemma 31 in L S C , also gives that the morphism is an S-local equivalence. By 2-out-of-3 property, the morphism is an S-local equivalence. Applying Ken Brown's lemma we obtain that Sym n (ρ) = Sym n (D → B ∧ map(A, X)) is a weak equivalence in C , so in L S C . Hence, by 2-out-of-3 property, Sym n (η) is an S-local equivalence. Then is an S-local equivalence by Corollary 30. Since E → RE is a weak equivalence between cofibrant objects in C , i.e. in the "old" model structure, it is a weak equivalence also in L S C . As a result, Sym n (j) = Sym n (X → E → RE) is an S-local equivalence. Theorem 28 is proved.
Notice that the above construction of R S X is functorial in X.

Some generalizations.
Let C be a simplicial closed symmetric monoidal left proper cellular model category, and let D be a model category. Let F : C −→ D be a functor, such that F respects trivial cofibrations between cofibrant objects, so that left derived LF exists. Let S be a set of morphisms in C satisfying the following (i) F (S) is in the class of weak equivalences in D; (ii) for any cofibration f between cofibrant objects in C , such that F (f ) is a weak equivalence in D, and any push-out g of f the morphism F (g) is again a weak equivalence in D; (iii) if f and g are two homotopy equivalent morphisms in C with the same domains and codomains, then F (f ) is homotopy equivalent to F (g).
Then F carries S-local equivalences between cofibrant objects to weak equivalences in D.
Clearly, the condition (ii) is nothing but a weakening of the property to be left Quillen functor.
Notice also that if the above three conditions are satisfied for the category C , then the second and the third conditions are satisfied also for the category L S C , because cofibrations and the notion of homotopy equivalence do not change when passing from C to L S C .

Symmetric powers in stable homotopy categories
In this section we start to apply the above gadgets to symmetric spectra and outline how to verify the symmetricity axioms for generating (trivial) cofibrations in the topological setting.
An abstract stable homotopy category, in our understanding, is the homotopy category of the category of symmetric spectra over a given simplicial model monoidal category C , stabilizing a smash-with-T functor for a certain cofibrant object T in C . Notice that the symmetricity of spectra is very essential here, as symmetric powers of ordinary non-symmetric spectra cannot be stabilzed.
6.1. Recollections on symmetric spectra. First we recall some basic milestones in constructing symmetric spectra, but it should be pointed out that the true understanding of this topic can be gained only by implementing a bulk of practical computations, including combinatorics.
Let C be a pointed closed symmetric monoidal model category which is, moreover, left proper and cellular model category. Recall that left properness means that the push-out 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 cofibraions are effective monomorphisms. Further details about the notions engaged here can be found in [9], [10] or [8].
Let now T be a cofibrant object in C . As it was shown in [10], with the above collection of structures imposed upon C there is a nice passage from C to a category S = Spt Σ (C , T ) of symmetric spectra over C stabilizing the functor Let us remind the basics of this construction for reader's sake. Let Σ be the coproduct or the category whose objects are non-negative numbers and morphisms are defined by the formula Let C Σ be the category of symmetric sequences over C . Since C is closed symmetric monoidal, so is the category C Σ with the monoidal product given by the formula , and obvious action of Σ n , see [12] or [10].
Let S(T ) be the free monoid on the symmetric sequence (∅, T, ∅, ∅, . . . ), i.e. the symmetric sequence where T 0 = ½ is the unit, T 1 = T and Σ n acts on T n by permutation of factors. The whole point is that the monoid S(T ) is commutative. Then S is nothing but the subcategory of modules over S(T ) in C Σ . In particular, any symmetric spectrum X is a sequence of objects X 0 , X 1 , X 2 , X 3 , . . .
in C together with Σ n -equivariant morphisms such that for all n, i ≥ 0 the composite Since now we will systematically apply the powerful machine developed in [10] in order to construct and to work with the model structure on S and the corresponding homotopy category It is important to emphasize that this machine is fairly universal and so applicable to both topological and motivic settings.
The model structure on S has been constructed in two steps -projective model structure coming from the model structure on C and its subsequent Bousfield localization using the main result of [8]. For any non-negative n consider the evaluation functor Ev n : S −→ C sending any symmetric X to its n-slice. The point is that each Ev n has a left adjoint F n : C −→ S , which is constructed as follows. First we define a naive functorF n taking any object X in C into the symmetric sequence (∅, . . . , ∅, Σ n × X, ∅, ∅, . . . ) .
On the second stage we set F n X =F n X ∧ S(T ) , see [10,Def.7.3].
Let now I T = ∪ n≥0 F n I and where F n I is the set of all the morphisms of type F n f , f ∈ I, and the same for F n J.
Recall that we are assuming the model structure in C is left proper and cellular. Provided these two assumptions, the projective model structure in S is left proper and cellular too and, moreover, I T and J T are the classes of generating cofibrations and trivial cofibrations for the projective model structure in S , see the Appendix in [10]. In particular, the class of cofibrations with regard to the projective model structure, is equal to the class I T -cof.
Let n and m be two non-negative integers, m ≥ n, and let X be an object in C . The group Σ m−n is canonically embedded into the group Σ n and acts on X ∧ T m−n permuting factors in T m−n . Then , see [10, §7]. In particular, induced by the canonical embedding of Σ 1 into Σ n+1 . Let S stab be the class of morphisms ζ QX n , where X runs trough domains and codomains of the generating cofibrations in C and Q is a cofibrant replacement in C . Then the stable model structure in S is defined to be the Bousfield localization of the projective model structure with respect to the class S stab . The importance of the stable model structure is that the functor − ∧ T is a Quillen autoequivalence of S with respect to this model structure.
6.2. Künneth towers for symmetric spectra. For the purposes of constructing Künneth towers in the model category S it is important and enough that the cofibrations in the stable model structure in S are the same as cofibrations in the projective model structure. Thus, Suppose in addition that the domains of morphisms in I are cofibrant. Then it follows from [10,Theorem 8.11] that the monoidal structure on C makes S into a closed symmetric monoidal model category.
In Section 3.2 we defined what is a symmetrizable cofibration. Working with symmetric spectra it is necessary to require more. A cofibration f : X → Y in C is called to be symmetrizable if the morphism˜ n n−1 (f ) → Sym n Y is a cofibration for all non-negative integers n. It implies that Sym n (f ) is a cofibration, Theorem 24. This is equivalent to say that there exists a lifting h, B for any (fibration) trivial fibration E → B in C , such that h is invariant under the action of Σ n on X ∧n . For any morphism g : A → B it is Σ nequivariant if Σ n acts on A and B, and g commutes with these actions. We will say that f is strongly symmetrizable if, for each n, there exists a Σ n -equivariant lifting h for any Σ n -equivariant (fibration) trivial fibration E → B in C .
Proposition 32. Assume that all morphisms in I (in J) are strongly symmetrizable. Then all morphisms in I T (in J T ) are symmetrizable.
Proof. To prove this proposition we need some more preparation. Let A be an object in C , and let m, p and m are non-negative integers, such that m ≥ pn .
Then the m-slice of the monoidal power F n (A) ∧p is where Σ m−pn permutes T ∧(m−pn) leaving A ∧p untouched, and Σ m−pn is embedded into Σ m as permutations permuting the last m − pn elements in the set {1, 2, . . . , m}.
The m-th slice of the symmetric spectrum Sym p F n (A) is where Σ m−pn permutes T ∧(m−pn) leaving A ∧p untouched, and Σ m−pn is embedded into Σ m as permutations permuting the last m−pn elements in Σ m , as above, but now we have some extra-action of Σ p . Namely, elements in Σ p permute A ∧p leaving the object T ∧(m−pn) untouched, and Σ p is embedded into Σ m as p-shuffle permuting first p blocks, in n elements each one, in the set {1, 2, . . . , m}. Now see that the m-slice of the canonical morphism into the colimit, is nothing but the quotient morphism This morphism can be also understood as a "factorization by a large subgroup map".
Let now f : X → Y be a strongly symmetrizable generating (trivial) cofibration in C . We need to show that the morphism is a (trivial) cofibration. Consider the following commutative diagram: Here E → B is a (fibration) trivial fibration, and the lifting h 0 is for free, as F n (f ) ∧p is a (trivial) cofibration by the fact that the category of symmetric spectra is model monoidal. All we need is to construct the missing lifting h.
For simplicity, let's consider the case p = 2. By the nature of F n , in order to construct a morphism from F n (A) to a symmetric spectra, for any object A, it is enough to construct this morphism on the first n-th slice. In our case, if p = 2, then m = 2n. In other words, it is enough to construct a lifting is just a coproduct of (2n)! copies of Y ∧ Y , and the action of Σ 2 is given by permuting Y in Y ∧ Y and 2-shuffle of the two blocks {1, . . . , n} and {2n + 1, . . . , 2n} in {1, . . . , 2n}, it is enough to construct a Σ 2 -equivariant lifting where the transposition of two elements, i.e. the only non-trivial element in Σ 2 , is embedded into Σ 2n as a 2-shuffle of two blocks in n elements each one. But this is provided by the strong symmetrizability of f . Now it is easy to prove the following result: Theorem 33. Let C be a pointed closed symmetric monoidal model category which is, moreover, left proper and cellular model category with generating class of cofibrations I such that the domains of morphisms in I are cofibrant. Let furthermore T be a cofibrant object in C and let S be the category of symmetric spectra over C with stabilized functor − ∧ T : C ∧ C → C , i.e. let S = Spt Σ (C , T ). Assume that all morphisms in I are strongly symmetrizable. Then for any two cofibrant symmetric spectra X and Y , any stable cofibration f : X −→ Y between them and any natural numbers n and i, such that i ≤ n, we have that the morphisms Proof. Combine Theorem 24 with Proposition 32. 6.3. Künneth towers in abstract stable homotopy categories. By an abstract stable homotopy category we would mean the homotopy category T of the category of symmetric spectra S = Spt Σ (C , T ) over a pointed simplicial closed symmetric monoidal model category C , which is left proper and cellular with generating class of symmetrizable cofibrations I, whose domains are cofibrant, stabilizing the multiplication by a cofibrant object T in it, such that the stable S 1 -suspension functor Σ S 1 : S → S induces an autoequivalence on T . In particular, T is triangulated closed monoidal category.
Notice that by Hovey's result, see [10], the homotopy category T = Ho(S ) is equivalent to the homotopy category of ordinary T -spectra provided the cyclic permutation on T ∧ T ∧ T is left homotopic to the identity morphism.
We expect that under some mild assumptions on C and T symmetric powers can be stabilized, i.e. there exist left derived functors LSym n in the homotopy category T . This is because the conditions of Theorem 28 are satisfied for the set S stab . Indeed, since any morphism ζ X n : F n+1 (X ∧ T ) −→ F n (X) is a product of the morphism with the identity morphism id Fn(X) , for a cofibrant X, it essential to analyze symmetric powers of the morphism ζ ½ 0 . Then, for any distinguished triangle in T and any natural n one would have two towers LSym n X =˜ n 0 (f ) →˜ n 1 (f ) → · · · →˜ n i (f ) → · · · →˜ n n (f ) = LSym n Y in T whose cones could be computed by Künneth's rule:

Possible applications
In this section we outline possible applications of the above theorems. The detailed account will be given in forthcoming papers. 7.1. Additivity of powers in distinguished triangles. Let T be a closed symmetric monoidal triangulated category, which is the homotopy category of a pointed simplicial closed symmetric monoidal model category C cofibrantly generated with symmetrizable generating cofibrations and trivial cofibrations. Then we have left derived symmetric powers and Künneth towers in T . The first natural question would be how to construct Koszul dual functors, i.e. alternating powers for objects in the triangulated category T . The idea is to use the simplicial suspension on the model level and, in a parallel way, the shift-functor in the triangulated category. This is suggested by the Künneth tower for the distinguished triangle where X is a object in T . Doing things in this way one can define left and right alternating powers by the formulas The following result is a direct consequence of Theorem 26, and is an integral version of the result obtained in [5].
Theorem 34. Let T be a closed symmetric monoidal triangulated category, which is the homotopy category of a pointed simplicial closed symmetric monoidal model category C , and C is cofibrantly generated with symmetrizable classes of generating cofibrations and trivial cofibrations. Then for any distinguished triangle the dual object to X. Then X is said to be dualizable, [20], if there exists corresponds, by adjunction, to the identity morphism on X (see also [17] and [11], where such objects are called finite and strongly dualizable, respectively). Notice that X is dualizable if and only if the natural morphism is an isomorphism for any object Y in T , see [17]. Let T ′ be the full subcategory generated by dualizable objects in T . It is shown in [17] that T is a monoidal subcategory. If T is, moreover, triangulated and the triangulation is compatible with the closed symmetric monoidal structure, [11], [21], then T ′ is thick triangulated and monoidal subcategory in T , and we can consider the Grothendieck group K 0 (T ′ ) of T ′ . Also we have the Euler characteristic homomorphism due to May, see [20] and [21]. Now suppose T is a closed symmetric monoidal triangulated category, which is the homotopy category of a pointed simplicial closed symmetric monoidal model category C , as in Theorem 26, so that Künneth towers exist. Let T ′′ be the full subcategory in T generated by all objects X such that for all n ≥ 0 the symmetric power LSym n X is in T ′ , i.e. is a dualizable object in T . By definition, T ′′ is a subcategory in T ′ . Using Künneth towers and monoidality of T ′ one can show that T ′′ is a triangulated subcategory in T ′ . It is an interesting question could, in general, T ′′ be different from T ′′ , i.e. can a symmetric power of a dualizable object be not dualizable?
We have the Euler characteristic homomorphism from K 0 (T ′′ ) to the same group End T (½). Let K 0 (T ′ )[[t]] be the ring of formal power series in variable t with coefficients in K 0 (T ′ ), and let K 0 (T ′ )[[t]] * be the multiplicative group of invertible elements in this ring. To each object X in T ′′ we can now associate its zeta-function Theorem 35. Let T be a closed symmetric monoidal triangulated category, which is the homotopy category of a pointed simplicial closed symmetric monoidal model category C satisfying the condition of Theorem 26. Then for any distinguished triangle In particular, rationality of zeta-functions is a 2-out-of-3 property in distinguished triangles in the category T ′′ , and Proof. Use Künneth towers and apply the arguments used in [5].
In other words, the zeta-function ζ X (t) depends only on the class of X in K 0 (T ′′ ) and defines a group homomorphism Notice that the reason to consider the categories T ′ and T ′′ is that K 0 (T ) = 0 because T has infinite coproducts, while K 0 (T ′ ) has a nontrivial Euler characteristic homomorphism.
A slightly different approach would be as follows. Suppose there exists another subcategory T ′′ in T ′ , such that for any natural n and any object X in T ′′ the symmetric power LSym n X is again in T ′′ . The arguments used in [6] show that the existence of Künneth towers implies that symmetric powers in T induce a λ-structure σ n : K 0 (T ′′ ) −→ K 0 (T ′′ ) , n = 0, 1, 2, . . . , for any object X in T ′′ . Notice that LAlt n L and LAlt n R coincide on K 0 (T ′′ ) bringing a λ-structure dual to σ * . The most reasonable candidate for T ′′ is the subcategory of small objects in T , see [11,Theorem 2.1.3 (c)], or a subcategory of cell-complexes with appropriate collection of cells having the property that any symmetric power of a cell-complex is again a cellcomplex.
7.3. Topological spectra. Let C be the category △ op Sets * of pointed simplicial sets. Then C is a pointed simplicial closed symmetric monoidal model category, which is left proper and cellular, and whose cofibrations are monomorphisms of simplicial sets. Let S be the category of symmetric spectra over C and let T be the homotopy category of S . Then T is nothing but the ordinary topological stable homotopy category SH, see [12]. Let us show that all (trivial) cofibrations in △ op Sets * are strongly symmetrizable.
First we recall that the monoidal product in △ op Sets * is constructed by the formula where X × Y is a product of (unpointed) simplicial sets and X ∨ Y is the bouquet of X and Y , i.e. the coproduct of X and Y in the pointed sense. Let Sets be the category of sets and let Sets * be the category of pointed sets. Notice that Sets * is closed symmetric monoidal category with analogously defined monoidal product. The category △ op Sets * can be considered as the category of presheaves on the simplicial category △ with values in the category Sets * . Therefore, all we need is to prove symmetrizibility of monomorphisms section-wise. But this can be shown by a straightforward set-theoretical verification.
As to trivial cofibrations, we notice that taking symmetric powers commutes with the geometric realization functor. This is why symmetric powers preserve weak equivalences between simplicial sets, because for CWcomplexes it is true by Lemma 2.
Moreover, since for any CW -complex X there exists a Σ n -equivariant cell-decomposition of X ∧n we see that the symmetrizability of (trivial) cofibrations is strong.
Provided symmetric powers of ζ ½ 0 are stable equivalences, we have left derived symmetric powers and Künneth towers for distinguished triangles in the topological stable homotopy category SH. Notice that in the topological stable homotopy category an object X is dualizable if and only if it is a finite spectrum, i.e. a spectrum built up by a finite collection of cells, because the category T is unital algebraic, see [11]. More explicitly, an object X in T is finite if and only if there is a finite cellular Postnikov tower is a finite direct sum of copies of ½[i]. Recall that ½[i] is the topological symmetric spectrum of the pointed sphere S i for i ≥ 0. It is well-known that (pointed) symmetric powers of spheres are finite CW-complexes. Thus, T ′′ = T ′ . Moreover, it follows from the definition of a finite spectrum that [½] is a generator in K 0 (T ′ ), χ(½) = 1 ∈ Z = End T (½), so that the Euler characteristic is an isomorphism. Furthermore, LSym n ½ = ½ for all n ≥ 0. Hence, Actually, this holds in any closed symmetric monoidal model category such that the permutation of factors ½ ∧ ½ → ½ ∧ ½ is the identity.
By Theorem 35, ζ X (t) depends only on the class [X] in group K 0 (T ′′ ) = K 0 (T ′ ) = Z, which is equal to the Euler characteristic χ(X) of X. Since ζ ½ = (1 − t) −1 and χ(½) = 1, we get, for a finite topological spectrum X, This is a kind of stable homotopy generalization of MacDonald's theorem saying that the zeta-function of a smooth manifold is rational, see [18].
Let us see how all this works on naive examples. Take any spectrum X in T and consider a distinguished triangle The corresponding Künneth tower for n = 2 consists of two morphisms LSym 2 X =˜ 2 0 −→˜ 2 1 −→˜ 2 2 = LSym 2 (0) = 0 , and we have two distinguished triangles . These two triangles give a distinguished triangle where the last morphism is nothing but the built-over canonical morphism from X ⊗ X onto LSym 2 X.
If now X = ½ is the unite in T then LSym 2 ½ = ½ and we obtain a of presheaves of sets on the Cartesian product of two categories (Sm/k)×△. It is well-known that C and, respectively, C * have several appropriate model structures such that the resulting homotopy category T = Ho(S ) of the category of symmetric spectra where T = S 1 ∧ G m , is the Morel-Voevodsky motivic stable homotopy category, which we denoted here by SH(S) , see [28] and [14]. Our aim is to pick up one, such that it would be cofibrantly generated by symmetrizable cofibrations, so that one could apply the theorems above. Now we are going to recall some necessary facts on the motivic model structure in C constructed by Jardine in [14] (see also [13]). For further details we refer to the systematic exposition in [24].
For any non-negative integer n let △ n = Hom △ (−, [n]) be the n-th simplex. If X is a smooth scheme over the base S let △ n X be a presheaf on Sm/S × △ sending any pair (U, [m]) to the Cartesian product of sets Hom Sm/S (U, X) × Hom △ ([m], [n]). Then we get a fully faithful embedding of the category of smooth schemes over S into the category of simplicial presheaves, P : Sm/S −→ C , by sending X to △ 0 X , and the same on morphisms.
Let now K be a simplicial set, i.e. a presheaf of sets on the simplicial category △. It induces another presheaf on Sm/S × △ by ignoring schemes and sending a pair (U, m) to the value K m of the functor K on the object [m] in △. This gives a functor △ op Sets −→ C , which allows to define a simplicial structure on the category C .
Let f : X → Y be a morphism in C . Following Jardine, [13], we say that f 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 topology on the category Sm/S. Let I be a class of monomorphisms of type X ֒→ △ n U for some simplicial presheaf X, smooth S-scheme U and n ≥ 0, and let J be the class of monomorphisms which are weak equivalences. Notice that in spite of that we are talking about presheaves, the Nisnevich topology is engaged seriously into the above model structure on C .
The 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]).
Then C together with the above weak equivalences and monomorphisms taken as cofibrations is a simplicial left proper and cellular closed symmetric monoidal model category cofibrantly generated by the class of generating cofibrations I and the class 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]. Cofibrations in this model structure are exactly section-wise monomorphisms of simplicial presheaves.
Such constructed model structure is called injective. It satisfies the assumptions of the main result in [8] which admits a Bousfield localization of the injective model structure by contraction of the affine line A 1 into a point. The localized model structure is called motivic and it is again simplicial left proper cellular closed symmetric monoidal model cofibrantly generated by a new localized class of generating cofibrations I ′ and a new localized class of generating trivial cofibrations J ′ .
The category C * inherits all the structures in C . Then the category S = Spt Σ (C * , T ) has a structure of a simplicial closed symmetric monoidal model category by Hovey's result, [10]. Moreover, the simplicial suspension Σ S 1 induces an autoequivalence in its homotopy category SH(S), so that SH(S) is a triangulated category.
The point is now that I ′ = I, i.e. the generating cofibrations in the localized motivic model structure remain to be the same usual monomorphisms of simplicial presheaves of sets. It is then easy to verify section-wise that all monomorphisms are symmetrizable, just as in the case of simplicial sets, and the generating cofibrations in C * are all symmetrizable by Lemma 23. Therefore, the categories C and C * have Künneth towers by Theorem 24, the category of motivic symmetric spectra S has Künneth towers by Theorem 33. However, we don't yet know whether the motivic stable homotopy category SH(S) has symmetric and alternating powers endowed with Künneth towers by Theorem 26. This is because we do not know the behaviour of symmetric powers under A 1 -localization and we do not understand the difference between E(Sym n X) and LSym n (E(X)) for an algebraic variety X over a field.

Open questions
In this last section we collect some naturally arising questions whose study was not included into the paper. We expect that possible answers will be given using appropriate Künneth towers in this or that way. The below reflections are made modulo assumption that there exist left derived functors LSym n in the homotopy category T . 8.1. Integral and rational coefficients. An abstract stable homotopy category T , being closed under all coproducts, is a category with splitting idempotents, see [23,Prop. 1.6.8]. Let now T Q be the Q-localized version of the additive category. Then one can easily construct symmetric and alternating powers LSym n Q and LAlt n Q with coefficients in Q using actions of Young symmetrizers on objects X n in T , in the same way as in [15]. The question is then do we have that (LSym n X) ⊗ Q = LSym n Q X and (LAlt n X) ⊗ Q = LAlt n Q X , where − ⊗ Q is the natural localization functor T → T Q . A possible approach here would be to represent the passing from integral to rational coefficients as a localization of model categories. Another approach could be to compare the splittings of the distinguished triangles in Künneth towers given in [5] with the present constructions. 8.2. Zeta-functions with coeff. in the Grothendieck-Witt group. Let k be a field and let S = Spec(k). The above approach to zeta-functions gives the following thing. It is well known that if char(k) = 2 the ring End(½) in SH(k) is isomorphic to the Grothendieck-Witt group GW (k) of quadratic forms over k, see [22]. The Euler characteristic associates to each dualizable motivic symmetric spectrum E its class of quadratic forms χ(E) in GW (k). For any scheme Noetherian scheme X over k let E(X) be its motivic symmetric spectrum considered as an object in SH(k). Assuming now that the motivic symmetric spectra of schemes and their symmetric powers are in an appropriate subcategory T ′′ ⊂ SH(k) we could associate, to any scheme X over k, its zeta-function ζ X (t) = 1 + [E(X)]t + [LSym 2 E(X)]t 2 + [LSym 3 E(X)]t 3 + . . . whose coefficients [Sym n E(X)] are in K 0 (T ′′ ). Applying now the Euler characteristic χ we obtain a new zeta-function ζ χ X (t) = 1 + χ(E(X))t + χ(LSym 2 E(X))t 2 + χ(LSym 3 E(X))t 3 + . . . with coefficients in GW (k), which seems to be an interesting object of arithmetical nature. Certainly, to provide the above zeta-function a geometrical sense we need to understand what could be the link between Sym n E(X) and E(Sym n X) -the question of critical importance for the whole theory presented here, see Section 8.5 below.
Let T p,q = S p ∧ G q m be the (p, q)-th Voevodsky's sphere in the model category of simplicial presheaves △ op P re(Sm/S). Applying the arguments used in Section 7.3 one can show that S p vanishes under symmetric or alternating powers in SH(k), depending on the parity of p. The Künneth tower for the Mayer-Vietoris associated with a punctured affine line G m = A 1 − {0} gives that G q m vanishes under the alternating powers in SH(k). Can we prove then that T p,q -cellular objects, in the sense of [4], all have rational zeta-functions with coefficients in GW (k)? 8.3. Finiteness condition with coefficients in Z and nilpotency theorems. Having constructed symmetric and alternating powers, endowed with Künneth towers, in an abstract stable homotopy category T it is natural to ask what would be a Z-coefficient analog of Kimura-O'Sullivan theory of finite-dimensional objects in T ? Certainly, under any reasonable finiteness condition a finite topological spectrum should satisfy that condition. Is there any connection between the Devinatz-Hopkins-Smith nilpotency theorem in stable topology, Rost's nilpotency theorem in algebraic geometry, and Kimura's nilpotency theorem for Chow-motives? Recall that topological nilpotency theorem says that if f : F → E is a morphism of topological spectra, where F is finite, then f is smash-nilpotent provided MU ∧ f is null homotopic, where MU is the complex cobordism spectrum, see [26]. Kimura's theorem asserts that any numerically trivial endomorphism of a Chow-motive (with rational coefficients) is nilpotent, see [15]. Notice that if we could reprove topological nilpotency theorem in "invariant" terms then, possibly, we could also be able to transmit the reasoning into the motivic setting replacing MU by the motivic cobordism spectrum MGL, see [16], and finite spectra by finite T p,q -cellular spectra in SH(k).
In our understanding, the meaning of motivic finite-dimensionality with integral coefficients likely consists of the following: while Q-rational motivic finite-dimensionality measures Chow groups, Z-coefficient motivic finitedimensionality measures not only algebraic cycles but also the luck of points on a variety rational over the ground field. For example, if X is a conic over a field k and X(k) = ∅ then M(X) = ½ ⊕ Ä, where Ä is the Lefschetz motive, so that dim Z (M(X)) = 2. However, if X is an anisotropic conic, the size of the motive M(X) must be bigger.
Perhaps, it is better to talk about a rank of a motive, while working with coefficients in Z.

8.4.
A 1 -localization of symmetric powers. As we have already mentioned above, it is not possible to construct symmetric powers in A 1homotopy categories without knowing how do symmetric powers behave under A 1 -localization. Our expectation is that they behave not very well. This is a problem even in the A 1 -unstable setting. And, certainly, this has some arithmetical meaning. 8.5. Symmetric powers of spectra and spectra of symmetric powers inétale topology. Recall that the question of what could be the link between LSym n E(X) and E(Sym n X) is of critical importance whose understanding would provide the geometrical meaning to the our approach. In [29] Voevodsky constructs symmetric powers on the ground of symmetric powers of schemes and develops the theory of motivic Eilenberg-MacLane spaces needed for the proof of Bloch-Kato conjecture. Understanding of relation between our symmetric powers and Voevodsky's ones would also give a uniform approach along that line. Besides, our approach is in coherence with the theory developed in [2] and [1]. Hopefully there should exist certain appropriate towers in SH connecting LSym n E(X) and E(Sym n X) in a nice way. In this section we collect some naive reasoning for sets with discrete topology and schemes with theétale topology to support this expectation.
The above functor P can be extended to a functor P : Sch/S −→ C from the category Sch/S of all separated schemes of finite type over S into C . The problem is that P obviously commutes with products but it does not commute with colimits. Let, for example, S = Spec(k) for a field k and let X be a smooth scheme over k. Its n-th symmetric power Sym n X is an object in Sch/k. Then the n-th symmetric power Sym n (P (X)) is not the same thing as P (Sym n X) in C . Consider the category of sets Sets as a category with discrete topology (everything is closed-open). Let P re(Sets) be the category of presheaves of sets on Sets and let Shv(Sets) be the category of sheaves with respect to the discrete topology on Sets. For any sheaf F in Shv(Sets) and any set X one has F (X) = x∈X F ({x}) .
Open neighbourhoods of points in Sets are points, and a stalk F x of a sheaf F at a point x is just F ({x}). Let h : Sets −→ P re(Sets) be the Yoneda embedding. For any set X the corresponding presheaf h X = Hom Sets (−, X) is a sheaf, so that h takes its values in Shv(Sets) in fact.
Let now G be a finite group acting on a set X. This can be viewed as a functor from G to Sets sending the unique object in G to X and any show that the canonical morphism of sets α R : X(R)/G −→ (X/G)(R) is an isomorphism. Let A R be the category ofétale algebras over R and let A l be the category ofétale algebras over l. As R is Henselian the residue homomorphism R → l induces an equivalence of categories Ψ : A R −→ A l .
Let f : Spec(R) → X/G be an element in (X/G)(R). Its preimage is a set of R-points of theétale R-algebra S, where Spec(S) is the pull-back Spec(S) As Ψ is an equivalence, we have that α −1 R (f ) = α −1 k (f ) , i.e. α −1 (f ) is bijective to l-points of theétale l-algebra L. Since R is strictly Henselian, the residue field l is separably closed, whence L = l ⊕ · · · ⊕ l n , where n is the order of the finite group G. Then the elementf can be identified with the quotient of the set of all l-points L by G. Then f is the same thing as the quotient of the set of all R-points S by G. It follows that α R is a bijection.
Notice that, if (h X /G) + N is is the sheaf associated with h X /G in the Nisnevich topology on Sm/k, the same argument shows that the canonical morphism (h X /G) + N is −→ h X/G is injective, i.e. a monomorphism section-wise.
What does all this mean? Consider the group Σ n acting on n-th selfproduct of a scheme X. One can take a stratification of X n such that the morphism from strata to Sym n X is the quotient over a free action of suitable subgroups in Σ n . Lemma 36 says that in any appropriateétale version of motivic symmetric spectra Sym n (E) will be the same as E(Sym n X). In Nisnevich topology the situation is more subtle but still not hopeless.