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(Co)Simplicial Descent Categories

In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD -->D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natural structure of Brown category of cofibrant objects on sD. We use these facts to produce cofiber sequences on the localized category of D by E, which give rise to a natural Verdier triangulated structure in the stable case.


Introduction
Given a category D, the category ∆ • D of simplicial objects in D supports a great deal of homotopic structure. To illustrate this fact, if ∆ • Set * denotes the category of pointed simplicial sets, observe that if D has coproducts and final object * , there is a natural functor This functor induces 'cofiber sequences' in ∆ • D, constructed as follows. The suspension ΣX of X is S 1 ⊗ X, while ∆ [1] ⊗ X is the cone of X. Then, the pushout of X → CX along a map f : X → Y is C(f ), the cone of f . This pushout always exists in ∆ • D, and it defines the cofiber sequence On the other hand, one of the fundamental observations of non-abelian homological algebra is that simplicial resolutions are a reasonable generalization to non-abelian settings of resolutions of an object by a chain complex. For instance, cosimplicial Godement resolutions provide resolutions of sheaves of spectra, and of sheaves of complexes with some additional data, as filtrations or multiplicative structures.
If our aim is to do homological algebra in D, one approach can be then to transport structure from ∆ • D to D, through a sort of 'simple' functor s : ∆ • D → D. Two classical examples are the following. The geometric realization ∆ • Set → T op, or more generally ∆ • T op → T op, has been used from the beginning of Algebraic Topology as a tool to develop homotopical and homological structure on T op. In the case of chain complexes over an additive or abelian category, the same can be said about the simple functor s : ∆ • Ch * (A) → Ch * (A) (that is just the total complex of a double complex). In the cosimplicial setting, Deligne's simple of cosimplicial mixed Hodge complexes [DeIII] is used to define a mixed Hodge structure over the cohomology of singular varieties. Also, Navarro's Thom-Whitney simple [N] of cosimplicial commutative differential graded algebras is a key tool to define the Thom-Whitney derived functors.
In the general context of Quillen models, a simplicial model category M is a simplicial category structure compatible with a model category structure on M. In this case, there is a satisfactory notion of simple functor s : ∆ • M → M. It can be made into a Quillen functor through the Reedy model structure on ∆ • M.
In [BK], the authors study the homotopy limit holim : ∆∆ • Set * → ∆ • Set * . In [T, section 5], the properties of holim are extended to cosimplicial fibrant spectra, in order to study cohomology of sheaves of fibrant spectra.
Returning to the general case of a category D, equipped with a class of equivalences, and with a simple functor s : ∆ • D → D, it is natural to ask Which properties of s : ∆ • D → D ensure that D inherits a suitable homotopic structure from ∆ • D?
More concretely, one would desire that the image under s of the sequences (2) behave as 'usual' cofiber sequences in HoD = D[E −1 ]. In particular, they should induce a triangulated category structure on HoD in the stable case. On the other hand, given a functor F : D → E, the expected result is that LF , the (left) derived functor of F , if it exists, could be computed as the simple s of a suitable simplicial resolution. Dually, for right derived functors one would use a cosimplicial simple s : ∆D → D.
In this paper we present the notion of simplicial descent category as an answer to these questions. Some of the most remarkable axioms we propose are the following. Coproducts: D has finite coproducts. The simple functor commutes with coproducts up to equivalence, and E ⊔ E ⊆ E. Exactness: The simple of a degree-wise equivalence is an equivalence. Acyclicity: The simple of the simplicial cone of a map f is acyclic if and only if f ∈ E. Eilenberg-Zilber: The iterated simple of a bisimplicial object is equivalent to the simple of its diagonal. We explain the axioms in the second section of this paper, after introducing some simplicial preliminaries in the first one.
Simplicial descent categories are inspired by the notion of cubical homological descent category introduced in [GN]. However, there are relevant differences between them. For instance, the Eilenberg-Zilber axiom (S4) has no analogous in the cubical setting, since cubical objects have no diagonal.
In order to produce new simplicial descent structures from known ones, we introduce in the third section the transfer lemma. The point is to use a functor ψ : D → D ′ to transfer a simplicial descent structure from a simplicial descent category D ′ to a general category D. The procedure is motivated by the singular chain functor from topological spaces to chain complexes of abelian groups. The singular chain functor is compatible with the (fat) geometric realization and the simple functor s of chain complexes [Dup,5.15]. Using this fact, one can transfer the descent structure from chain complexes to topological spaces. The simplicial transfer lemma is similar to the transfer lemma in the cubical setting [GN].
We also prove in sections 4 and 5 that the notion of simplicial descent category includes all examples of 'simple functors' cited in the first part of the introduction. The forgetful functor from commutative differential graded algebras, Cdga, to vector spaces is compatible with the Thom-Whitney simple and the simple of cochain complexes [N]. This, together with the transfer lemma, yields a cosimplicial descent structure on Cdga. On the other hand, the descent structure on mixed Hodge complexes is indeed obtained from two different simplicial descent structures on filtered cochain complexes, that are related through the 'decalage' functor of a filtration. In one of them the equivalences are the filtered quasi-isomorphisms. In the other, they are the maps inducing isomorphism in the second terms of the associated spectral sequences. Finally, the proofs of the axioms of simplicial descent category for fibrant spectra together with the homotopy limit as simple functor is essentially done in [T, section 5]. We also include more examples not mentioned before, as differential graded algebras over a commutative ring, or DG-modules over a fixed DG-category, just to cite a few.
A simplicial descent structure on (D, E) yields the following properties of HoD = D[E −1 ]. First, HoD can be described by means of calculus of left fractions over a quotient category of D modulo homotopy. Second, the image under the simple functor of the sequences (2) provides cofiber sequences in HoD that satisfy the 'non-stable' axioms of triangulated categories. When HoD is pointed, functor (1) is indeed an action (up to equivalence) of pointed simplicial sets on ∆ • D. In this case, the suspension of any object is a cogroup object in HoD. Therefore, the following theorem holds THEOREM [R1, 5.3] If D is an stable simplicial descent category -that is, the induced suspension functor is an equivalence of categories-then HoD becomes a Verdier's triangulated structure. This triangulated structure is 'natural' in the following sense. Assume the suspension functor is an equivalence of categories Σ : D → D. Then, for any small category I, F un(I, D) is again a stable simplicial descent category. Therefore F un(I, D)[E −1 ] is triangulated, and u * : (Co)simplicial descent categories are also a suitable setting to derive functors. We treat this question in the last section of this paper. We prove the existence of derived functors (in the sense of Quillen [Q]) with the help of the Cartan-Eilenberg categories machinery [GNPR]. Resolutions are computed through a given triple, compatible with the descent structure. More concretely, if T is a triple on D, denote by T X → X the augmented cosimplicial object produced by T in the usual way. We prove here the THEOREM 6.5 Let (D, E, s, µ, λ) be a cosimplicial descent category and T a compatible triple. Consider a functor F : D → E such that F s is an isomorphism for every s ∈ E.

Simplicial objects
Let D be a category. A simplicial object in D is a functor X : ∆ op → D. It is determined by the data where d i and s j are called the face and degeneracy maps, and satisfy the (dual) simplicial identities. A map f : X → Y of simplicial objects is a natural transformation of functors, or equivalently a collection of maps {f n : X n → Y n } of D compatible with the face and degeneracy maps of X. We denote by ∆ • D the category of simplicial objects in D.
Dually, ∆D is the category of cosimplicial objects in D. Its objects are functors X : ∆ → D, determined by the data We can associate to any object A in D the constant simplicial object A × ∆, with (A × ∆) n = A for all n, and with identities as face and degeneracy maps. In this way we get the constant functor − × ∆ : D → ∆ • D (− × ∆ : D → ∆D in the cosimplicial case). When understood, we will denote also by A the simplicial object A × ∆.
The bisimplicial objects in D are the simplicial objects in ∆ • D. They form the category ∆ • ∆ • D. The category ∆∆D of bicosimplicial objects in D is defined similarly. The diagonal functor D : Finally, ∆ e denotes the strict simplicial category. It is the subcategory of ∆ with its same objects, and strict monotone maps as morphisms. As before, we can define the categories ∆ • e D and ∆ e D of strict simplicial and cosimplicial objects in D.

Simplicial cones and cylinders
If D has (finite) coproducts, then simplicial (finite) sets acts on ∆ • D. More concretely, consider X ∈ ∆ • D and a simplicial (finite) set K. Then, K ⊠ X ∈ ∆ • D is equal to Kn X n in degree n, the coproduct of copies of X n indexed over K n . The face and degeneracy maps of K ⊠ X are obtained by those of K and X.
Recall that ∆[k] is the simplicial finite set with ∆[k] n = Hom ∆ ([n], [k]). The cylinder object of Next we introduce a combinatorial lemma, that will be used later. Consider the following commutative Applying Cyl by rows and columns we obtain Let ψ : Cyl(γ ′ , γ) → Cyl(δ ′ , δ) and ψ ′ : Cyl(f ′′ , g ′′ ) → Cyl( f , g) be the respective images under Cyl of: where each I means the corresponding canonical map.
Proof. The proof of the lemma is an easy exercise left to the reader. One can write down the coproducts defining Cyl(δ ′ , δ) n and Cyl( f , g) n and observe that they are the same after reordering the terms, and this reordering preserves the face and degeneracy maps. Also, one can deduce the lemma from the invariance of a double pushout by index interchange, and the natural isomorphism ∆ [1] Assuming that D has a final object * , the simplicial cone of f : X → Y in ∆ • D, C(f ), is by definition the double mapping cylinder of f and X → * .
Dually, if D has (finite) products, ∆D has a natural coaction by simplicial (finite) sets. If X ∈ ∆D and K is a simplicial finite set, then X K is equal to Kn X n in degree n. Then X ∆ [1] is called the path object of X, and again we have natural maps d 0 , d 1 : X ∆[1] → X. As before, we can define the (double mapping) path object associated to f : Y → X and g : Z → X, and the dual of lemma 1.2 is satisfied.
In particular, if f : X → Y is a map of cosimplicial objects, then the fibre F (f ) of f is the pullback in ∆D of the maps d 0 × d 1 : X ∆[1] → X × X and f × 0 : X × 0 → X × X (provided 0 is a initial object in D).

Simplicial Descent Categories
Before giving the general axioms of simplicial descent categories, let us illustrate them in the category of chain complexes. Next we recall the definition and basic properties of 'the simple functor' from simplicial chain complexes to chain complexes, induced by the total complex of a double complex.
(2.1) Let A be an additive category and Ch * (A) be the category of (unbounded) chain complexes in A.
, each X n is a chain complex {X n,p , d Xn } p∈Z . Hence X induces a double complex KX with (KX) n,p = X n,p . The boundary maps are d Xn : X n,p → X n,p−1 and ∂ : X n,p → X n−1,p , In the unbounded case the above coproduct may be not finite. Therefore, we assume the existence of countable sums in A, assumption that can be dropped in the uniformly bounded below case (for instance the positive chain complexes case). The following properties hold.
• Normalization: Each chain complex A ∈ Ch * (A), is canonically a direct summand of RA = s(A × ∆), in such a way that the projection λ A : RA → A and the inclusion ρ A : A → RA are inverse homotopy equivalences.
• Eilenberg-Zilber [DP,2.15 , the Alexander-Whitney map µ Z : sDZ → ssZ and the 'shuffle' or Eilenberg-Zilber map ν Z : ssZ → sDZ are inverse homotopy equivalences. In degree n, The shuffle map is defined in degree n as the sum of the maps where the sum is indexed over the (i, j)-shuffles (α, β), and ǫ(α, β) is the sign of (α, β) [EM].
In the abelian case, we also have • Exactness [B, p. 98, ex. 1] If X is a simplicial object such that X n is acyclic for all n, then sX is so.
In addition, if f : A → B is a morphism of chain complexes, it is easy to check that the simple of the Then, we deduce the following acyclicity property from the homology long exact sequence associated with f . • Acyclicity: A map f of chain complexes is a quasi-isomorphism if and only if the simple of its simplicial cone is acyclic.
Next we introduce some notations. Let C and D be two categories, E a class of morphisms in D and F un(C, D) the category of functors from C to D. Denote also by E the class of morphisms τ : between the corresponding localized categories. By abuse of notation, we will write ev Recall that the class E is said to be saturated if it holds that f ∈ E if and only if f is an isomorphism in D[E −1 ]. In particular, E contains all isomorphisms of D, is closed by retracts and satisfies the 2-out-of-3 property.
DEFINITION 2.2. A simplicial descent category is the data (D, E, s, µ, λ) where (S 1) D is a category with finite coproducts, initial object 0 and final object * . (S 2) E is a saturated class of morphisms in D, stable by coproducts (that is E ⊔ E ⊆ E). Morphisms in E will be called equivalences, and an object A such that A → * is in E will be called acyclic. (S 3) s : ∆ • D → D is a functor, called the simple functor, which commutes with coproducts up to equivalence.
An additive simplicial descent category is a simplicial descent category which is also additive, and such that the simple functor is an additive functor. Also, µ must be an isomorphism in F un ad (∆ The opposite of an (additive) simplicial descent category is by definition an (additive) cosimplicial descent category. (S 4) provides an isomorphism of HoD between s(n → s(m → Z n,m )) and s(m → s(n → Z n,m )), since the diagonals of {Z n,m } and {Z m,n } are the same. 2.-In many examples of simplicial descent categories, the simple functor is a coend where C ∈ ∆D is fixed and ⊠ is a monoidal structure on D.
A map from C to the unit element u for ⊠ provides a natural transformation λ : s(− × ∆) → Id, while a quasi-inverse of λ comes usually from the choice of an isomorphism between C 0 and the unit element u for On the other hand, a coalgebra structure C n → C n ⊠ C n provides a natural transformation µ : sD → ss compatible with λ. This is the case of (commutative) differential graded algebras and topological spaces. If R is a commutative ring, the simple functor of chain complexes of R-modules can be written in this way as well.
3.-If D is an arbitrary category, one can always consider trivial simple functors s : ∆ • D → D, as sX = X k where k ≥ 0 is fixed. They always provide trivial simplicial descent structures on D, where the equivalences are forced to be all morphisms of D (since (S 7) holds). As all maps are equivalences, the associated homotopy category D[E −1 ] is equivalent to the trivial category with one object and one map. 4.-In [R1, 3.6] it is proven that s sends simplicial homotopy equivalences in ∆ • D to equivalences in D. Also, if f is a morphism of ∆ • ∆ • D such that sf n, * ∈ E for all n, then (S 4) and (S 6) imply that sDf ∈ E. Then, the class {f ∈ ∆ • D | sf ∈ E} is a ∆-closed class in the sense of [Vo].
EXAMPLE 2.5. Let D = Ch * (A) be the category of chain complexes over an abelian category A (with countable sums in the unbounded case). Set W = quasi-isomorphisms as class of equivalences in D. Then D, W together with the data s, µ, λ given in (2.1) is a simplicial descent category, that we denote by Q Ch * (A). The exactness property given there implies (S 6). Indeed, assume f n ∈ E for all n. Then, by (S 7), so is sC(f n ) → * , as well as T = s(n → sC(f n )) → * . By the Eilenberg-Zilber property, If A is additive, then (Ch * (A), λ, µ, ρ) with the class of equivalences S = {homotopy equivalences} is also a simplicial descent category. It will be referred to as H Ch * (A). The axioms can be proved as in the abelian case, except the exactness axiom, whose proof is the same as for first quadrant double complexes (see for instance [B, §5]).
The notion of simplicial descent category is natural in the sense of the following proposition, whose straightforward proof is left to the reader.
We finish this section with some formal consequences of the axioms. From now on (D, E, s, µ, λ) denotes a simplicial descent category.
such that sα, sβ and sγ are equivalences. Then the induced morphism sCyl(f, g) → sCyl(f ′ , g ′ ) is also in E. This is the case, for instance, when α n , β n and γ n are equivalences for all n.
Proof. The case α n , β n , γ n ∈ E for all n follows directly from (S 2) and (S 6), and the general case follows from the following result.
HoD. This isomorphism commutes with the respective canonical maps from RsY , RsZ, sY , and sZ to L and sCyl(f, g). (S 4) provides an isomorphism in HoD relating sCyl(f, g) to ssT . But for each n, s(m → T n,m ) is equivalent to Cyl((sf ) × ∆, (sg) × ∆) n by (S 3). Therefore, (S 6) ensures the existence of ν : sCyl(f, g) On the other hand, last assertion follows by definition and by the compatibility between λ and µ.
Proof. Denote by θ : X → * × ∆ the trivial map. By (S 5), s( * × ∆) → * ∈ E, and by lemma 2.7 there is an equivalence between sC(sf × ∆) and L = sCyl(sf × ∆, sθ × ∆). But, as we have seen, L ≃ sCyl(f, θ) = sCf in HoD. Hence sC(sf × ∆) and sCf are isomorphic in HoD, so (S 7) ensures that sf ∈ E if and only if sCf is acyclic. To see a) and b), we can assume that f, g are maps in D, since sCyl(f, g) ≡ sCyl((sf ) × ∆, (sg) × ∆) and Rh is an equivalence if and only if h is. To relax the notations we write h × ∆ = h for a map h of D. It suffices to prove that f ∈ E if and only if s(I Z : Z → Cyl(f, g)) is so. Indeed, it is easy to check that Υ(Cyl(f, g)) ≡ Cyl(g, f ) if f, g are constant maps. Then, b) can be formally deduced from a) using (S 8).
To prove a), consider the diagram By lemma 1.2 we get an isomorphism in ∆ • D between the cone of I Z : Z → Cyl(f, g), and Cyl(f, α). Here α is the map from X to CZ induced by g, and CZ is the cone of Id Z : Z → Z. By (S 7), sCZ is acyclic, so sα is equivalent to the trivial map X → * . Then, it follows from lemma 2.7 that sCyl(sf, sα) is isomorphic in HoD to sC(sf ) ≃ sCf . We conclude that sCf is isomorphic in HoD to sCI Z , then the result follows from the first assertion.
3 Descent functors and the transfer lemma DEFINITION 3.1. If (D, s, E, µ, λ) and (D ′ , E ′ , s ′ , µ ′ , λ ′ ) are simplicial descent categories, a descent functor between them is a functor ψ : There exists an isomorphism Θ : In addition Θ must be compatible with λ, λ ′ and with µ, µ ′ in the following sense: The normalization theorem [May,22.3] implies that s and s N are naturally homotopic, so the identity functor induces descent functors between (Ch * (A), W, s, µ, λ) and (Ch * (A), W, s N , µ N , λ N ). The same holds if we replace W by the homotopy equivalences, or µ, µ N by ν, ν N . 3.-If F : I → J is a functor of small categories and D is a simplicial descent category, then F * : JD → ID is a descent functor with respect to the object-wise descent structures given in proposition 2.6. Also, if i ∈ I, then functor ev i : ID → D with ev i (f ) = f (i) is a descent functor. In addition, if ψ : D → D ′ is a descent functor, so is the induced functor ψ : ID → ID ′ .
We state now the transfer lemma, that will be widely used in next section.   [R1, proposition 2.6] it is proven that any morphism of C[E ′−1 ] is represented by a roof → ∼ ←. Therefore, as E ′ is saturated, each isomorphism Θ of C[E ′−1 ] is represented by a roof formed by two equivalences. 2.-In (S 4) ′ we can assume also that µ : ss → sD. In this case the compatibility with λ means that λ sX = µ ∆×X and sλ X = µ X×∆ in D, for all X ∈ ∆ • D.

Simplicial Examples
Our first example of simplicial descent category is the one of chain complexes 2.5. Using the transfer lemma, we deduce new simplicial descent structures, following the picture below

Simplicial objects in additive/abelian categories
Let Ch + A be the category of positive chain complexes over an additive or abelian category A. As in 2.1, the functor K : ∆ • A → Ch + A is defined by taking the alternate sum of the face maps of X as boundary map of KX. The Eilenberg-Zilber-Cartier theorem [DP] means in the simplicial descent context that functor K transfers the descent structure from Ch + A to ∆ • A. Define a descent structure on ∆ • A by: Simple functor: The simple functor is the diagonal functor D : Equivalences: We consider two classes of equivalences. On one hand, S = {simplicial homotopy equivalences}.
On the other, W = {quasi-isomorphisms (that is, those maps inducing isomorphism on homology)}.
Transformations λ and µ: The natural transformations λ and µ are defined as the corresponding identity natural transformations. Proof. It follows from the Eilenberg-Zilber-Cartier theorem [DP] that K : * ∆ • A → * Ch + A satisfies the transfer lemma 3.3 for * = H, Q, taking Θ = Alexander-Whitney map, which is a natural homotopy equivalence KD → sK. On the other hand, the equality {f | Kf is a homotopy equivalence} = S is deduced from the classical homotopy theory of simplicial objects in A [May].

Simplicial Sets
Let Set be the category of sets, mod R the one of modules over the ring R, and L : Set → mod R the functor that maps T to the free R-module with basis T . Recall that the homology of a simplicial set W (with coefficients in R) is the homology of the simplicial R-module L(W ). That is, it is the homology of the chain complex KL(W ). Define a descent structure on ∆ • A by: Simple functor: The simple functor is the diagonal D : The class E of equivalences consists of those morphisms that induce isomorphism on homology (with coefficients in R).
Transformations λ and µ: The natural transformations λ and µ are defined as the identity natural transformations.
The statement of the transfer lemma for L :

Topological Spaces
Let △ m ⊂ R m+1 be the standard m-dimensional simplex. Define the following simplicial descent structure on the category T op of topological spaces and continuous maps. Simple functor: s : ∆ • T op → T op is the 'fat' geometric realization [S, appendix A]. A simplicial topological space X induces the bifunctor ∆ e × ∆ e → T op , ([n], [m]) → X n × △ m . The fat geometric realization of X is the coend of this bifunctor [ML]: More specifically, sX is the quotient space of n≥0 X n × △ n modulo the equivalence relation generated by We will write [x, t] for the equivalence class of an element (x, t) ∈ n≥0 X n × △ n . Equivalences: The class E consists of those maps that induce isomorphism on singular homology with coefficients in a fixed commutative ring R. The proof is basically [Dup,5.15]. See [R, 5.5] for more details.
REMARK 4.4. i.-Fat geometric realization is exact with respect to quasi-isomorphisms [Dup,5.16]. If we consider the usual geometric realization | · | : ∆ • T op → T op, it turns out that given a map f : X → Y such that f n is a quasi-isomorphism for all n, we need to impose some extra conditions in order to have that |f | is again a quasi-isomorphism (see, for instance, [M, 11.13]). This is the reason why we do not consider | · | as simple functor.
On the other hand, recall that the canonical map sX → |X| is a homotopy equivalence in case the simplicial topological space X is such that its degeneracy maps are closed cofibrations (see [S, appendix A]).
ii.-It can be proved as well that geometric realization | · | : ∆ • Set → T op and fat geometric realization s : ∆ • Set → T op are descent functors [R, 5.5].

Cosimplicial Examples
Recall that a cosimplicial descent category is by definition the dual category of a simplicial descent category. Hence, cochain complexes form a cosimplicial descent category. We describe this structure in our first cosimplicial example. The remaining examples are deduced from the dual of transfer lemma 3.3.

Cochain complexes
Let Ch * (A) be the category of (unbounded) cochain complexes over an additive or abelian category A with countable products. From equality (Ch * (A op )) op = Ch * (A) we get the data (Ch * A, s, µ, λ). Together with the homotopy equivalences or quasi-isomorphisms as equivalences, this is in fact a cosimplicial descent category by definition.
More concretely, if X ∈ ∆Ch * (A) then (sX) q = p+n=q X n,p , and the boundary map is d = (−1) p ∂ + d X n : p+n=q X n,p −→ p+n=q+1 X n,p . On the other hand, a cochain complex A is canonically a direct summand of s(A×∆), and λ A : A → s(A×∆) is the inclusion. Finally, µ : ss → sD and ν : sD → ss are the (dual) Alexander-Whitney and Eilenberg-Zilber maps, respectively. (5.1) As in the case of chain complexes, we can define the normalized simple s N : ∆Ch * (A) → Ch * (A). It is given by (s N A) n = p+q=n p−1 i=0 Ker{s i : A p,q → A p−1,q }. Now λ N is the identity natural transformation, and µ N , ν N are induced by the previous ones.

Differential graded algebras
Consider a commutative (associative and unitary) ring R. Let Ch + R be the category of positive cochain complexes of R-modules and Dga(R) the one of differential graded R-algebras (not necessarily commutative, and positively graded).
Recall that s : ∆Ch + R → Ch + R is the composition of functors K and T ot, which are monoidal with respect to the tensor product of R-modules. Then so is s. Given cosimplicial cochain complexes X and Y , the Künneth morphism k : sX ⊗ sY −→ s(X ⊗ Y ) is induced by the Alexander-Whitney map. In degree n, k n is the sum of the morphisms This Kï¿ 1 2 nneth morphism preserves degenerate parts, so it gives rise to a Kï¿ 1 2 nneth morphism for the normalized simple, k : Hence s N induces s AW : ∆Dga(R) → Dga (R), the (normalized) Alexander-Whitney simple [N, 3.1], which is part of the following descent structure. Simple functor: Given A ∈ ∆Dga(R), consider the cosimplicial cochain complex UA ∈ ∆Ch * R obtained by forgetting the multiplicative structure of A. Then s N (UA) is a differential graded algebra through s N (UA) ⊗ s N (UA) → s N (UA), defined as the composition where k is the Kï¿ 1 2 nneth morphism and τ n : UA n ⊗ UA n → UA n is the structural morphism of the differential graded algebra A n . The Alexander-Whitney simple is the functor s AW : ∆Dga(R) → Dga (R) obtained in this way. Transformations λ AW and µ AW : The transformation λ AW is equal to the identity natural transformation. On the other hand, µ AW : s AW D → s AW s AW is obtained from the Eilenberg-Zilber map ν N at the level of cochain complexes. In fact, it is a morphism of algebras [EM,p. 232].
REMARK 5.2. We use the Eilenberg-Zilber map ν N as the datum µ AW in the cosimplicial descent structure of Dga(R) because it is symmetric, so it preserves the algebra structure. This is not true for the Alexander-Whitney map, which is just symmetric up to homotopy.
The following proposition follows directly from the transfer lemma.
PROPOSITION 5.3. The category Dga(R) together with the normalized Alexander-Whitney simple and the quasi-isomorphisms as equivalences is a cosimplicial descent category. In addition, the forgetful functor from Dga(R) to Q Ch * R is a descent functor.

Commutative differential graded algebras
Denote by Cdga(k) the category of commutative differential graded algebras (or cdg algebras) over a fixed field k of characteristic 0. Navarro's Thom-Whitney simple [N] gives rise to a cosimplicial descent structure on Cdga(k), which we now describe. Simple functor: Let L ∈ ∆ • Cdga(k) [BG] be where Λ(x 0 , . . . , x n , dx 0 , . . . , dx n ) is the free cdg algebra generated by {x k } k in degree 0 and by {dx k } k in degree 1. The boundary map is the unique derivation such that d(x k ) = dx k , d(dx k ) = 0. The face maps d i : L n+1 → L n and the degeneracy maps s j : L n → L n+1 are defined as Recall that ∆ e is the strict simplicial category (with no degeneracy maps). Then, the Thom-Whitney simple is the end [ML] s T W (A) = n L n ⊗ A n .

Equivalences:
The class E consists of the quasi-isomorphisms, that is to say, those morphisms of cdg algebras which induce isomorphism on cohomology.
where π is the iterated projection and τ p : L p ⊗ L p → L p is the structural map of the cdg algebra L p , which is a morphism of cdg algebras since L p is commutative.
Let k[p] be the cochain complex with (k[p]) p = k and (k[p]) n = 0 if n = p. As in [N, (2.2)], we denote by △ p : L p → k[p] the map of cochain complexes which in degree p is the integral over the simplex △ p , L p p → k.
PROPOSITION 5.4. The category Cdga(k) together with the Thom-Whitney simple and the quasi-isomorphisms is a cosimplicial descent category. In addition, the forgetful functor from Cdga(k) to the descent category Q Ch * k of cochain complexes of k-vector spaces is a descent functor.
To see that the forgetful functor U : Cdga(k) → Ch * k satisfies the dual hypothesis of proposition 3.3, let us check (FD2) op . By [N, 2.15], the diagram commutes up to a natural homotopy equivalence I : Us T W → sU. Given A ∈ ∆Cdga(k), I A : Us T W A → sUA is in degree n the map (s T W A) n → (sUA) n = p+q=n A p,q whose projection onto the component A p,q is given by where π denotes the projection. The compatibility between λ T W : Id Cdga(k) → s T W (− × ∆), λ : Id Ch * k → s(− × ∆) and I = Θ follows from the equality I A×∆ U(λ T W A ) = λ UA . It remains to see that µ T W : s T W s T W → s T W D, µ : ss → sD and Θ are compatible. If Z ∈ ∆∆Cdga(k), I claim that Θ DZ U(µ T W ) Z is equal, up to homotopy, to µ UZ sΘ Z Θ sT W Z . Recall that the transformation µ comes from the Alexander-Whitney map.
Using the homotopy inverse E of Θ given in [N], one can obtain a homotopy inverse E ′ of sΘ Z Θ sT W Z . Then, it is enough to see that Θ DZ U(µ T W ) Z E ′ coincides with µ UZ up to homotopy. As in [N, (3.4)], one can check that Θ DZ U(µ T W ) Z E ′ is the total complex of a map of double complexes which is equal to the Alexander-Whitney map in degree 0, so the statement follows from Eilenberg-Zilber-Cartier theorem [DP].

DG-modules over a DG-category
Let R be a fixed commutative ring. We follow the notations given in [K] for DG-categories. Recall that Dif R is the DG-category whose objects are (unbounded) cochain complexes of R-modules, and CA is the category of differential graded modules (or DG-modules) over a fixed DG-category A. Recall that a DG-module M is a functor of DG-categories M : A op → Dif R. A morphism of DG-modules is a natural transformation λ : M → N such that λ A : M A → N A is a morphism of cochain complexes for all A ∈ A.
Next we endow CA with a cosimplicial descent category structure. To this end, denote by Q Ch * R = (Ch * R, R W, R s, R µ, R λ) the descent category of cochain complexes of R-modules.
Simple functor: Equivalences: The equivalences are those morphisms ρ : Next proposition follows from the fact that Q Ch * R is a cosimplicial descent category.

Filtered cochain complexes
Given an abelian category A, let CF + A be the category of filtered positive cochain complexes. Its objects are pairs (A, F), where A is a positive cochain complex over A and F is a decreasing birregular filtration. We endow here CF + A with two different descent structures. In the first one, the equivalences are the filtered quasi-isomorphisms, and they are the E 2 -isomorphism in the second one. Both are related through the 'decalage' functor [DeII,I.3.3]. We briefly recall the graded and decalage functors. Let (Ch + A, s, W, µ, λ) be the descent category of positive cochain complexes over A.
For each k ∈ Z, the graded functor Gr k : CF + A → Ch + A is given by F Gr k A = F k A F k+1 A, where (A, F) ∈ CF + A. Let (Ch + A) Z be the category of graded cochain complexes, whose objects are families, indexed over Z, of positive cochain complexes. The graded functor Gr : CF + A → (Ch + A) Z applied to (A, F) is in degree k the complex F Gr k A. Let Dec : CF + A → CF + A be the functor that maps (A, F) to the filtered complex (A, Dec(F)), where Next we describe the first descent structure on CF + A. Simple functor: If (A, F) ∈ ∆CF + A, let sF be the filtration of sA defined as (sF) k (sA) = s(F k A). Define (s, s) : ∆CF + A → CF + A as (s, s)(A, F) = (sA, sF). Equivalences: The class E consists of the filtered quasi-isomorphisms, that is, those f such that Gr k (f ) is a quasi-isomorphism for all k ∈ Z. Transformations λ and µ: Transformations λ and µ in Ch + A preserve the filtrations, inducing λ and µ for (s, s).
PROPOSITION 5.6. (CF + A, (s, s), E, λ, µ) is an additive cosimplicial descent category. In addition, Gr : Proof. Note that (Ch + A) Z is a cosimplicial descent category by 2.6. The transfer lemma is trivially satisfied for the graded functor Gr : CF + A → (Ch + A) Z . Indeed, sGr(A, F) ≡ Gr(sA, sF) since s is exact. Now we endow CF + A with a second descent structure. Simple functor: Define (s, δ) : ∆CF + A → CF + A as follows. If (A, F) ∈ CF + A, then (s, δ)(A, F) = (sA, δF), where sA is the usual simple of cochain complexes. On the other hand, δF is the diagonal filtration over sA, given by Equivalences: The class of equivalences E 2 consists of the E 2 -isomorphisms, that is, those morphisms that induce isomorphism on the second term of the spectral sequences associated with (A, F) and (B, G).
REMARK 5.8. I. If k is a fixed integer, then previous proposition can be proved in a similar way for the category CF k A of filtered (by a birregular filtration) cochain complexes (A, F) such that A n = 0 when n < k. II. The above simplicial descent structures can be adapted as well to the normalized case, replacing s by the normalized simple s N .

Mixed Hodge Complexes
We define here a category of mixed Hodge complexes and endow it with a structure of cosimplicial descent category. The simple functor agrees with Deligne's construction [DeIII,8.I.15] on objects. It becomes a functor since the comparison morphisms of our mixed Hodge complexes are genuine filtered quasi-isomorphisms instead of maps in the corresponding filtered derived category. We denote by Q and C the respective categories of Q and C-vector spaces.
DEFINITION 5.9. A mixed Hodge complex is the data ((K Q , W), (K C , W, F), α), where a) (K Q , W) is a positive cochain complex of Q-vector spaces with finite dimensional cohomology, filtered by the increasing filtration W. b) (K C , W, F) is a positive bifiltered cochain complex of C-vector spaces, where W and F are respectively increasing and decreasing birregular filtrations, called the weight and Hodge filtrations. c) α is the data (α 0 , α 1 , ( K, W)), where ( K, W) is an object of CF + C and α i , i = 0, 1, is a filtered quasiisomorphism The following axiom must be satisfied (MHC) For each n the boundary map of W Gr n K C is compatible with the induced filtration F, and ( W Gr n H k K C , F) is a Hodge structure of weight n + k. That is, REMARK 5.10. (1. -) The filtered quasi-isomorphisms have a calculus of fractions in the category of filtered complexes up to filtered homotopy [IlI,p.271]. Hence any isomorphism in the filtered derived category can be represented by a zig-zag as in c). Moreover, any arbitrary zig-zag of filtered quasi-isomorphisms can be reduced to a length 2 zig-zag in a functorial way, [R, 6.5.17], using a 'quasi-pullback' procedure (constructed through the path object coming from the cosimplicial structure on CF + C). In [Be] the comparison zig-zag α has the shape · ← · → ·, because they use a 'reduction' procedure based on the filtered cone object, instead of the filtered path one. Another possibility is to allow zig-zags α of arbitrary length, as in [PS]. All constructions and results developed here can be adapted to this situation.
(2.-) Except for the Z-part, a mixed Hodge complex as above is a mixed Hodge complex in the sense of Deligne, thinking in α as an isomorphism in the filtered derived category. Also, applying the decalage functor Dec to the weight filtration W we get a mixed Hodge complex as defined in [H] and [Be].
(3.-) We dropped the Z-part of a mixed Hodge complex by simplicity, but all results in this section are also valid for mixed Hodge complexes with Z-coefficients.
EXAMPLE 5.11. [DeIII,8.I.8] Let j : U → X be an open immersion of complex smooth varieties, where X is proper and Y = X\U is a normal crossing divisor. If F is a sheaf on T , set RΓ(T, F ) = Γ(T, C God F ), where C God F is the Godement resolution of F . Analogously, if F is a bounded below complex of sheaves on T (eventually filtered), set RΓ(T, F ) = Γ(T, T ot(C God F )), where T ot means the total complex of a double complex. The point is that RΓ has values in the category of (filtered) complexes instead of in the derived category, and the (hyper) cohomology H · (T, F ) may be computed as the cohomology of RΓ(T, F ). Let (Ω X Y , W, F) be the logarithmic De Rham complex of X along Y [DeII, 3.I]. W is the so-called 'weight filtration', and F is the 'Hodge filtration', that is the filtration 'bête' associated with Ω X Y . Let (RΓj * Q, W) be the filtered complex of sheaves of Q-vector spaces on X, where W = τ ≤ is the 'canonical' filtration. A general argument shows that there is a zig-zag of filtered quasi-isomorphisms relating RΓ(j * Q U , W) ⊗ C to RΓ(Ω X Y , W) (see [H, p. 66] or [PS,4.11]). It is basically the result [DeII,3.I.8] relating Ω X Y to j * Ω U , together with Poincaré lemma (that is, Ω U is a resolution of the constant sheaf C U ). This zig-zag can be reduced to a length 2 zig-zag in a natural way (through the path object). Therefore (RΓ(j * Q, W), RΓ(Ω X Y , W, F)) is a mixed Hodge complex in the sense of previous definition.
DEFINITION 5.12. A morphism of mixed Hodge complexes consists of morphisms f Q : (K Q , W) → (K ′ Q , W ′ ) and f C : (K C , W, F) → (K ′ C , W ′ , F ′ ) of (bi)filtered complexes. If α = (α 0 , α 1 , K, W) and α ′ = (α ′ 0 , α ′ 1 , K ′ , W ′ ) are the respective zig-zags is a morphism of bifiltered complexes such that squares I and II in the diagram below In this way we obtain the category Hdg of mixed Hodge complexes.
Next we endow Hdg with a structure of cosimplicial descent category. Simple functor: If K = ((K Q , W), (K C , W, F), α) is a cosimplicial mixed Hodge complex, let s Hdg K be the mixed Hodge complex ((sK Q , δW), (sK C , δW, sF), sα), where s denotes the usual simple of cochain complexes and δW is the diagonal filtration. More concretely If α = (α 0 , α 1 , ( K, W)) then sα denotes the zig-zag It follows from general Hodge theory that an equivalence f : K → K ′ induces isomorphism on the resulting Hodge structures. Transformation λ: λ Hdg : Id Hdg → s Hdg (− × ∆) is λ Hdg induced by the transformations λ Q and λ C of Ch + Q and Ch + C respectively. Transformation µ: Similarly, µ Hdg s Hdg s Hdg Z → s Hdg DZ. PROPOSITION 5.13. The category (Hdg, s Hdg , E Hdg , µ Hdg , λ Hdg ) is an additive cosimplicial descent category. In addition, the forgetful functor U : Hdg → Ch + Q given by U((K Q , W), (K C , W, F), α) = K Q is a functor of additive cosimplicial descent categories.
Proof. Note that s Hdg = (s, δ, s) : ∆Hdg → Hdg is indeed a functor. Given K ∈ ∆Hdg, then s Hdg K is a mixed Hodge complex by [DeIII,8.I.15 i], and s Hdg is functorial with respect to the morphisms of ∆Hdg by definition. The us see that transfer lemma if satisfied for U : Hdg → Ch + Q. To see (SDC 4) ′op and (SDC 5)' op , let K = ((K Q , W), (K C , W, F), α) be a mixed Hodge complex. Clearly λ Q K Q , λ C K C and λ C e K preserve the filtrations. Set K = K × ∆. As the following diagram commutes in CF + C o o s(α1) / / (s(K Q ⊗ C), δ(W ⊗ C)) ∼ / / (sK Q , δ(W)) ⊗ C then λ Hdg is a morphism in Hdg. It can be argued similarly with µ Hdg

Fibrant Spectra
We describe now a cosimplicial descent structure on fibrant spectra. Essentially all axioms of cosimplicial descent category for fibrant spectra are proved in [T, section 5], taking the homotopy limit as simple functor, and the stable equivalences as equivalences. Indeed, the proofs in loc. cit. are based in the properties of the homotopy limit of cosimplicial simplicial sets developed in [BK]. Recall that a fibrant spectrum X is a family of fibrant pointed simplicial sets {X n } n∈N together with maps S 1 ∧ X n → X n+1 that are weak (homotopy) equivalences of simplicial sets. Morphisms of spectra are defined in the obvious way. We denote by Sp the category of fibrant spectra. In this case the 'stable equivalences', i.e., the morphisms of spectra inducing isomorphism on the stable homotopy groups, are the same as the degree-wise weak homotopy equivalences. Then, set E = {f : X → Y such that f n is a weak (homotopy) equivalence in∆ • Set * for all n} The simple functor is the homotopy limit, holim : ∆Sp → Sp, given degree-wise as the homotopy limit of (pointed) cosimplicial simplicial sets [BK], (holimX) n = holim(X n ), that is defined as follows. The 'function space' hom * : (∆∆ • Set * ) op × ∆∆ • Set * → ∆ • Set * is in degree n hom n (S, T ) = Hom ∆∆ • Set * (S × ∆[n], T ) Define ∆/− ∈ ∆∆ • Set * as N (∆/[n]) in cosimplicial degree n. N (∆/[n]) denotes the nerve of the category ∆/[n]. If Z ∈ ∆∆ • Set * , then holimZ = hom * (∆/−, Z). If we replace ∆ by an arbitrary small category I, we get a 2-functor holim I : ISp → Sp [BK, T].
PROPOSITION 5.14. The category Sp together with holim as simple functor and E as equivalences is a cosimplicial descent category.
2. sν Y θ 2 Y = θ Y ν sY for all Y ∈ ∆D, where θ 2 is the iteration of θ given below. The iterates θ i : T i s −→ sT i of θ are the natural transformations defined as θ i X = θ T i−1 X T (θ i−1 X ), if X ∈ ObD. Note that θ i X is also in E. THEOREM 6.3. Let (D, E, s, µ, λ) be a cosimplicial descent category and T a compatible triple. Set S = E, F = sT and W = F −1 (E). Then (D, S, W) is a right Cartan-Eilenberg category. The fibrant objects are those isomorphic to FX for some object X in D.
The above result generalizes [GNPR,theorem 5.1.5], where D is the category of positive complexes over an additive category. LEMMA 6.4. Let (D, s, E, µ, λ) be a cosimplicial descent category. If α : X −1 × ∆ → X is a coaugmentation with an extra degeneracy then sα ∈ E.