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A monoidal Dold-Kan correspondence for comodules

We present the notion of fibrantly generated model categories. Cofibrantly generated model categories are generalizations of CW-approximations which provide an inductive cofibrant replacement. We provide examples of inductive fibrant replacements constructed as Postnikov towers. These provide new types of arguments to compute homotopy limits in model categories. We provide examples for simplicial and differential graded comodules. Our main application is to show that simplicial comodules and connective differential graded comodules are Quillen equivalent and their derived cotensor products correspond.


Introduction
Let R be a commutative ring. The Dold-Kan correspondence established in [Dol58,Kan58] is an equivalence between the category sMod R of simplicial R-modules and the category Ch ≥0 R of non-negative chain complexes over R. The normalization functor N : sMod R → Ch ≥0 R is symmetric monoidal with respect to the graded tensor product of chains, the levelwise tensor product of simplicial modules and the Eilenberg-Zilber map. The inverse equivalence Γ : Ch ≥0 R → sMod R also has a (non-symmetric) monoidal structure coming from the Alexander-Whitney map. The categories are not isomorphic as symmetric monoidal categories. Nevertheless, the homotopy categories of sMod R and Ch ≥0 R have isomorphic symmetric monoidal structures when endowed with their respective derived tensor product. Formally, it was shown in [SS03] that there is a weak symmetric monoidal Quillen equivalence between the symmetric monoidal model categories of sMod R and Ch ≥0 R . The equivalence lifts to the associated categories of algebras and modules. In [Pér20a,4.1], we proved that the underlying ∞-categories of sMod R and Ch ≥0 R (in the sense of [Lur17, 1.3.4.15, 4.1.7.4]) are equivalent as symmetric monoidal ∞-categories. In particular, any homotopy coherent (co)algebras and (co)modules between sMod R and Ch ≥0 R are equivalent. In [Pér20c, 1.1], for R = k a finite product of fields, and C a simply connected strictly cocommutative and coassociative coalgebra in Ch ≥0 k , we proved that homotopy coherent comodules in Ch ≥0 k over C correspond to strictly coassociative comodules in Ch ≥0 k . The cotensor product of comodules equalize the coactions. When over a finite product of fields, it remains a comodule. The category CoMod C (Ch ≥0 k ) of connective differential graded comodules over C is endowed with a model structure in which weak equivalences are quasi-isomorphisms between comodules. The model structure is compatible with respect to the cotensor product. This allowed us to endow a symmetric monoidal structure with the derived cotensor product on the underlying ∞-category of CoMod C (Ch ≥0 k ), see [Pér20c,1.2]. Similar arguments hold for simplicial comodules over a simply connected simplicial coalgebra. Although homotopy coherent simplicial comodules correspond to homotopy coherent connective differential graded comodules, it is not immediate that their derived cotensor product correspond. Our main result in this paper shows that they are indeed equivalent.
Theorem 1.1 (Theorem 5.1, Theorem 5.3). Let k be a finite product of fields. Let C be a simply connected cocommutative differential graded coalgebra over k. Then there is a Quillen equivalence: compatible with the derived cotensor products. In particular, the ∞-categories of homotopy coherent simplicial comodules over Γ(C) and homotopy coherent connective differential graded comodules over C are equivalent as symmetric monoidal ∞-categories.
It is important to notice that at the level of model categories, the results of [SS03] do not imply any Quillen equivalences on model categories of coalgebras or comodules. In fact, in [Sor19], it was shown that the Dold-Kan correspondence does not lift to a Quillen equivalence between coassociative and counital coalgebras.
We dualize the methods in [SS03]. Following [Hes09] and [BHK + 15], we introduce the notion of fibrantly generated model categories. CW-complexes provide an inductive cofibrant replacement for spaces. This was generalized into the notion of cofibrantly generated model categories. We instead provide Postnikov towers as an inductive fibrant replacement. Because of the lack of "cosmallness" in practice, we cannot apply the dual of the small object argument. Instead we provide ad-hoc Postnikov presentations: (acyclic) fibrations are retracts of maps built from pullbacks and transfinite towers of generating (acyclic) fibrations. Moreover, the cotensor product on comodules behaves well with fibrant objects instead of cofibrant objects. We introduce the notion of symmetric comonoidal model categories that allows us to derive the cotensor product and obtain a symmetric monoidal structure on the homotopy category and the underlying ∞category. Dualizing [SS03], we construct weak comonoidal Quillen equivalences that induce an equivalence of the derived cotensor product.
Previously, Postnikov presentation were introduced to lift model structures from a left adjoint, see [Hes09,BHK + 15]. A different and more efficient approach was given [HKRS17,GKR18]. Our paper shows that Postnikov presentations of model categories still provide an efficient tool to study homotopy limits and derived comodules. One major disadvantage is that we must specify by hand these Postnikov presentations and they are not automatically constructed in practice.
Outline. In Section 2, we introduce all the dual notions and methods of [SS03]: fibrantly generated model categories, Postnikov presentations of a model category, left-induced model categories, comonoidal model categories and weak comonoidal Quillen equivalences between them. In Section 3, we provide an example of a Postnikov presentation for unbounded chain complexes. It provides inductive methods to compute homotopy limits of chain complexes. In Section 4, we remind some of the results in [Pér20c] and show that comodules also admit an efficient Postnikov presentation. Finally, we apply our methods to prove our main result on Dold-Kan correspondence for comodules in Section 5.
Acknowledgment. The results here are part of my PhD thesis [Pér20b], and as such, I would like to express my gratitude to my advisor Brooke Shipley for her help and guidance throughout the years. Special thanks to Kathryn Hess for many fruitful conversations.
Notation. We present here the notation and terminology used throughout this paper.
(1) Given a model category M with class of weak equivalence W, we denote M c and M f its subcategories spanned by cofibrant and fibrant objects respectively. We denote the underlying ∞-category (2) The letter k shall always denote a finite product of fields, i.e. a commutative ring k such that it is a product in rings: where each k i is a field, for some 1 ≤ n < ∞. In the literature, such rings are referred to as commutative semisimple Artinian rings.
(3) Let Ch k denote the category of unbounded chain complexes of k-modules (graded homologically).
Denote Ch ≥0 k the category of non-negative chain complexes over k. Both categories are endowed with a symmetric monoidal structure. The tensor product of two (possibly non-negative) chain complexes X and Y is defined by: with differential given on homogeneous elements by: Let sMod k be the category of simplicial k-modules. It is endowed with a symmetric monoidal structure with tensor product defined dimensionwise. We denote the tensor of all three categories above simply as ⊗. The monoidal unit is denoted k, and is either the chain complex k concentrated in degree zero, or the simplicial constant module on k. (4) The category Ch k is endowed with a model structure in which weak equivalences are quasi-isomorphisms, cofibrations are monomorphisms, fibrations are epimorphisms. The category Ch ≥0 k is endowed with a model structure in which weak equivalence are quasi-isomorphisms. cofibrations are monomorphisms, and fibrations are positive levelwise epimorphisms. The category sMod k is endowed with a model structure in which weak equivalences are weak homotopy equivalences, cofibrations are monomorphisms and fibrations are Kan fibrations. These are all combinatorial and symmetric monoidal model categories, see [Hov99] and [SS03] for details. Notice that every object is cofibrant and fibrant. (5) Denote N : sMod k → Ch ≥0 k the normalization functor which is an equivalence of categories. Its inverse is denoted Γ : Ch ≥0 k → sMod k . From [SS03], we obtain two weak monoidal Quillen equivalences: and: In the adjunction (1.1), the normalization functor N is a right adjoint and is lax symmetric monoidal via the Eilenberg-Zilber map. In the adjunction (1.2), the normalization functor N is a left adjoint and is lax (but not symmetric) comonoidal via the Alexander-Whitney map.

Postnikov Presentations of Model Categories
One of the main tool of model categories is to assume the structure to be cofibrantly generated by a pair of sets (see definition in [Hov99, 2.1.17]). If in addition the category is presentable, we say it is combinatorial. In this case, cofibrations and acyclic cofibrations are retracts of maps built out of pushouts and transfinite compositions, and we can inductively construct a cofibrant replacement. Given a symmetric monoidal model category M (as in [Hov99,4.2.6]) in which the tensor product interplays well with the generating acyclic cofibrations (see [SS00,3.3]), then one can guarantee a nice model structure for algebras and modules in M, see [SS00, 4.1]. Moreover, given a weak monoidal Quillen equivalence (as in [SS03,3.6]) between symmetric monoidal model categories, the equivalence lifts to a Quillen equivalence between algebras and modules, see [SS03,3.12]. This induces an equivalence of symmetric monoidal ∞-categories on the underlying ∞categories of the monoidal model categories, see [Pér20a,2.13].
In this section, following [BHK + 15], we introduce the dual concept of fibrantly generated model categories. We make no requirement of "cosmallness" as it is rarely satisfied in practice. Instead when fibrations and acyclic fibrations are retracts of maps built out of pullbacks and towers, we say the model category admits a Postnikov presentation. In [HKRS17,GKR18], model structures on monoidal categories are lifted to categories of coalgebras and comodules. However there is no guarantee that Postnikov presentations are lifted. We present here the notion of comonoidal model category which ensures that the homotopy category is endowed with a monoidal structure via a right derived tensor product. We also introduce the notion of weak comonoidal Quillen equivalences which provides a compatibility of the right derived tensor products.
2.1. Postnikov Presentation. We present the definition of Postnikov presentations, introduced by Kathryn Hess, which is dual to cellular presentations and appeared in [Hes09], [HS14] and [BHK + 15]. We begin with the dual notion of relative cell complex [Hov99, 2.2.9].
Definition 2.1 ([Hes09, 5.12]). Let P be a class of morphisms in a category C closed under pullbacks. Let λ be an ordinal. Given a functor Y : λ op → C such that for all β < λ, the morphism Y β+1 → Y β fits into the pullback diagram: where X ′ β+1 → X β+1 is some morphism in P, and Y β → X β+1 is a morphism in C, and we denote: for any limit ordinal γ < λ. We say that the composition of the tower Y : Proposition 2.3. Let R : C → D be a right adjoint between complete categories. Let P be a class of morphisms in C. Then we have: R(Post P ) ⊆ Post R(P) .
We also recall the dual notion of small object in a category.
Definition 2.4. Let D be a subcategory of a complete category C. We say an object A in C is cosmall relative to D if there is a cardinal κ such that for all κ-filtered ordinals λ (see [Hov99, 2.1.2]) and all λ-towers Y : λ → D op , the induced map of sets: is a bijection. We say that A is cosmall if it is cosmall relative to C itself.
Example 2.5. The terminal object, if it exists, is always cosmall. In procategories, every object is cosmall. A category C is copresentable is its opposite Example 2.6. A category C is copresentable if its opposite category C op is presentable. Therefore in copresentable categories, every object is cosmall. However, if C is presentable then C is not copresentable unless C is a complete lattice, see [AR94, 1.64].
Example 2.7. As noted after [Hov99, 2.1.18], the only cosmall objects in the category of sets are the empty set and the one-point set. In practice, objects in a presentable categories are rarely cosmall.
The dual of the small object argument [Hov99, 2.1.14] can is stated below. As per our above discussion, in practice, the result is not applicable in most categories of interest.
Proposition 2.8 (The cosmall object argument). Let C be a complete category and P be a set of morphisms in C. If the codomains of maps in P are cosmall relative to Post P , then every morphism f of C can be factored functorially as: where δ(f ) is a P-Postnikov tower and γ(f ) admits the left lifting property with respect to all maps in P.
Notation 2.9. Given a class of morphisms A in C, we denote A its closure under formation of retracts.
Definition 2.10. A Postnikov presentation (P, Q) of a model category M is a pair of classes of morphisms P and Q such that the class of fibrations is Post P , the class of acyclic fibrations is Post Q , and for any morphism f : X → Y in M: (a) the morphism f factors as: where i is a cofibration and q is a Q-Postnikov tower; (b) the morphism f factors as: where j is an acyclic cofibration and p is a P-Postnikov tower.
We say in this case that the model category M is Postnikov presented by (P, Q).
Remark 2.11. Since we do not require sets, every model category is trivially Postnikov presented by the classes of all fibrations and acyclic fibrations. Although it was noted in [BHK + 15, 2.13, 2.14] that this trivial presentation can occasionally be useful (see also [Pér20b,B.4.1]), we use more interesting subclasses in this paper, see Theorems 3.10 and 4.13.
Definition 2.12. Let M be a complete model category that admits a Postnikov presentation (P, Q). Given any object X in M, we can provide an inductive fibrant replacement F X as follows. Let * be the terminal object of M. There is an object F X in M that factors the trivial map: where j : X ∼ ֒→ F X is an acyclic cofibration in M, and p is a P-Postnikov tower. This means that F X can be defined as iterated maps of pullbacks along P, starting with (F X) 0 = * .
In our examples, the above inductive fibrant replacement is a countable tower and its limit is a homotopy limit.
Notation 2.13. Denote N the poset {0 < 1 < 2 < · · · }. Let C be any complete category. Objects in C N are diagrams of shape N and can be represented as (countable) towers in C: We denote such object by {X(n)} = (X(n), f n ) n∈N . The limit of the tower is denote lim n X(n).  2.2. Fibrantly Generated Model Categories. We introduce the notion of fibrantly generated as in [BHK + 15]. Our definition of fibrantly generated makes no assumption of cosmallness and is not the dual definition of cofibrantly generated model categories.
Definition 2.15. A model category is fibrantly generated by (P, Q) if the cofibrations are precisely the morphisms that have the left lifting property with respect to Q, and the acyclic cofibrations are precisely the morphisms that have the left lifting property with respect to P. We call P and Q the generating fibrations and generating acyclic fibrations respectively.
Remark 2.16. Just as for Remark 2.11, since we allow P and Q to be classes, any model category is trivially fibrantly generated by its fibrations and acyclic fibrations. In practice, it is useful to have a smaller class to study the model structure.
Proposition 2.17. If a model category is Postnikov presented by a pair of classes (P, Q), then it is fibrantly generated by (P, Q).
Proof. Direct consequence of the retract argument (see [Hov99,     If M is fibrantly generated by (P, Q), then A is fibrantly generated by (R(P), R(Q)).
Remark 2.22. If M is Postnikov presented by (P, Q), there is no reason to expect that A is Postnikov presented by (R(P), R(Q)), as we made no assumption of cosmallness in general.
2.4. Weak Comonoidal Quillen Pair. If (C, ⊗) is a symmetric monoidal model category, then it endows the associated homotopy category Ho(C) with a symmetric monoidal structure endowed with the derived functor ⊗ L , see [Hov99,4.3.2]. Essentially the conditions ensure that the tensor is closed on cofibrant objects and preserves weak equivalence on each variable. Therefore we can total left derive the bifunctor − ⊗ − : C × C → C. If we denote C c the full subcategory of cofibrant objects in C, then the underlying We provide dual conditions such that the homotopy category Ho(C) and the underlying ∞-category N (C f ) W −1 are endowed with a symmetric monoidal structure. We shall weaken the dual definition of [Hov99, 4.2.6] as follows.
Definition 2.23. A (symmetric) comonoidal model category C is a category endowed with both a model structure and a (symmetric) monoidal structure (C, ⊗, I), such that: (i) for any fibrant object X in C, the functors X ⊗ − : C → C and − ⊗ X : C → C preserve fibrant objects and weak equivalences between fibrant objects; (ii) for any fibrant replacement I → f I of the unit, the induced morphism X ∼ = I ⊗ X → f I ⊗ X is a weak equivalence, for any fibrant object X.

The requirement (ii) is automatic if I is already fibrant
Example 2.24. If C is a symmetric monoidal model category C, then C op is a symmetric comonoidal model category. The converse is not true. In particular, we do not claim that the functor X ⊗ − : C → C is a right Quillen functor. It may not even be a right adjoint.
Example 2.25. Any symmetric monoidal model category C in which each object is fibrant and cofibrant is also a symmetric comonoidal model category. For instance C = sMod k , Ch ≥0 k , Ch k . Recall from [Lur17, 2.0.0.1] that to any symmetric monoidal category C, one can define its operator category C ⊗ , such that the nerve N (C ⊗ ) is a symmetric monoidal ∞-category whose underlying ∞-category is N (C), see [Lur17, 2.1.2.21]. Given C a symmetric comonoidal model category with fibrant unit, then C f is a symmetric monoidal category. Apply [Lur17, 4.1.7.6] to obtain a symmetric monoidal structure on the homotopy category of C (just as in [Hov99,4.3.2]) that is coherent up to higher homotopies.
where W is the class of weak equivalences restricted to fibrant objects in C.
We can dualize the notion of a weak monoidal Quillen pair from [SS03, 3.6].
Definition 2.27. Let (C, ⊗, I) and (D, ∧, J) be symmetric comonoidal model categories. A weak comonoidal Quillen pair consists of a Quillen adjunction: where R is lax monoidal such that the following two conditions hold.
(i) For all fibrant objects X and Y in D, the monoidal map: is a weak equivalence in C. (ii) For some (hence any) fibrant replacement λ : J ∼ −→ f J in D, the composite map: is a weak equivalence in C, where the unlabeled map is the natural monoidal structure of R.
A weak comonoidal Quillen pair is a weak comonoidal Quillen equivalence if the underlying Quillen pair is a Quillen equivalence.
Example 2.28. Given any weak monoidal Quillen pair: is a weak comonoidal Quillen pair. Because of our choice of definitions, the converse is not true.
Example 2.29. The Dold-Kan weak monoidal Quillen equivalences (1.1) and (1.2) are also weak comonoidal Quillen equivalences when R = k a finite product of fields. This follows from the fact that their unit and counit in the adjunctions are isomorphisms.
In general, given a weak comonoidal Quillen equivalence, we do not guarantee that it lifts to a Quillen equivalence on associated categories of coalgebras and comodules. In other words, we do not provide a dual version of [SS03,3.12]. The essential reason being that there is no guarantee in general to obtain a Postnikov presentation on left-induced model structures of coalgebras or comodules such that the limit of the Postnikov towers behaves well with respect to the forgetful functor.
Instead, we show that homotopy coherent (co)algebras and (co)modules are equivalent as ∞-categories. In Section 5, we provide an example of a weak comonoidal Quillen pair that lifts to the associated categories of comodules. One can adapt the proof in [Pér20a] to obtain the following result.
where C f ⊆ C and D f ⊆ D are the full subcategories of fibrant objects. If L and R form a weak comonoidal Quillen equivalence, then R is a symmetric monoidal equivalence of ∞-categories.

Example: Unbounded Chain Complexes
We show here that the category Ch k of unbounded chain complexes over k is fibrantly generated by a pair of sets (Theorem 3.4) and admits an efficient Postnikov presentation on its model structure (Theorem 3.10). In particular, this provides new methods to compute homotopy limits in Ch k (Remark 3.11) and thus in the ∞-category of modules over the Eilenberg-Mac Lane spectrum Hk. The result was proved in [Hes09] and [BHK + 15] for finitely generated non-negative chain complexes over a field.
3.1. The Generating Fibrations. We present here sets of generating fibrations and acyclic fibrations for Ch k . A similar result was proved in early unpublished versions of [Sor16] in [Sor10, 3.1.11, 3.1.12] for non-negative chain complexes over a field.
Definition 3.1. Let V be a k-module. Let n be any integer. Denote the n-sphere over V by S n (V ), the chain complex that is V concentrated in degree n and zero elsewhere. Denote the n-disk over V by D n (V ), the chain complex that is V concentrated in degree n − 1 and n, with differential the identity. We obtain the epimorphisms D n (V ) → S n (V ): This defines functors: The map defined above is natural, i.e. we have a natural transformation D n (−) ⇒ S n (−), for all n ∈ Z. When V = k, we simply write D n and S n . Notice that we have a short exact sequence of chain complexes: Definition 3.2. Define P and Q to be the following sets of maps in Ch k : We thicken the sets P and Q to classes P ⊕ and Q ⊕ of morphisms in Ch k : Clearly, the maps in P and P ⊕ are fibrations in Ch k and the maps in Q and Q ⊕ are acyclic fibrations in Ch k .
Remark 3.3. When k is a field, as every k-module is free, we get: Theorem 3.4. The model category of unbounded chain complexes Ch k is fibrantly generated by the pair of sets (P, Q).
We prove the above theorem with Lemmas 3.8 and 3.9 below. We first prove the following well-known result. Given any chain complex X, we denote the n-boundaries of X by B n (X) and the n-cycles of X by Z n (X).
Proposition 3.5. Let X be a chain complex in Ch k . Then X is split as a chain complex and we have a non-canonical decomposition: In particular, any chain complex X can be decomposed non-canonically as product of disks and spheres: Proof. We have the following short exact exact sequences of k-modules: Since any short exact sequence splits, we can choose sections (the dashed maps denoted above), such that we obtain the following isomorphism of k-modules: An investigation of the differentials X n → X n−1 give the desired result.
Notation 3.6. Given any class of maps A in a category C, we denote LLP(A) the class of maps in C having the left lifting property with respect to all maps in A.
Proof. Since P ⊆ P ⊕ , we get LLP(P ⊕ ) ⊆ LLP(P) . Suppose now f is in LLP(P), let us argue it also belongs in LLP(P ⊕ ). Suppose we have a diagram: for some basis λ. In particular, the choice of the splitting V ⊕ W → V induces the commutative diagram: where D n α and S n α are a copies of D n and S n . Since f is in LLP(P), we obtain a lift ℓ α : Y → D n α , for each α. It induces a lift ℓ : Y → α D n α which restricts to Y → D n (V ) via the retracts (dashed maps in the diagram). Proof. Let f : X → Y be a map in LLP(Q), let us show it is a cofibration in Ch k , i.e. a monomorphism. Following Proposition 3.5, we decompose X as: Lemma 3.9. Maps in the set P are the generating fibrations in Ch k .
Proof. Notice that LLP(P) ⊆ LLP(Q) as any lift Y → D n in the following commutative diagram induces the dashed lift: In particular, Lemma 3.8 shows that maps in LLP(P) are monomorphisms. Let f : X → Y be map in LLP(P) and let us show it is a quasi-isomorphism. Since f is a monomorphism, there is an induced short exact sequence in Ch k : It remains to show that K is acyclic. Notice first that K is defined as the pushout: and so, since f is in LLP(P), then 0 → K is in LLP(P). Following Proposition 3.5, decompose K as: where V n = H n (K). Then we obtain a map by projection: that factors through the non-trivial map: But this is only possible when V n = 0, hence K must be acyclic. Thus f is a quasi-isomorphism.

A Postnikov Presentation.
In Definition 3.2 and Theorem 3.4, we introduced sets of generating fibrations and acyclic fibrations for Ch k . We now show they induce a Postnikov presentation on Ch k .
Theorem 3.10. The pair (P ⊕ , Q) is a Postnikov presentation of the model category of unbounded chain complex Ch k .
We shall prove Theorem 3.10 with Lemmas 3.14 and 3.15 below.
Remark 3.11. Our arguments in Theorem 3.10 can be generalized to the category Ch k D , the category of functors D → Ch k , where D is a small category. Indeed, endow Ch k D with its injective model structure with levelwise weak equivalences and cofibrations (it exists by [HKRS17,3.4.1]). Using the same notation as [BHK + 15,4.10], the pair (P ⊕ ⋔ D, Q ⋔ D) is a Postnikov presentation for Ch k D . This provides inductive arguments to compute homotopy limits in Ch k .
Remark 3.12. We were not able to restrict ourselves to the set P and had to consider the class P ⊕ . We note here a few basic results.
The maps S n → 0 are in Post P as they are obtained as pullbacks: Similarly, for any k-module V , the maps S n (V ) → 0 are in Post P⊕ . (iii) Since D n → 0 is the composite D n −→ S n −→ 0, we see that Q ⊆ Post P , and thus Post Q ⊆ Post P ⊆ Post P⊕ by Proposition 2.2. (iv) Although P ⊕ Post P , we have P ⊕ ⊆ Post P (see Notation 2.9). Indeed, for any k-module V , any map D n (V ) → S n (V ) is the retract of a map D n (F ) → S n (F ) where F is a free k-module. Then, for λ a basis of F , we have the retract in Ch k : induced by the split short exact sequence in k-modules: Lemma 3.13. Let X be any chain complex over k. Then the trivial map X → 0 is a P ⊕ -Postnikov tower. If X is acyclic, the trivial map is a Q ⊕ -Postnikov tower.
Lemma 3.14. Every acyclic fibration in Ch k is a retract of a Q-Postnikov tower. Every map in Ch k factors as a cofibration followed by a Q-Postnikov tower.
Proof. We provide two proofs by presenting two different factorizations. The first one has the advantage to be functorial but harder to compute. The second is not functorial but is easier to compute. Let us do the first possible factorization. By Theorem 3.4, the set Q of maps in Ch k is the set of generating acyclic fibrations. Their codomain is the terminal object in Ch k and is thus cosmall. We can then apply the cosmall object argument (Proposition 2.8) to obtain the desired factorization. For the second possible factorization, start with any morphism f : X → Y in Ch k . Choose a decomposition of X by using Proposition 3.5: X ∼ = n∈Z S n (V n ) ⊕ D n (W n ), for some collection of k-modules V n and W n . The inclusions S n (V n ) ֒→ D n+1 (V n ) define then a monomorphism in Ch k : Since V n and W n are projective k-modules, they can be embedded into free k-modules, say F n and G n , with basis λ n and γ n . Then we obtain the following monomorphisms in Ch k : Thus, we get the monomorphism in Ch k : X n∈Z λn,γn D n+1 ⊕ D n . Denote Z the acyclic chain complex n∈Z λn,γn D n+1 ⊕D n . We obtain the desired second factorization in Ch k : where the map q is the projection onto Y , which is indeed a Q-Postnikov tower by Proposition 2.2.
Lemma 3.15. Every fibration in Ch k is a retract of a P ⊕ -Postnikov tower and every map in Ch k factors as an acyclic cofibration followed by a P ⊕ -Postnikov tower.
We prove the above lemma at the end of this section.
Remark 3.16. Unlike Lemma 3.14, we cannot use the cosmall object argument in order to prove Lemma 3.15. Indeed, as noted in [Sor10], the codomains S n of maps in the set P are not cosmall relative to Post P . Indeed, let Y k = S n for all k ≥ 0 and Y k+1 → Y k be the zero maps. Let Y = lim k≥0 Y k be the limit in Ch k . The set map: colim is not a bijection. Indeed, the map is equivalent to the map: which is never a bijection. A similar argument can be applied to show that the codomains S n (V ) of the maps in the class P ⊕ are not cosmall relative to Post P⊕ , for any k-module V .
Lemma 3.17. Let X be a chain complex in Ch k . Let V be a k-module. Let n be an integer in Z. Given a surjective linear map f n : X n → V non-trivial only on n-cycles, there is a map of chain complexes f : X → S n (V ), and the pullback chain complex P in the following diagram: has homology: H i (P ) ∼ = ker (H n (f )) i = n, H i (X) i = n, and we have P i = X i for i = n − 1 and P n−1 = X n−1 ⊕ V .
Proof. By construction, since pullbacks in Ch k are taken levelwise, for i = n, n − 1, we have the pullbacks of k-modules: fn Thus P n−1 ∼ = X n−1 ⊕ V and P i = X i for any i = n − 1. The differential P n → P n−1 is the linear map X n X n−1 ⊕ V, dn⊕f and the differential P n−1 → P n−2 is the linear map: where the unlabeled map is the natural projection. All the differentials P i → P i−1 for i = n, n − 1 are the differentials X i → X i−1 of the chain complex X. Clearly, we get H i (P ) = H i (X) for i = n, n − 1. For i = n − 1, by Proposition 3.5, we can choose a decomposition: The differential d n : X n → X n−1 sends the factor B n−1 (X) in X n to itself, and the factor H n (X) ⊕ B n (X) to zero. By definition, the map f n : X n → V sends the factor H n (X) in X n to the image of f n , which is V since f n is surjective, and the factor B n−1 (X) ⊕ B n (X) to zero. Thus the image of the differential P n −→ P n−1 , is precisely B n−1 (X) ⊕ V . Therefore, we obtain: For i = n, notice that the n-boundaries of P are precisely the n-boundaries of X, the n-cycles of P are the n-cycles x in X such that f n (x) = 0. Since f n : X n → V is entirely defined on the copy H n (X) in X n , we get from the commutative diagram: fn Hn(f ) that H n (P ) ∼ = ker(H n (f )).
Lemma 3.18. Let j : X → Y be a monomorphism in Ch k (i.e. a cofibration), such that it induces a monomorphism in homology in each degree. Let n be a fixed integer in Z. Then the map j factors in Ch k as: where F n (Y ) is a chain complex built with the following properties.
• The chain map F n (p j ) : F n (Y ) → Y is a P ⊕ -Postnikov tower.
• The chain map F n (j) : X → F n (Y ) is a monomorphism (i.e. a cofibration in Ch k ).
• The k-module (F n (Y )) i differs from Y i only in degree i = n − 1.
• In degrees i = n in homology, we have H i (F n (Y )) ∼ = H i (Y ) and the maps: In particular, the maps H i (F n (j)) are monomorphisms. Moreover, if the maps H i (j) are isomorphisms, then so are the maps H i (F n (j)). • In degree n in homology, the map H n (F n (j)) : H n (X) Proof. We construct below the chain complex F n (Y ) explicitly using Lemma 3.17. By Proposition 3.5, we can decompose Y n as: where Y n is the direct sum of the copies of the boundaries. Denote the k-module V = coker(H n (j)) and define the linear map f n : Y n → V to be the natural projection. In particular, the map f n sends n-boundaries of Y to zero. This defines a chain map: f : Y −→ S n (V ). Notice that since j : X → Y is a monomorphism, we get j(X n ) ⊆ Y n , and so, by construction of f , we get that the composite: X Y S n (V ), j f is the zero chain map. We obtain F n (Y ) as the following pullback in Ch k , with a chain map F n (j) induced by the universality of pullbacks: By construction, the induced chain map F n (p j ) : F n (Y ) → Y is in Post P⊕ . From the commutativity of the diagram: since j is a monomorphism, so is F n (j). Since H i (j) is a monomorphism for i ∈ Z, then so is H i (F n (j)). By Lemma 3.17, we get H i (F n (Y )) ∼ = H i (Y ) for all i = n. For i = n, we get: as we have the short exact sequence of k-vector spaces: since V = coker(H n (j)). Thus H n (F n (j)) is an isomorphism as desired.
Proof of Lemma 3.15. The first statement follows from the second using the retract argument. Given a chain map f : X → Y , we build below a chain complex W as a tower in Ch k using Lemma 3.18 repeatedly so that f factors as: where j is a monomorphism and a quasi-isomorphism, and all the vertical maps and p are in Post P⊕ . The composition of all the vertical maps and p is a chain map W → Y which is in Post P⊕ , by Proposition 2.2. We first start by noticing the following factorization: induced by the following pullback in Ch k : The map p is in Post P⊕ by Lemma 3.13 and Proposition 2.2. By commutativity of the upper triangle, we see that the monomorphism j induces a monomorphism in homology. We denote W = X ⊕ Y . The second step is to replace the map j : X → W by a chain map j + : X → W + that remains a cofibration, a monomorphism in homology in negative degrees, and an isomorphism in homology in non-negative degrees. We construct W + as the limit lim n≥0 (G + n (W )) in Ch k of the tower of maps: where each G + n (p) is in Post P⊕ . The map j + : X → W + is induced by the monomorphisms G + n (j) : X → G + n (W ) which are compatible with the tower: and G + n (j) induces an isomorphism in homology in degrees i, for 0 ≤ i ≤ n, and a monomorphism otherwise. We construct the chain complexes G + n (W ) of the tower inductively as follows. • For the initial step, apply Lemma 3.18 to the monomorphism j : X → W , for n = 0. Denote G + 0 (W ) := F 0 (W ). The cofibration G + 0 (j) defined as the chain map: , is an isomorphism in homology in degree 0, and a monomorphism in other degrees. The chain map G + 0 (p) defined as the map: • For the inductive step, suppose, for a fixed integer n ≥ 0, the chain complex G + n (W ) is defined, together with a cofibration G + n (j) : X → G + n (W ) inducing an isomorphism in homology for degrees i, where 0 ≤ i ≤ n, and a monomorphsim in homology for other degrees. Apply Lemma 3.18 to the monomorphism G + n (j) for the degree n + 1. Denote: ). The cofibration G + n+1 (j) defined as the chain map: , is an isomorphism in homology in degrees i where 0 ≤ i ≤ n + 1, and a monomorphism in other degrees. We obtain a P ⊕ -Postnikov tower G + n+1 (p) defined as the chain map: , such that the following diagram commutes: The induced map j + : X → W + is a monomorphism of chain complexes. Indeed, for any fixed i ∈ Z, we have: , for all i ∈ Z, and so j + is a monomorphism in negative degrees in homology, and an isomorphism in homology in non-negative degrees.
The last step is to replace the map j + : X → W + by the desired chain map j : X → W that is an acyclic cofibration. We construct W similarly as W + (inductively applying Lemma 3.18) but in negative degrees. We build W as the limit lim n≥0 (G − n (W + )) in Ch k of the tower of maps: where each G − n (p) is in Post P⊕ . The map j : X → W is induced by the monomorphisms G − n (j) : X → G − n (W + ) which are compatible with the tower: and G − n (j) induces an isomorphism in homology in degrees i, for i ≥ −n, and a monomorphism otherwise. Similarly as the positive case, the map j : X → W can be shown to be a monomorphism and quasiisomorphism, hence an acyclic cofibration, as desired.

Example: Simplicial and Connective Comodules
We introduce here, in Theorem 4.13, a Postnikov presentation for simplicial and connective differential graded comodules, over simply connected coalgebras (Definition 4.11). This provides us with an efficient inductive fibrant replacement as a Postnikov tower (Corollary 4.21). We begin to adapt our results on Ch k in previous section to Ch ≥0 k and sMod k . and: We also obtain sets and classes in sMod k via the equivalence Γ : Ch ≥0 k → sMod k . For n ≥ 0, we denote K(V, n) := Γ(S n (V )) which corresponds precisely to the simplicial model of the Eilenberg-Mac Lane space of V in degree n. For n ≥ 1, we denote P K(V, n) := Γ(D n (V )) which corresponds to the based path of K(V, n). Define: and: The Quillen adjunction: Ch ≥0 k Ch k : τ ≥0 ⊥ shows that the model structure on Ch ≥0 k is in fact leftinduced from Ch k . From Theorem 3.4 and Proposition 2.21, we obtain that Ch ≥0 k and sMod k are fibrantly generated. As pointed out in Remark 2.22, it is not automatic that we obtain Postnikov presentations for Ch ≥0 k and sMod k from the one in Theorem 3.10. However, we claim we can adapt the arguments in previous section to obtain the following result. Alternatively, one can apply Theorem 4.13 below with C = k. (i) The model category of non-negative chain complexes Ch ≥0 k is fibrantly generated by the pair of set (P ≥0 , Q ≥0 ). The pair (P ≥0 ⊕ , Q ≥0 ) is a Postnikov presentation of Ch ≥0 k . (ii) The model category of simplicial k-modules sMod k is fibrantly generated by the pair of sets (P ∆ , Q ∆ ).
The pair (P ∆ ⊕ , Q ∆ ) is a Postnikov presentation of sMod k .

Left-Induced Model Structures on Comodules.
We present here the model structures for comodules in sMod k and Ch ≥0 k , due to [HKRS17,GKR18]. We first introduce general definitions. Definition 4.3. Let (C, ⊗, I) be a symmetric monoidal category. A coalgebra (C, ∆, ε) in C consists of an object C in C together with a coassociative comultiplication ∆ : C → C ⊗ C, such that the following diagram commutes: and admits a counit morphism ε : C → I such that we have the following commutative diagram: The coalgebra is cocommutative if the following diagram commutes: where τ is the twist isomorphism from the symmetric monoidal structure of C. A morphism of coalgebras f : (C, ∆, ε) → (C ′ , ∆ ′ , ε ′ ) is a morphism f : C → C ′ in C such that the following diagrams commute: Definition 4.4. Let (C, ⊗, I) be symmetric monoidal category. Let (C, ∆, ε) be a coalgebra in C. A right comodule (X, ρ) over C, or a right C-comodule, is an object X in C together with a coassociative and counital right coaction morphism ρ : X → X ⊗ C in C, i.e., the following diagram commutes: The category of right C-comodules in C is denoted CoMod C (C). Similarly, we can define the category of left C-comodules where objects are endowed with a left coassociative counital coaction X → C ⊗ X and we denote the category by C CoMod(C).
Remark 4.5. If C is a cocommutative comonoid in C the categories of left and right comodules over C are naturally isomorphic: C CoMod(C) ∼ = CoMod C (C). In this case, we omit to mention if the coaction is left or right.
Remark 4.6. Since a coalgebra in C is an algebra in C op , then we can define the category of right comodules as modules in the opposite category: CoMod C (C) = (Mod C (C op )) op , and similarly for the left case.
Remark 4.7. All the results for this paper will be stated for right comodules. But each statement remains valid for left comodules, and in fact more generally for any bicomodule over the same coalgebra.
Definition 4.8. For any object X in C, we say that X ⊗ C is the cofree right C-comodule generated by X. Similarly, we can define the cofree left C-comodule generated by X as C ⊗ X.
Proposition 4.9 ([Pér20c, 2.12]). Let M be either the monoidal model category sMod k or Ch ≥0 k . Let C be a coalgebra in M. Then the category of right C-comodules in M admits a combinatorial model category left-induced from the forgetful-cofree adjunction: In particular, U preserves and reflects cofibrations and weak equivalences. The model structure is combinatorial and simplicial. Every object is cofibrant.
Corollary 4.10. Let k be a finite product of fields.
(i) Let C be a coalgebra in Ch ≥0 k . The model category of right C-comodules CoMod C (Ch ≥0 k ) is fibrantly generated by the pair of sets (P ≥0 ⊗ C, Q ≥0 ⊗ C).
(ii) Let C be a coalgebra in sMod k . The model category of right C-comodules CoMod C (sMod k ) is fibrantly generated by the pair of sets (P ∆ ⊗ C, Q ∆ ⊗ C).

Postnikov Presentations.
We now introduce a Postnikov presentation for comodules in sMod k and Ch ≥0 k . We follow the approach of [Hes09] in which the finitely generated case was treated. We first need to restrict the coalgebras considered.
Definition 4.11. Let R be any commutative ring. A chain complex X over R is simply connected if: X 0 = R, X 1 = 0 and X i = 0 for all i < 0. A simplicial R-module X is simply connected if X 0 = R and there are no non-degenerate 1-simplices.
Over a finite product of fields k, every simply connected simplicial modules or chain complexes is weakly equivalent to a simply connected object in the usual sense.
Remark 4.12. A simplicial R-module X is simply connected if and only if N(X) is a simply connected chain complex. A chain complex X over R is simply connected if and only if Γ(X) is a simply connected simplicial module.
For any chain complex C and any k-module V , we see that the i-th term of the chain complex S n (V ) ⊗ C is the k-module V ⊗ C i−n . If we choose C to be a simply connected differential graded k-coalgebra, we get: Thus, around the n-th term, the chain complex S n (V ) ⊗ C is similar to S n (V ). We can therefore modify the homology of a C-comodule for a specific degree without modifying the lower degrees. Therefore the arguments of previous section can be applied.
Theorem 4.13. Let k be a finite product of fields.
Using the vocabulary of [HS14], we essentially show that the comonad − ⊗ C is tractable and allows the inductive arguments and thus by [HS14,5.8] we indeed have that (P ≥0 ⊕ ⊗ C, Q ≥0 ⊗ C) and (P ∆ ⊕ ⊗ C, Q ∆ ⊗ C) are Postnikov presentations. We shall prove (i) of Theorem 4.13 with Lemmas 4.14 and 4.15 below. We prove (ii) of Theorem 4.13 with Lemmas 4.14 and 4.16. This will provide us with a very explicit inductive fibrant replacement for comodules as we will see in Corollary 4.21.
We now start proving Theorem 4.13. The following lemma follows from [Hes09, 1.15] and does not require any connectivity assumption.
Lemma 4.14. Let k be a finite product of fields.
(i) Let C be a coalgebra in Ch ≥0 k . Every acyclic fibration in CoMod C (Ch ≥0 k ) is a retract of a (Q ≥0 ⊗ C)-Postnikov tower. Every map in CoMod C (Ch ≥0 k ) factors as a cofibration followed by a (Q ≥0 ⊗ C)-Postnikov tower.
(ii) Let C be a coalgebra in sMod k . Every acyclic fibration in CoMod C (sMod k ) is a retract of a (Q ∆ ⊗ C)-Postnikov tower. Every map in CoMod C (sMod k ) factors as a cofibration followed by a (Q ∆ ⊗ C)-Postnikov tower.
Proof. We only do the case Ch ≥0 k . The simplicial case is proved in an identical way. We provide two proofs by presenting two different factorizations. The first one has the advantage to be functorial but harder to compute. The second is not functorial but is easier to compute. Let us do the first possible factorization. By Theorem 3.4, the set Q ≥0 ⊗ C of maps in CoMod C (Ch ≥0 k ) is the set of generating acyclic fibrations. Their codomain is the terminal object in CoMod C (Ch ≥0 k ) and is thus cosmall. We can then apply the cosmall object argument (Proposition 2.8) to obtain the desired factorization.
For the second possible factorization, start with any morphism f : X → Y in CoMod C (Ch ≥0 k ). As in Lemma 3.14, we can obtain a monomorphism of chain complexes U (X) ֒→ Z where Z is an acyclic chain complex. By adjointness, this defines a map of C-comodules ι : X ֒→ Z ⊗ C that remains a monomorphism. We obtain the desired second factorization in CoMod C (Ch ≥0 k ): where the map q is the projection onto Y , which is indeed a (Q ≥0 ⊗ C)-Postnikov tower by Proposition 2.2.
We can argue completely similarly for the simplicial case.
Lemma 4.16. Let C be a simply connected coalgebra in sMod k . Every fibration in CoMod C (sMod k ) is a retract of a (P ∆ ⊕ ⊗ C)-Postnikov tower. Any morphism in CoMod C (sMod k ) factors as a cofibration followed by a (P ∆ ⊕ ⊗ C)-Postnikov tower.
Proof. We argue as in [Pér20c, A.6] and thus show less details. Any morphism X → Y in CoMod C (sMod k ) factors through W = (U (X) ⊗ C) ⊕ Y . We build now a tower {G n (W )} of C-comodules that stabilizes in each degree. For the case n = 0, it is easy to adapt [Pér20c, A.9] in the simplicial case. Suppose we have constructed the cofibraton X → G n (W ). Then define G n+1 (W ) as follows. Define V to be the cokernel of the (n + 1)-th homology of: Then we obtain a map N(U (G n (W ))) → S n+1 (V ) and thus a map in sMod k : U (G n (W )) → K(V, n + 1).
We now clarify how to compute limits of certain towers in comodules. The towers constructed in Lemmas 4.15 and 4.16 are similar to the one of Lemma 3.15. Eventually in each degree, the layers of the tower have the correct homology. We make the idea precise in the following definition and results.  Lemma 4.19. Let M be either sMod k or Ch ≥0 k . Let {X(n)} be a tower in M that stabilizes in each degree. Let C be any object M. Then the tower X(n) ⊗ C in M also stabilizes in each degree and we have: Proof. We prove only the case M = Ch ≥0 k . For all n ≥ 0, and all 0 ≤ i ≤ n, we have: as 0 ≤ a ≤ i ≤ n. This argument generalizes in higher degrees and thus shows that the desired tower stabilizes in each degree. For all i ≥ 0, notice that both lim n X(n) ⊗ C i and lim n X(n) ⊗ C i are isomorphic to Corollary 4.20. Let M be either sMod k or Ch ≥0 k . Let C be a coalgebra in M. Let {X(n)} be a tower in CoMod C (M) that stabilizes in each degree. Then the natural map: is an isomorphism in M.
Proof. This follows directly from Lemma 4.19 as U preserves and reflects a limit precisely when the comonad −⊗C : M → M preserves that limit. In detail, if we denote X := lim n U (X(n)), then the coaction X → X ⊗C is constructed as follows. For each degree i ≥ 0, the map X i → (X ⊗ C) i is entirely determined by the coaction X(i + 1) → X(i + 1) ⊗ C.
The following crucial result follows directly from Lemma 4.15 where we apply the factorization to a trivial map of right C-comodule X → 0. We recall that we define homotopy limits of towers as limits of fibrant towers as in Proposition 2.14.
Corollary 4.21. Let M be either sMod k or Ch ≥0 k . Let C be a simply connected coalgebra in M. Let X be any right C-comodule in M. Then there exists a countable tower {X(n)} in CoMod C (M) with limit X := lim C n X(n) where the right C-comodules X(n) are built inductively as follows. • Define X(0) to be the trivial C-comodule 0.
• Suppose X(n) was constructed for a certain n ≥ 1. Then there exists a certain k-module V n such that X(n + 1) is defined as the following pullback in CoMod C (M): and we obtain the short exact sequence in CoMod C (Ch ≥0 k ): and we obtain the short exact sequence in CoMod C (sMod k ): The tower {X(n)} enjoys the following properties.
There exists an acyclic cofibration of right C-comodules X X.
(iv) The tower {X(n)} stabilizes in each degree. In particular: U ( X) = U (lim C n X(n)) ∼ = lim n (U (X(n))). (v) Each map X(n + 1) → X(n) for n ≥ 0 is a fibration in CoMod C (M), and its underlying map U (X(n + 1)) → U (X(n)) is also a fibration in M. In particular X is the homotopy limit of {X(n)} in CoMod C (M) and we have: U ( X) ≃ U (holim C n X(n)) ≃ holim n (U (X(n))). Remark 4.23. The Postnikov tower construction is not functorial. Given a map of C-comodules X → Y we do not obtain natural maps X(n) → Y (n).

4.3.
A Comonoidal Model Structure. We defined a symmetric monoidal structure on comodules. For simplicity, we work over cocommutative coalgebras, but one can work instead with non-cocommutative coalgebra and obtain a non-symmetric monoidal structure on bicomodules over the coalgebra.
Definition 4.24. Let M be either sMod k or Ch ≥0 k . Let X and Y be C-comodules in M. Define their cotensor product X C Y to be the following equalizer in M: where the two parallel morphisms are induced by the coactions X → X ⊗ C and Y → C ⊗ Y . Since the tensor product preserves equalizers, we obtain a C-coaction on X C Y .
We record some facts on the cotensor product. . For any C-comodule X, we have a natural isomorphism of C-comodules:   (lim C n X(n)) C Y ∼ = lim C n (X(n) C Y ). Proof. Equalizers of towers that stabilize in each degree also stabilize in each degree. Then the result follows from Lemma 4.19.
The following results were proved in [Pér20c] for M = Ch ≥0 k . We claim we can adapt the arguments to obtain the same results for M = sMod k .  Let M be either sMod k or Ch ≥0 k . From Proposition 2.26, we see that we can (right) derive the cotensor product of comodules and endow the homotopy category of C-comodules with a symmetric monoidal structure. More generally, we can endow a symmetric monoidal structure to the underlying ∞-category of M. From [Pér20c, 1.1, 1.2], this endows a symmetric monoidal structure to the ∞-category of homotopy coherent comodules over C in D ≥0 (k), the ∞-category of connective comodules over the Eilenberg-Mac lane spectrum Hk. A priori, the structure depends on the choice of M as the cotensor products of simplicial comodule and connective differential graded comodules are not isomorphic. The next section will now prove the monoidal structure is independent from the choice of M.

Application: Dold-Kan Correspondence For Comodules
We prove here in Theorem 5.1 that the Dold-Kan correspondence lifts to a Quillen equivalence on the categories of (right) comodules, over a simply connected coalgebra. Moreover, we show in Theorem 5.3 that the derived cotensor products induced by Theorem 4.30 are equivalent. A crucial tool in our arguments is the Postnikov tower (Definition 4.22) as a fibrant replacement for our model categories of comodules 5.1. Lifting The Dold-Kan Correspondence. We dualize the construction appearing in [SS03,3.3]. Let L : (C, ⊗, I) → (D, ∧, J) be a lax comonoidal functor between symmetric monoidal categories, with right adjoint R. Let C be a coalgebra in C. Then the functor L lifts to (right) comodules: However, if we suppose R to only be lax monoidal, then R does not lift to comodules. Suppose cofree objects exist and equalizers exist. We define the correct right adoint R C on comodules by the following commutative diagram of adjoints: In particular, for any object M in D, we obtain: More specifically, the functor R C is defined as the following equalizer in CoMod C (C): where X is a right L(C)-comodule in D. We have omitted the forgetful functor U for simplicity. The first parallel morphism in the equalizer is induced by the coaction X → X ∧ L(C). The second parallel morphism is induced by the universal property of cofree C-comodules on the map in C: where the first map is induced by the unit of adjunction between L and R, and the second map is induced by the lax monoidal structure of R. We apply the above construction to the adjunctions (1.1) and (1.2) of the Dold-Kan correspondence.
Theorem 5.1. Let C be a simply connected coalgebra in Ch ≥0 k . Let D be a simply connected coalgebra in sMod k . Then the induced adjunctions: and: are Quillen equivalences.
Remark 5.2. Using the rigidification result of [Pér20c], we can prove Theorem 5.1 using ∞-categories. However, we believe there is value to present a proof that does not depend on the rigidification, and in any cases, both rely on Corollary 4.21. Nevertheless, let us explain how to prove Theorem 5.1 using ∞-categories. Let W dg be the class of quasi-isomorphisms in Ch ≥0 k . Let W ∆ be the class of weak homotopy equivalences in sMod k . Let C be a simply connected coalgebra in Ch ≥0 k . Denote W dg,CoMod denote the class of quasiisomorphisms between differential graded C-comodules. Let W ∆,CoMod denote the class of weak homotopy equivalences between simplicial Γ(C)-comodules. Given C a symmetric monoidal ∞-category, and T an A ∞ -coalgebra in C, we denote CoMod T (C) the ∞-category of right T -comodules in C, i.e., right T -modules in the opposite C op . We obtain the following diagram of ∞-categories: The vertical maps are equivalences of ∞-categories by [Pér20c, 1.1]. The top horizontal map is an equivalence by [Pér20a, 4.1]: it states that the Dold-Kan correspondence induces a symmetric monoidal equivalence on the underlying ∞-categories of sMod k and Ch ≥0 k . In particular, homotopy coherent comodules are equivalent. Therefore, the bottom horizontal map is an equivalence of ∞-categories. In particular, the left adjoint of the adjunction (5.1) is a Quillen equivalence. The proof is similar for the other adjunction.
Proof of Theorem 5.1. We shall only prove the statements for the adjunction (5.1). The other one has an entirely similar proof.
Since the functor Γ : Ch ≥0 k → sMod k is left Quillen, and cofibrations and weak equivalences are determined by the underlying categories, then the left adjoint functor Γ : is also left Quillen. It also clearly reflects weak equivalences between (cofibrant) objects. Thus, by [Hov99,1.3.16] we only need to show that the counit: ΓN C (X) −→ X, is a weak homotopy equivalence, for any fibrant Γ(C)-comodule X.
For this matter, let {X(n)} be the Postnikov tower of X, and denote X the (homotopy) limit of the tower, see Corollary 4.21. Then X is a retract of X and thus we only need to show the weak homotopy equivalence when applied on X. We prove by induction on the tower. Clearly the map ΓN C (X(0)) −→ X(0) is a weak homotopy equivalence as it is the trivial map.
Suppose X is a cofree comodule M ⊗ Γ(C), for M in sMod k . Then the formula: combined with the weak homotopy equivalence: induced by the fact that Γ : Ch ≥0 k → sMod k is part of a weak monoidal Quillen equivalence shows that ΓN C (M ⊗Γ(C)) −→ M ⊗Γ(C) is a weak homotopy equivalence. In particular, we obtain that ΓN C (X(1)) −→ X(1) is a weak homotopy equivalence. Now suppose we have shown ΓN C (X(n)) −→ X(n) for some n ≥ 1. Since X(n + 1) is the homotopy pullback in CoMod Γ(C) (sMod k ) (and in sMod k ): for some epimorphism P → Q in sMod k , then as N C is a right Quillen functor, it preserves homotopy pullbacks and thus we get the following homotopy pullback in CoMod C (Ch ≥0 k ) (and in Ch ≥0 k ): as N preserves fibrations. In particular, notice that the induced maps N C (X(n + 1)) → N C (X(n)) are fibrations in CoMod C (Ch ≥0 k ) and Ch ≥0 k . Moreover, the tower {N C (X(n))} stabilizes in each degree (as it is the equalizer of towers that stabilizes in each degree by Lemma 4.19). Thus the homopy limits of {N C (X(n))} in CoMod C (Ch ≥0 k ) and in Ch ≥0 k are equivalent to N C ( X). Consider the above pullback in Ch ≥0 k , then as Γ : Ch ≥0 k → sMod k is a right Quillen functor, we obtain the following homotopy pullback in sMod k : Applying the weak homotopy equivalences: induced again by Γ : Ch ≥0 k → sMod k being part of a weak monoidal Quillen equivalence, we obtain by [Hov99, 5.2.6] the homotopy pullback in sMod k : By induction, since Γ N(X(n)) → X(n) is a weak homotopy equivalence, we obtain that Γ N(X(n + 1)) → X(n + 1) is a weak homotopy equivalence. We conclude that: is a weak homotopy equivalence by noticing that (we omit the underlying functor): ≃ Γ(holim C n N C (X(n))) ≃ Γ(holim n N C (X(n))) ≃ holim n ΓN C (X(n)) ≃ holim n X(n) ≃ X.
We have used that the homotopy limits of {X(n)} and {N C (X(n))} can be computed in their underlying categories and that N C is a right Quillen functor.

A Weak Comonoidal Quillen
Equivalence. Let C be a cocommutative coalgebra in Ch ≥0 k . Consider Γ as a lax symmetric comonoidal left adjoint functor induced by the Eilenberg-Zilber map, as in the adjunction (1.1). Let X and Y be C-comodules. The comonoidal structure on Γ induces a unique dashed map on the equalizers: Thus Γ lifts to a functor CoMod C (Ch ≥0 k ) → CoMod Γ(C) (sMod k ) that is lax symmetric comonoidal with respect to their cotensor products.
Recall that we denote by D ≥0 (k) the ∞-category of connective comodules over the Eilenberg-Mac Lane spectrum Hk. It is equivalent to the underlying ∞-categories of sMod k and Ch ≥0 k .
Theorem 5.3. Let k be a finite product of fields. Let C be a simply connected cocommutative coalgebra in Ch ≥0 k . Then the induced adjunction: is a weak comonoidal Quillen equivalence. In particular, we obtain an equivalence of symmetric monoidal ∞-categories: with respect to their derived cotensor product of comodules.
Remark 5.4. From the adjunction (1.2), we can endow N with a lax comonoidal left adjoint structure and repeat our above arguments. However, if D is a cocommutative coalgebra in sMod k , then there is no guarantee that N(D) is also cocommutative. Thus CoMod N(D) (Ch ≥0 k ) is not a symmetric monoidal category. Thus we cannot promote the adjunction (5.2) of Theorem 5.1 to be a weak comonoidal Quillen equivalence. Alternatively, we can consider bicomodules over non-cocommutative coalgebras D and N(D). In this case we obtain by similar arguments a weak comonoidal (but not symmetric) equivalence. For simplicity, we do not give details but the proof is entirely similar as of Theorem 5.3.
Proof of Theorem 5.3. We have N C (Γ(C)) ∼ = C, thus we only need to show that the lax monoidal map: is a weak homotopy equivalence for X and Y fibrant Γ(C)-comodules. Let {X(n)} be the Postnikov tower of X as in Definition 4.22. Let X be its (homotopy) limit. It is enough to show that: is a weak homotopy equivalence.
We show inductively that N C (X(n)) C N C (Y ) → N C (X(n) Γ(C) Y ) is a weak homotopy equivalence. For n = 0 the case is vacuous as the map is trivial. For n = 1, then X(1) is a cofree Γ(C)-comodule M ⊗ Γ(C), for some M in sMod k . Then from the formula N C (M ⊗ Γ(C)) ∼ = N(M ) ⊗ C, we get: Then by induction and our above argument on cofree objects, we obtain by [Hov99,5.2.6] that N(M ) ⊗ N C (Y (n + 1)) → N C (M ⊗ Y (n + 1)) is a weak homotopy equivalence. Now we can conclude as: We have used the fact that N C and M ⊗ − and N(M ) ⊗ − preserve towers that stabilize in each degree and they remain fibrant in the sense of Proposition 2.14.
We have just shown N C (X(1)) C N C (Y ) → N C (X(1) Γ(C) Y ). Let us show the higher cases by induction. Recall that X(n + 1) is determined as a homotopy pullback in CoMod Γ(C) (sMod k ) and in sMod k : X(n + 1) P ′ ⊗ Γ(C) X(n) Q ′ ⊗ Γ(C), for some epimorphism P ′ → Q ′ in sMod k . Since N C and − C N C (Y ) preserves fibrations that are epimorphisms and pullbacks, we obtain the following two homotopy pullbacks in CoMod Γ(C) (sMod k ): N C (X(n + 1)) C N C (Y ) N(P ′ ) ⊗ N C (Y ) and: By induction and our previous argument, this shows: N C (X(n + 1)) C N C (Y ) → N C (X(n + 1) Γ(C) Y ), is a weak homotopy equivalence.
Since N C preserves towers that stabilize in each degree and by Lemma 4.28, we can conclude by the following string of weak homotopy equivalences: Thus N C ( X) C N C (Y ) −→ N C ( X Γ(C) Y ) is a weak homotopy equivalence.
Remark 5.5. There is an alternative Dold-Kan correspondence for comodules. Let us dualize another construction from [SS03]. Let CoCAlg(C) denote the category of commutative coalgebras in C a symmetric monoidal category. Let L : (C, ⊗, I) → (D, ∧, J) be a lax symmetric comonoidal functor between symmetric monoidal categories, with right adjoint R. Then L lifts to a functor L : CoCAlg(C) → CoCAlg(D), with a right adjoint R, that can be constructed when the categories admit equalizers and cofree cocommutative coalgebras. Let D be a cocommutative coalgebra in D. The counit LR(D) → D induces an adjunction from the change of coalgebras: CoMod D (D). for any simply connected cocommutative coalgebra D in sMod k . Just as in the proof of Theorem 5.3, we can show that the above adjunction is a weak comonoidal Quillen pair. If we suppose ΓN(D) → D to be a weak homotopy equivalence, then one can prove that the above adjunction is a weak comonoidal Quillen equivalence. However, from [Sor19], we expect ΓN(D) → D to not be a weak homotopy equivalence in most cases. In particular the functor N : CoCAlg(sMod k ) → CoCAlg(Ch ≥0 k ) does not represent the correct right adjoint of the left derived functor of Γ on the ∞-categories of E ∞ -coalgebras in the underlying ∞-categories of sMod k and Ch ≥0 k . Thus we do not expect the above Quillen pair to be a Quillen equivalence. We conjecture that the problems disappear if we consider simply connected conilpotent cocommutative coalgebras.