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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 [P´er20c] 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 [P´er20b], 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 Mc and Mf its subcategories spanned by cofibrant and fibrant objects respectively. We denote the underlying ∞-category of M −1 −1 by N (Mc)[W ] as in [Lur17, 1.3.4.15], or simply N (M)[W ] if every object is cofibrant. If M −1 is a symmetric monoidal model category (in the sense of [Hov99, 4.2.6]), then N (Mc)[W ] is a symmetric monoidal ∞-category via the derived tensor product, see [Lur17, 4.1.7.4]. The notation stems from a localization N (M) → N (M)[W−1] of ∞-categories, where N denotes the nerve of a category. When M is combinatorial, the underlying ∞-category of M is also equivalent to −1 N (Mf )[W ] by [Lur17, 1.3.4.16]. (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:

k = k1 ×···× kn,

where each ki is a field, for some 1 ≤ n < ∞. In the literature, such rings are referred to as commutative semisimple Artinian rings. (3) Let Chk denote the category of unbounded chain complexes of k-modules (graded homologically). ≥0 Denote Chk 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:

(X ⊗ Y )n = Xi ⊗k Yj , i+j=n M 2 with differential given on homogeneous elements by:

d(x ⊗ y)= dx ⊗ y + (−1)|x|x ⊗ dy.

Let sModk 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 Chk is endowed with a model structure in which weak equivalences are quasi-isomor- ≥0 phisms, cofibrations are monomorphisms, fibrations are epimorphisms. The category Chk is en- dowed with a model structure in which weak equivalence are quasi-isomorphisms. cofibrations are monomorphisms, and fibrations are positive levelwise epimorphisms. The category sModk 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. ≥0 (5) Denote N : sModk → Chk the normalization functor which is an equivalence of categories. Its inverse ≥0 is denoted Γ : Chk → sModk. From [SS03], we obtain two weak monoidal Quillen equivalences:

Γ ≥0 ChR ⊥ sModR, (1.1) N and: N ≥0 sModR ⊥ ChR . (1.2) Γ 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.

2. 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 [P´er20a, 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]. 3 Definition 2.1 ([Hes09, 5.12]). Let P be a class of morphisms in a category C closed under pullbacks. Let op λ be an ordinal. Given a functor Y : λ → C such that for all β < λ, the morphism Yβ+1 → Yβ fits into the pullback diagram: ′ Yβ+1 Xβ+1 y

Yβ Xβ+1, ′ where Xβ+1 → Xβ+1 is some morphism in P, and Yβ → Xβ+1 is a morphism in C, and we denote:

Yγ := lim Yβ, β<γ for any limit ordinal γ < λ. We say that the composition of the tower Y :

limYβ −→ Y0, λop

if it exists, is a P-Postnikov tower. The class of all P-Postnikov towers is denoted PostP. + Proposition 2.2 ([BHK 15, 2.10]). If C is a complete category, the class PostP is the smallest class of morphisms in C containing P closed under composition, pullbacks and limits indexed by ordinals. Proof. See dual statements in [Hov99, 2.1.12, 2.1.13].  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(PostP) ⊆ PostR(P). Proof. Right adjoints preserve limits.  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 → Dop, the induced map of sets:

colim (HomC(Yβ, A)) −→ HomC lim Yβ, A , β<λ β<λ   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 Cop 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 PostP, then every morphism f of C can be factored functorially as: f A B

γ(f) δ(f) Cf , 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. 4 b 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: f X Y

i q V where i is a cofibration and q is a Q-Postnikov tower; (b) the morphism f factors as: f X Y

j p W 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 [P´er20b, 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 FX as follows. Let ∗ be the terminal object of M. There is an object FX in M that factors the trivial map:

X ∗

j p FX,

∼ where j : X ֒→ FX is an acyclic cofibration in M, and p is a P-Postnikov tower. This means that FX can be defined as iterated maps of pullbacks along P, starting with (FX)0 = ∗. In our examples, the above inductive fibrant replacement is a countable tower and its limit is a homotopy limit. N Notation 2.13. Denote N the poset {0 < 1 < 2 < ···}. Let C be any complete category. Objects in C are diagrams of shape N and can be represented as (countable) towers in C:

f f f ··· 3 X(2) 2 X(1) 1 X(0).

We denote such object by {X(n)} = (X(n),fn)n∈N. The limit of the tower is denote limnX(n). Proposition 2.14 ([GJ99, VI.1.1]). Let M be a model category. Then the category of towers MN can be endowed with the Reedy model structure, where a map {X(n)}→{Y (n)} is a weak equivalence (respectively a cofibration), if each map X(n) → Y (n) is a weak equivalence (respectively a cofibration) in M, for all n ≥ 0. An object {X(n)} is fibrant if and only if X(0) is fibrant and all the maps X(n + 1) → X(n) in the tower are fibrations in M. Moreover, if we denote ι : M → MN the functor induced by the constant diagram, N then we obtain a Quillen adjunction ι : M ⊥ M : limn.

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. 5 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, 1.1.9]).  Remark 2.18. The converse of Proposition 2.17 is true if P and Q are sets that permit the cosmall object argument. However, this rarely happens in context of interest as seen in Example 2.7. 2.3. Left-Induced Model Categories. We remind now the terminology of [HKRS17, GKR18]. Classically, under some assumptions, given a pair of adjoint functors L : M⊥ A : R where M is endowed with a model structure, we can lift a model structure to A in which weak equivalences and fibrations are created by the right adjoint. See [Hir03, 11.3.2]. Definition 2.19. Let M be a model category and A be any category, such that there is a pair of adjoint functors: L : AM⊥ : R. We say that the left adjoint L : A → M left-induces a model structure on A if the category A can be endowed with a model structure where a morphism f in A is defined to be a cofibration (respectively a weak equivalence) if L(f) is a cofibration (respectively a weak equivalence) in M. This model structure on A, if it exists, is called the left-induced model structure from M. The following result guarantees that left-inducing from a combinatorial model category gives back a combinatorial model category. Proposition 2.20 ([BHK+15, 2.23],[HKRS17, 3.3.4]). Suppose A is a model category left-induced by a model category M. Suppose both A and M are presentable. If M is cofibrantly generated by a pair of sets, then A is cofibrantly generated by a pair of sets. Proposition 2.21 ([BHK+15, 2.18]). Suppose A is a model category left-induced by a model category M, via an adjunction L : AM⊥ : R. 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 Cc the full subcategory of cofibrant objects in C, then the underlying −1 L ∞-category N (Cc) W is symmetric monoidal via the derived tensor product ⊗ , see [Lur17, 4.1.7.4]. We provide dual conditions such that the homotopy category Ho(C) and the underlying ∞-category −1   N (Cf ) W 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 → fI of the unit, the induced morphism X =∼ I ⊗ X → fI ⊗ X is a weak equivalence, for any fibrant object X. 6 The requirement (ii) is automatic if I is already fibrant Example 2.24. If C is a symmetric monoidal model category C, then Cop 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 ≥0 also a symmetric comonoidal model category. For instance C = sModk, Chk , Chk. 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 Cf 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. Proposition 2.26 ([Lur17, 4.1.7.6], [NS18, A.7]). Let (C, ⊗, I) be a symmetric comonoidal model category. −1 Suppose that I is fibrant. Then the underlying ∞-category N (Cf )[W ] of C can be given the structure of a N ⊗ symmetric monoidal ∞-category via the symmetric monoidal localization of (Cf ):

N ⊗ N −1 ⊗ (Cf ) (Cf )[W ] ,

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:

L : (C, ⊗, I)⊥ (D, ∧, J): R, 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:

R(X) ⊗ R(Y ) R(X ∧ Y ),

is a weak equivalence in C. ∼ (ii) For some (hence any) fibrant replacement λ : J −→ fJ in D, the composite map:

R(λ) I R(J) R(fJ),

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:

L (C, ⊗, I)⊥ (D, ∧, J), R their opposite adjunction: Rop op op (C , ⊗, I)⊥ (D , ∧, J), Lop 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. 7 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 [P´er20a] to obtain the following result. Theorem 2.30 ([P´er20a, 2.13]). Let (C, ⊗, I) and (D, ∧, J) be symmetric comonoidal model categories with fibrant units. Let WC and WD be the classes of weak equivalence in C and D respectively. Let:

L : (C, ⊗, I)⊥ (D, ∧, J): R, be a weak comonoidal Quillen pair. Then the derived functor of R : D → C induces a symmetric monoidal functor between the underlying ∞-categories: N −1 N −1 R : (Df ) WD (Cf ) WC ,

where Cf ⊆ C and Df ⊆ 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.

3. Example: Unbounded Chain Complexes

We show here that the category Chk 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 Chk (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 Chk. 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 Sn(V ), the chain complex that is V concentrated in degree n and zero elsewhere. Denote the n-disk over V by Dn(V ), the chain complex that is V concentrated in degree n − 1 and n, with differential the identity. We obtain the epimorphisms Dn(V ) → Sn(V ): Dn(V ) ··· 0 V V 0 ···

Sn(V ) ··· 0 0 V 0 ··· . This defines functors: n n S (−): Modk Chk,D (−): Modk Chk.

The map defined above is natural, i.e. we have a natural transformation Dn(−) ⇒ Sn(−), for all n ∈ Z. When V = k, we simply write Dn and Sn. Notice that we have a short exact sequence of chain complexes: 0 Sn−1(V ) Dn(V ) Sn(V ) 0.

Definition 3.2. Define P and Q to be the following sets of maps in Chk:

n n n P = {D −→ S }n∈Z, Q = {D −→ 0}n∈Z.

We thicken the sets P and Q to classes P⊕ and Q⊕ of morphisms in Chk: n n P⊕ := D (V ) −→ S (V ) | V any k-module , n∈Z n 8 o n Q⊕ := D (V ) −→ 0 | V any k-module . n∈Z n o Clearly, the maps in P and P⊕ are fibrations in Chk and the maps in Q and Q⊕ are acyclic fibrations in Chk. Remark 3.3. When k is a field, as every k-module is free, we get:

n n P⊕ = D −→ S | λ any ordinal , ( λ λ ) M M n∈Z

n Q⊕ = D −→ 0 | λ any ordinal . ( λ ) M n∈Z Theorem 3.4. The model category of unbounded chain complexes Chk 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 Bn(X) and the n-cycles of X by Zn(X).

Proposition 3.5. Let X be a chain complex in Chk. Then X is split as a chain complex and we have a non-canonical decomposition:

Xn =∼ Hn(X) ⊕ Bn(X) ⊕ Bn−1(X). In particular, any chain complex X can be decomposed non-canonically as product of disks and spheres:

n n X =∼ S (Vn) ⊕ D (Wn), n Z Y∈ where Vn = Hn(X) and Wn = Bn−1(X). Proof. We have the following short exact exact sequences of k-modules:

0 Bn(X) Zn(X) Hn(X) 0,

0 Zn(X) Xn Bn−1(X) 0. dn 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:

Xn =∼ Zn(X) ⊕ Bn−1(X)

=∼ Hn(X) ⊕ Bn(X) ⊕ Bn−1(X).

An investigation of the differentials Xn → Xn−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.

Lemma 3.7. We have the equalities of classes: LLP(P⊕)= LLP(P) and LLP(Q⊕)= LLP(Q) in Chk.

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:

XDn(V )

f

Y Sn(V ), 9 for some k-module V . Since V is projective, there is another k-module W such that V ⊕ W is free. Thus ∼ V ⊕ W = λ k, for some basis λ. In particular, the choice of the splitting V ⊕ W → V induces the commutative diagram: L

n n n n XD (V ) Dα Dα Dα α λ α λ M∈ Y∈ f

n n n n Y S (V ) Sα Sα Sα, α λ α λ M∈ Y∈ n n n n P n where Dα and Sα are a copies of D and S . Since f is in LLP( ), we obtain a lift ℓα : Y → Dα, for each n n α. It induces a lift ℓ : Y → α Dα which restricts to Y → D (V ) via the retracts (dashed maps in the diagram).  Q Lemma 3.8. Maps in the set Q are the generating acyclic fibrations in Chk.

Proof. Let f : X → Y be a map in LLP(Q), let us show it is a cofibration in Chk, i.e. a monomorphism. Following Proposition 3.5, we decompose X as:

n n X =∼ S (Vn) ⊕ D (Wn). n Z Y∈ n n+1 :Then the canonical inclusions S (Vn) ֒→ D (Vn) induce a monomorphism ι

n n n+1 n ι : S (Vn) ⊕ D (Wn) D (Vn) ⊕ D (Wn). n Z n Z Y∈ Y∈ Since f is in LLP(Q), then there is a map ℓ such that ι = ℓ ◦ f. Hence f must be a monomorphism. 

Lemma 3.9. Maps in the set P are the generating fibrations in Chk. Proof. Notice that LLP(P) ⊆ LLP(Q) as any lift Y → Dn in the following commutative diagram induces the dashed lift: XDn

Y Sn

Y 0. 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 Chk: f 0 XYK 0, where K = coker(f). It remains to show that K is acyclic. Notice first that K is defined as the pushout:

X 0 f p YK,

and so, since f is in LLP(P), then 0 → K is in LLP(P). Following Proposition 3.5, decompose K as:

n n K =∼ S (Vn) ⊕ D (Wn), n∈Z Y 10 where Vn = Hn(K). Then we obtain a map by projection:

n n n S (Vn) ⊕ D (Wn) S (Vn), n Z n Z Y∈ Y∈ that factors through the non-trivial map:

n n D (Vn) S (Vn) n Z n Z Y∈ Y∈

as 0 → K is in LLP(P)= LLP(P⊕). But this is only possible when Vn = 0, hence K must be acyclic. Thus f is a quasi-isomorphism.  3.2. A Postnikov Presentation. In Definition 3.2 and Theorem 3.4, we introduced sets of generating fibrations and acyclic fibrations for Chk. We now show they induce a Postnikov presentation on Chk.

Theorem 3.10. The pair (P⊕, Q) is a Postnikov presentation of the model category of unbounded chain complex Chk. We shall prove Theorem 3.10 with Lemmas 3.14 and 3.15 below. D Remark 3.11. Our arguments in Theorem 3.10 can be generalized to the category Chk , the category of D functors D → Chk, where D is a small category. Indeed, endow Chk with its injective model structure with levelwise weak equivalences and cofibrations (it exists by [HKRS17, 3.4.1]). Using the same notation + D as [BHK 15, 4.10], the pair (P⊕ ⋔ D, Q ⋔ D) is a Postnikov presentation for Chk . This provides inductive arguments to compute homotopy limits in Chk.

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. P P (i) As ⊆ ⊕, we get PostP ⊆ PostP⊕ . n (ii) The maps S → 0 are in PostP as they are obtained as pullbacks:

Sn Dn+1 y

0 Sn+1. n Similarly, for any k-module V , the maps S (V ) → 0 are in PostP⊕ . n n n (iii) Since D → 0 is the composite D −→ S −→ 0, we see that Q ⊆ PostP, and thus PostQ ⊆ PostP ⊆

PostP⊕ by Proposition 2.2.

(iv) Although P⊕ * PostP, we have P⊕ ⊆ Post\P (see Notation 2.9). Indeed, for any k-module V , any map Dn(V ) → Sn(V ) is the retract of a map Dn(F ) → Sn(F ) where F is a free k-module. Then, for λ a basis of F , we have the retract in Chk:

Dn Dn Dn λ λ λ M Y M

Sn Sn Sn, λ λ λ M Y M induced by the split short exact sequence in k-modules:

0 k ι k coker(ι) 0, λ λ M Y .where ι : k ֒→ k is the natural monomorphism λ λ M Y 11 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. Proof. Follows from Proposition 3.5, (ii) of Remark 3.12, and Proposition 2.2. 

Lemma 3.14. Every acyclic fibration in Chk is a retract of a Q-Postnikov tower. Every map in Chk 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 Chk is the set of generating acyclic fibrations. Their codomain is the terminal object in Chk 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 Chk. Choose a decomposition ∼ n n of X by using Proposition 3.5: X = n∈Z S (Vn) ⊕ D (Wn), for some collection of k-modules Vn and Wn. n n+1 :The inclusions S (Vn) ֒→ D (Vn) define then a monomorphism in Chk Q n n n+1 n X =∼ S (Vn) ⊕ D (Wn) D (Vn) ⊕ D (Wn). n Z n Z Y∈ Y∈ Since Vn and Wn are projective k-modules, they can be embedded into free k-modules, say Fn and Gn, with basis λn and γn. Then we obtain the following monomorphisms in Chk:

n+1 n n+1 n D (Vn) ⊕ D (Wn) D ⊕ D n Z n Z λ γ Y∈ Y∈ Mn Mn

n+1 n Thus, we get the monomorphism in Chk: X D ⊕ D . Denote Z the acyclic chain com- n Z λ ,γ Y∈ Yn n n n ι⊕f q D +1⊕D . We obtain the desired second factorization in Chk: XZ YY, plex n∈Z λn,γn ⊕ where the map q is the projection onto Y , which is indeed a Q-Postnikov tower by Proposition 2.2.  Q Q Lemma 3.15. Every fibration in Chk is a retract of a P⊕-Postnikov tower and every map in Chk 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 n 3.15. Indeed, as noted in [Sor10], the codomains S of maps in the set P are not cosmall relative to PostP. n Indeed, let Yk = S for all k ≥ 0 and Yk+1 → Yk be the zero maps. Let Y = limYk be the limit in Chk. The k≥0 set map: n n colim (HomChk (Yk,S )) −→ HomChk (Y,S ) k≥0 is not a bijection. Indeed, the map is equivalent to the map: ∗ k −→ k ,   k k M≥0 Y≥0 which is never a bijection. A similar argument can be applied to show that the codomains Sn(V ) of the P maps in the class ⊕ are not cosmall relative to PostP⊕ , for any k-module V . Lemma 3.17. Let X be a chain complex in Chk. Let V be a k-module. Let n be an integer in Z. Given a surjective linear map fn : Xn → V non-trivial only on n-cycles, there is a map of chain complexes f : X → Sn(V ), and the pullback chain complex P in the following diagram: PDn(V ) y

X Sn(V ), 12 has homology: ker (Hn(f)) i = n, Hi(P ) =∼ Hi(X) i 6= n,  and we have Pi = Xi for i 6= n − 1 and Pn−1 = Xn−1 ⊕ V .

Proof. By construction, since pullbacks in Chk are taken levelwise, for i 6= n,n − 1, we have the pullbacks of k-modules: Pn−1 V Pn V Pi 0 y y y

fn Xn−1 0, Xn V, Xi 0.

Thus Pn−1 =∼ Xn−1 ⊕ V and Pi = Xi for any i 6= n − 1. The differential Pn → Pn−1 is the linear map dn⊕f Xn Xn−1 ⊕ V, and the differential Pn−1 → Pn−2 is the linear map:

dn−1 Xn−1 ⊕ V Xn−1 Xn−2, where the unlabeled map is the natural projection. All the differentials Pi → Pi−1 for i 6= n,n − 1 are the differentials Xi → Xi−1 of the chain complex X. Clearly, we get Hi(P ) = Hi(X) for i 6= n,n − 1. For i = n − 1, by Proposition 3.5, we can choose a decomposition:

Xn =∼ Hn(X) ⊕ Bn−1(X) ⊕ Bn(X).

The differential dn : Xn → Xn−1 sends the factor Bn−1(X) in Xn to itself, and the factor Hn(X)⊕Bn(X) to zero. By definition, the map fn : Xn → V sends the factor Hn(X) in Xn to the image of fn, which is V since fn is surjective, and the factor Bn−1(X) ⊕ Bn(X) to zero. Thus the image of the differential Pn −→ Pn−1, is precisely Bn−1(X) ⊕ V . Therefore, we obtain:

ker(Pn−1 → Pn−2) Hn−1(P ) = im(Pn → Pn−1) Zn−1(X) ⊕ V =∼ Bn−1(X) ⊕ V

Zn−1(X) =∼ Bn−1(X) = Hn−1(X). 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 fn(x) = 0. Since fn : Xn → V is entirely defined on the copy Hn(X) in Xn, we get from the commutative diagram:

fn Zn(X) Xn V

Hn(f)

Hn(X), that Hn(P ) =∼ ker(Hn(f)). 

Lemma 3.18. Let j : X → Y be a monomorphism in Chk (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 Chk as: j X Y

Fn(j) Fn(pj ) Fn(Y )

where Fn(Y ) is a chain complex built with the following properties.

• The chain map Fn(pj ): Fn(Y ) → Y is a P⊕-Postnikov tower. 13 • The chain map Fn(j): X → Fn(Y ) is a monomorphism (i.e. a cofibration in Chk). • The k-module (Fn(Y ))i differs from Yi only in degree i = n − 1. • In degrees i 6= n in homology, we have Hi(Fn(Y )) =∼ Hi(Y ) and the maps:

Hi(Fn(j)) : Hi(X) −→ Hi(Fn(Y )) =∼ Hi(Y ),

are precisely the maps Hi(j): Hi(X) → Hi(Y ). In particular, the maps Hi(Fn(j)) are monomor- phisms. Moreover, if the maps Hi(j) are isomorphisms, then so are the maps Hi(Fn(j)). ∼= • In degree n in homology, the map Hn(Fn(j)) : Hn(X) −→ Hn(Fn(Y )) is an isomorphism.

Proof. We construct below the chain complex Fn(Y ) explicitly using Lemma 3.17. By Proposition 3.5, we can decompose Yn as:

Yn =∼ Hn(Y ) ⊕ Yn =∼ im Hn(j) ⊕ coker Hn(j) ⊕ Yn,     where Yn is the direct sum of the copies of the boundaries. Denote the k-module V = coker(Hn(j)) and define the linear map fn : Yn → V to be the natural projection. In particular, the map fn sends n-boundaries of Y to zero. This defines a chain map: f : Y −→ Sn(V ). Notice that since j : X → Y is a monomorphism, j f n we get j(Xn) ⊆ Yn, and so, by construction of f, we get that the composite: X Y S (V ), is the zero chain map. We obtain Fn(Y ) as the following pullback in Chk, with a chain map Fn(j) induced by the universality of pullbacks:

0 X Fn(j) ∃! n Fn(Y ) D (V ) y Fn(pj ) j Y Sn(V ). f

By construction, the induced chain map Fn(pj ): Fn(Y ) → Y is in PostP⊕ . From the commutativity of the diagram:

Fn(j) X Fn(Y )

Fn(pj ) j Y,

since j is a monomorphism, so is Fn(j). Since Hi(j) is a monomorphism for i ∈ Z, then so is Hi(Fn(j)). By Lemma 3.17, we get Hi(Fn(Y )) =∼ Hi(Y ) for all i 6= n. For i = n, we get:

Hn(Fn(Y )) =∼ ker Hn(f) =∼ Hn(X),   as we have the short exact sequence of k-vector spaces:

Hn(j) Hn(f) 0 Hn(X) Hn(Y ) V 0,

since V = coker(Hn(j)). Thus Hn(Fn(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 Chk using Lemma 3.18 repeatedly so that 14 f f factors as:

W

f . .

− + G1 (W )

− G1 (p) + ej W

. − + . G1 (j ) .

j+ + G1 (W )

+ G+ p G1 (j) 1 ( ) + G0 (W ) + G0 (j) + G0 (p) Xj Wp Y f

where j is a monomorphism and a quasi-isomorphism, and all the vertical maps and p are in PostP⊕ . The composition of all the vertical maps and p is a chain map W → Y which is in PostP⊕ , by Proposition 2.2. Wee first start by noticing the following factorization: f f XY,

j p X ⊕ Y induced by the following pullback in Chk:

X j

X ⊕ Y X y f p Y 0.

The map p is in PostP⊕ 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(Gn (W )) in Chk of the tower of maps: n≥0

+ + + + G2 (p) + G1 (p) + G0 (p) ··· G2 (W ) G1 (W ) G0 (W ) W, 15 + + + + where each Gn (p) is in PostP⊕ . The map j : X → W is induced by the monomorphisms Gn (j): X → + Gn (W ) which are compatible with the tower: + Gn (W )

+ Gn (j) + Gn (p)

+ X+ Gn−1(W ), Gn−1(j) + and Gn (j) induces an isomorphism in homology in degrees i, for 0 ≤ i ≤ n, and a monomorphism otherwise. + We construct the chain complexes Gn (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 + + G0 (W ) := F0(W ). The cofibration G0 (j) defined as the chain map: + F0(j): X −→ F0(W )= G0 (W ), is an isomorphism in homology in degree 0, and a monomorphism in other degrees. The chain map + G0 (p) defined as the map: + F0(pj ): G0 (W )= F0(W ) −→ W,

is a P⊕-Postnikov tower. + • For the inductive step, suppose, for a fixed integer n ≥ 0, the chain complex Gn (W ) is defined, + + together with a cofibration Gn (j): X → Gn (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 Gn (j) for the degree n + 1. Denote: + + Gn+1(W ) := Fn+1(Gn (W )). + The cofibration Gn+1(j) defined as the chain map: + + Fn+1(Gn(j)) : X −→ Fn+1(Gn (W )) = Gn+1(W ), is an isomorphism in homology in degrees i where 0 ≤ i ≤ n + 1, and a monomorphism in other P + degrees. We obtain a ⊕-Postnikov tower Gn+1(p) defined as the chain map: + + + F p + : G (W )= F (G (W )) −→ G (W ), n+1 Gn (j) n+1 n+1 n n such that the following diagram commutes:

G+ j n ( ) + X Gn (W )

G+ (j) + n+1 Gn+1(p) + Gn+1(W ). The induced map j+ : X → W + is a monomorphism of chain complexes. Indeed, for any fixed i ∈ Z, we have: + ∼ + ∼ + ∼ Gi+1(W ) i = Gi+2(W ) i = Gi+3(W ) i = ··· . + ∼ + + + Thus (W )i = Gi+1(W ) i. Therefore
the linear map
(j )i : Xi → (W
)i is the linear map: G+ j X G+ W ,
n+1( ) i : i −→ i+1( ) i + ∼ + + which is a monomorphism. Similarly, we get:
Hi(W )= Hi(Gi+1
(W )), 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(Gn (W )) in Chk of the tower of maps: e f n≥f0 G−(p) G−(p) f − + 2 − + 1 − + + ··· G2 (W ) G1 (W ) G0 (W )= W , 16 − − where each Gn (p) is in PostP⊕ . The map j : X → W is induced by the monomorphisms Gn (j): X → − + Gn (W ) which are compatible with the tower: e f − Gn (W )

− Gn (j) − Gn (p)

− X− Gn−1(W ), Gn−1(j)

− and Gn (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 quasi- isomorphism, hence an acyclic cofibration, as desired.  e f 4. 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 Chk ≥0 in previous section to Chk and sModk.

≥0 Definition 4.1. Denote τ≥0 : Chk → Chk the 0-th (homological) truncation (see [Wei94, 1.2.7]). From the sets and classes of Definition 3.2, we denote their image under the truncation by:

≥0 n n 0 ≥0 n P = {D −→ S }n≥1 ∪{0 → S }, Q = {D −→ 0}n≥1, and: P≥0 n n ⊕ := D (V ) −→ S (V ) | V any k-module n≥1 n o 0 −→ S0(V ) | V any k-module .

[ n ≥0 o We also obtain sets and classes in sModk via the equivalence Γ : Chk → sModk. For n ≥ 0, we denote K(V,n) := Γ(Sn(V )) which corresponds precisely to the simplicial model of the Eilenberg-Mac Lane space of V in degree n. For n ≥ 1, we denote PK(V,n) := Γ(Dn(V )) which corresponds to the based path of K(V,n). Define: ∆ P = {PK(k,n) −→ K(k,n)}n≥1 ∪{0 → K(k, 0)}, ∆ Q = {PK(k,n) −→ 0}n≥1, and: P∆ ⊕ := PK(V,n) −→ K(V,n) | V any k-module n≥1 n o 0 −→ K(V, 0) | V any k-module . [ n o ≥0 ≥0 The Quillen adjunction: Chk ⊥ Chk : τ≥0 shows that the model structure on Chk is in fact left- ≥0 induced from Chk. From Theorem 3.4 and Proposition 2.21, we obtain that Chk and sModk are fibrantly generated. As pointed out in Remark 2.22, it is not automatic that we obtain Postnikov presentations for ≥0 Chk and sModk 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. Theorem 4.2. Let k be a finite product of fields. ≥0 (i) The model category of non-negative chain complexes Chk is fibrantly generated by the pair of set P≥0 Q≥0 P≥0 Q≥0 ≥0 ( , ). The pair ( ⊕ , ) is a Postnikov presentation of Chk . (ii) The model category of simplicial k-modules sModk is fibrantly generated by the pair of sets (P∆, Q∆). P∆ Q∆ The pair ( ⊕ , ) is a Postnikov presentation of sModk. 17 4.1. Left-Induced Model Structures on Comodules. We present here the model structures for como- ≥0 dules in sModk and Chk , 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: C∆ C ⊗ C

∆ idC ⊗∆ ∆⊗id C ⊗ CC C ⊗ C ⊗ C, and admits a counit morphism ε : C → I such that we have the following commutative diagram: id ⊗ε ε⊗id C ⊗ CC C ⊗ I =∼ C =∼ I ⊗ CC C ⊗ C

∆ C. ∆ The coalgebra is cocommutative if the following diagram commutes: C ⊗ Cτ C ⊗ C

∆ ∆ C, 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:

f f C C′ C C′

∆ ∆′ ε′ ε f⊗f C ⊗ C C′ ⊗ C′, I.

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,ora 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: ρ ρ X X ⊗ C XX ⊗ C

ρ ρ⊗idC idX ⊗ε X ⊗ C X ⊗ C ⊗ C, X ⊗ I idX ⊗∆ =∼ X.

The category of right C-comodules in C is denoted CoModC (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) =∼ CoModC (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 Cop, then we can define the category of right comodules op op as modules in the opposite category: CoModC (C) = (ModC (C )) , 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. 18 ≥0 Proposition 4.9 ([P´er20c, 2.12]). Let M be either the monoidal model category sModk or Chk . 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:

U CoModC (M) ⊥ M. −⊗C In particular, U preserves and reflects cofibrations and weak equivalences. The model structure is combina- torial and simplicial. Every object is cofibrant. Corollary 4.10. Let k be a finite product of fields. ≥0 ≥0 (i) Let C be a coalgebra in Chk . The model category of right C-comodules CoModC (Chk ) is fibrantly generated by the pair of sets (P≥0 ⊗ C, Q≥0 ⊗ C). (ii) Let C be a coalgebra in sModk. The model category of right C-comodules CoModC (sModk) is fibrantly generated by the pair of sets (P∆ ⊗ C, Q∆ ⊗ C).

4.2. Postnikov Presentations. We now introduce a Postnikov presentation for comodules in sModk and ≥0 Chk . 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: X0 = R, X1 = 0 and Xi = 0 for all i< 0. A simplicial R-module X is simply connected if X0 = 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 Sn(V ) ⊗ C is the k-module V ⊗ Ci−n. If we choose C to be a simply connected differential graded k-coalgebra, we get: 0 i

≥0 Proof. We only do the case Chk . 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. ≥0 ≥0 Let us do the first possible factorization. By Theorem 3.4, the set Q ⊗ C of maps in CoModC(Chk ) ≥0 is the set of generating acyclic fibrations. Their codomain is the terminal object in CoModC (Chk ) and is thus cosmall. We can then apply the cosmall object argument (Proposition 2.8) to obtain the desired factorization. ≥0 For the second possible factorization, start with any morphism f : X → Y in CoModC (Chk ). 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 ≥0 We obtain the desired second factorization in CoModC (Chk ):

ι X

(Z ⊗ C) ⊕ YZ ⊗ C y f q Y 0,

where the map q is the projection onto Y , which is indeed a (Q≥0 ⊗ C)-Postnikov tower by Proposition 2.2. 

Lemma 4.15 ([P´er20c, A.6]). Let C be a simply connected differential graded k-coalgebra. Every fibration ≥0 P≥0 ≥0 in CoModC (Chk ) is a retract of a ( ⊕ ⊗ C)-Postnikov tower. Any morphism in CoModC(Chk ) factors P≥0 as a cofibration followed by a ( ⊕ ⊗ C)-Postnikov tower.

We can argue completely similarly for the simplicial case.

Lemma 4.16. Let C be a simply connected coalgebra in sModk. Every fibration in CoModC (sModk) is a P∆ retract of a ( ⊕ ⊗ C)-Postnikov tower. Any morphism in CoModC (sModk) factors as a cofibration followed P∆ by a ( ⊕ ⊗ C)-Postnikov tower.

Proof. We argue as in [P´er20c, A.6] and thus show less details. Any morphism X → Y in CoModC (sModk) factors through W = (U(X) ⊗ C) ⊕ Y . We build now a tower {Gn(W )} of C-comodules that stabilizes in each degree. For the case n = 0, it is easy to adapt [P´er20c, A.9] in the simplicial case. Suppose we have constructed the cofibraton X → Gn(W ). Then define Gn+1(W ) as follows. Define V to be the cokernel of the (n + 1)-th homology of:

N(U(X)) → N(U(Gn(W )).

n+1 Then we obtain a map N(U(Gn(W ))) → S (V ) and thus a map in sModk:

U(Gn(W )) → K(V,n + 1). 20 By adjointness, we get Gn(W ) → K(V,n + 1) ⊗ C and we define Gn+1(W ) as the following pullback:

0 X

Gn+1(W ) PK(V,n + 1) ⊗ C y

Gn(W ) K(V,n + 1) ⊗ C.

Then πn+1(X) =∼ πn+1(Gn+1(W )) as we can apply a simplicial version of [P´er20c, A.7], the comodule analogue of Lemma 3.17. Indeed, a pullback:

PPKy (V,n) ⊗ C

ZK(V,n) ⊗ C,

is a homotopy pullback in sModk, and thus N preserves homotopy pullbacks and thus as π∗(P ) =∼ H∗(N(P )) and N(K(V,n) ⊗ C) ≃ Sn(V ) ⊗ N(C), and N(C) remains simply connected, we can apply [P´er20c, A.7]. 

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.

≥0 Definition 4.17. Let M be either sModk or Chk . A tower {X(n)} in M stabilizes in each degree if for each degree i ≥ 0, the tower {X(n)i} of k-modules stabilizes for n ≥ i + 1, i.e., for all n ≥ 0, and all 0 ≤ i ≤ n, we have:

X(n + 1)i =∼ X(n + 2)i =∼ X(n + 3)i =∼ ··· .

Let C be a coalgebra in M. A tower {X(n)} in CoModC (M) stabilizes in each degree if the underlying tower {U(X(n))} in M stabilizes in each degree.

≥0 Remark 4.18. A tower {X(n)} in sModk stabilizes in each degree if and only if {N(X(n))} in Chk stabilizes ≥0 in each degree. A tower {X(n)} in Chk stabilizes in each degree if and only if {Γ(X(n))} in sModk stabilizes in each degree.

≥0 Lemma 4.19. Let M be either sModk or Chk . 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: n o limnX(n) ⊗ C =∼ limn X(n) ⊗ C .

 ≥0    Proof. We prove only the case M = Chk . For all n ≥ 0, and all 0 ≤ i ≤ n, we have:

X(n + 1) ⊗ C =∼ X(n + 1)a ⊗ Cb i a b i   M+ = =∼ X(n + 2)a ⊗ Cb a b i M+ = =∼ X(n + 2) ⊗ C , i   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 limnX(n) ⊗ C and limn X(n) ⊗ C are i i isomorphic to X(i + 1)a ⊗ Cb.        a+b=i M 21 ≥0 Corollary 4.20. Let M be either sModk or Chk . Let C be a coalgebra in M. Let {X(n)} be a tower in CoModC (M) that stabilizes in each degree. Then the natural map:

C ∼= U(limn X(n)) −→ limnU(X(n)) 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 := limnU(X(n)), then the coaction X → X ⊗C is constructed as follows. For each degree i ≥ 0, the map Xi → (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. ≥0 Corollary 4.21. Let M be either sModk or Chk . 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 CoModC(M) with limit C X := limn X(n) where the right C-comodules X(n) are built inductively as follows. • Define X(0) to be the trivial C-comodule 0. e • Define X(1) to be the cofree C-comodule U(X) ⊗ C. The map X(1) → X(0) is trivial. • Suppose X(n) was constructed for a certain n ≥ 1. Then there exists a certain k-module Vn such that X(n + 1) is defined as the following pullback in CoModC (M): ≥0 – for M = Chk : n X(n + 1) D (Vn) ⊗ C y

n X(n) S (Vn) ⊗ C,

≥0 and we obtain the short exact sequence in CoModC (Chk ):

n−1 0 S (Vn) ⊗ C X(n + 1) X(n) 0

– for M = sModk:

X(n + 1) PK(Vn,n) ⊗ C y

X(n) K(Vn,n) ⊗ C,

and we obtain the short exact sequence in CoModC (sModk):

0 K(Vn,n − 1) ⊗ C X(n + 1) X(n) 0 The tower {X(n)} enjoys the following properties. P≥0 ≥0 P∆ (i) The map X −→ 0 is a ( ⊕ ⊗ C)-Postnikov tower if M = Chk or a ( ⊕ ⊗ C)-Postnikov tower if ≃ M = sModk. There exists an acyclic cofibration of right C-comodules X X. e (ii) If X is a fibrant right C-comodule, then X is a retract of X. ≥0 (iii) For all n ≥ 1, for all 0 ≤ i ≤ n − 1, we have Hi(X(n)) =∼ Hi(X) if M = Chk eor πi(X(n)) =∼ πi(X) if M = sModk. e (iv) The tower {X(n)} stabilizes in each degree. In particular: C ∼ U(X)= U(limn X(n)) = limn(U(X(n))).

(v) Each map X(n + 1) → X(n) for n ≥ 0 is a fibration in CoModC (M), and its underlying map e U(X(n + 1)) → U(X(n)) is also a fibration in M. In particular X is the homotopy limit of {X(n)} C in CoModC (M) and we have: U(X) ≃ U(holimn X(n)) ≃ holimn(U(X(n))). 22 e e ≥0 Definition 4.22. Let M be either sModk or Chk . Let C be a simply connected coalgebra in M. Let X be a right C-comodule in M. The Postnikov tower of X is the tower {X(n)} in CoModC (M) built in Corollary 4.21. 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. ≥0 Definition 4.24. Let M be either sModk or Chk . Let X and Y be C-comodules in M. Define their cotensor product XCY to be the following equalizer in M:

XCY X ⊗ Y X ⊗ C ⊗ Y, 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 XCY . We record some facts on the cotensor product. Lemma 4.25 ([EM66, 2.2]). For any C-comodule X, we have a natural isomorphism of C-comodules:

XCC =∼ X =∼ CC X. Lemma 4.26 ([EM66, 2.1]). Let M be an object in M. Then for any cofree comodule M ⊗C we have natural isomorphisms of C-comodules:

(M ⊗ C)C X =∼ M ⊗ X and XC(C ⊗ M) =∼ X ⊗ M.

Proposition 4.27 ([P´er20c, 4.6]). Let X be a C-comodule. Then XC− : CoModC (M) → CoModC (M) is a left exact functor that preserves finite limits and filtered colimits.

In general, the functor XC− : CoModC (M) → CoModC(M) does not preserve non-finite limits and is neither a left nor a right adjoint. Nevertheless, the cotensor product behaves well with the Postnikov towers of Definition 4.22. ≥0 Lemma 4.28. Let M be either sModk or Chk . Let C be a simply connected coalgebra in M. Let {X(n)} be a Postnikov tower of a C-comodule X. Let Y be any C-comodule. Then {X(n)CY } stabilizes in each degree and: C  ∼ C  (limn X(n)) CY = limn (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.  ≥0 The following results were proved in [P´er20c] for M = Chk . We claim we can adapt the arguments to obtain the same results for M = sModk. ≥0 Proposition 4.29 ([P´er20c, 4.7]). Let M be either sModk or Chk . Let C be a coalgebra in M. The cotensor product defines a symmetric monoidal structure on C-comodules and we shall denote it (CoModC (M), C, C). ≥0 Theorem 4.30 ([P´er20c, 4.11]). Let M be either sModk or Chk . Let C be a simply connected cocommutative coalgebra in M. Then (CoModC (M), C, C) is a symmetric comonoidal model category with fibrant unit. ≥0 Let M be either sModk or Chk . From Proposition 2.26, we see that we can (right) derive the coten- sor 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 [P´er20c, 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. 23 5. 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:

L : CoModC (C) −→ CoModL(C)(D). 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 RC on comodules by the following commutative diagram of adjoints: L CoModC (C) ⊥ CoModL(C)(D), RC

−⊗C ⊢ U −∧L(C) ⊢ U

L C⊥ D. R In particular, for any object M in D, we obtain: RC (M ∧ L(C)) =∼ R(M) ⊗ C. C More specifically, the functor R is defined as the following equalizer in CoModC (C):

RC (X) R(X) ⊗ C R(X ∧ L(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: R(X) ⊗ C R(X) ⊗ RL(C) R(X ∧ L(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. ≥0 Theorem 5.1. Let C be a simply connected coalgebra in Chk . Let D be a simply connected coalgebra in sModk. Then the induced adjunctions:

Γ ≥0 CoModC (Chk ) ⊥ CoModΓ(C)(sModk), (5.1) NC and: N ≥0 CoModD(sModk) ⊥ CoModN(D)(Chk ), (5.2) ΓD are Quillen equivalences. Remark 5.2. Using the rigidification result of [P´er20c], 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. ≥0 Let Wdg be the class of quasi-isomorphisms in Chk . Let W∆ be the class of weak homotopy equivalences ≥0 in sModk. Let C be a simply connected coalgebra in Chk . Denote Wdg,CoMod denote the class of quasi- isomorphisms 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 24 A∞-coalgebra in C, we denote CoModT (C) the ∞-category of right T -comodules in C, i.e., right T -modules in the opposite Cop. We obtain the following diagram of ∞-categories:

≥0 −1 ≃ −1 CoMod N (Ch ) W CoMod N (sModk) W C k dg Γ Γ(C) ∆  h i  
≃ ≃

N ≥0 −1 Γ N −1 CoModC (Chk ) Wdg,CoMod CoModΓ(C)(sModk) W∆,CoMod .   h i h i The vertical maps are equivalences of ∞-categories by [P´er20c, 1.1]. The top horizontal
map is an equivalence by [P´er20a, 4.1]: it states that the Dold-Kan correspondence induces a symmetric monoidal equivalence on ≥0 the underlying ∞-categories of sModk and Chk . 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. ≥0 Since the functor Γ : Chk → sModk is left Quillen, and cofibrations and weak equivalences are determined ≥0 by the underlying categories, then the left adjoint functor Γ : CoModC (Chk ) → CoModΓ(C)(sModk) 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: ΓNC (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 mape ΓNC(X(0)) −→ X(0) is a weak homotopy equivalence as it is the trivial map.e Suppose X is ae cofree comodule M ⊗ Γ(C), for M in sModk. Then the formula: NC(M ⊗ Γ(C)) =∼ N(M) ⊗ C, combined with the weak homotopy equivalence: Γ(N(M) ⊗ C)≃ Γ(N(M)) ⊗ Γ(C) =∼ M ⊗ Γ(C),

≥0 induced by the fact that Γ : Chk → sModk is part of a weak monoidal Quillen equivalence shows that ΓNC (M ⊗Γ(C)) −→ M ⊗Γ(C) is a weak homotopy equivalence. In particular, we obtain that ΓNC(X(1)) −→ X(1) is a weak homotopy equivalence. Now suppose we have shown ΓNC (X(n)) −→ X(n) for some n ≥ 1. Since X(n + 1) is the homotopy pullback in CoModΓ(C)(sModk) (and in sModk): X(n + 1) P ⊗ Γ(C) y

X(n) Q ⊗ Γ(C),

C for some epimorphism P → Q in sModk, then as N is a right Quillen functor, it preserves homotopy ≥0 ≥0 pullbacks and thus we get the following homotopy pullback in CoModC (Chk ) (and in Chk ): NC(X(n + 1)) N(P ) ⊗ C y

NC(X(n)) N(Q) ⊗ C,

as N preserves fibrations. In particular, notice that the induced maps NC (X(n + 1)) → NC (X(n)) are ≥0 ≥0 C fibrations in CoModC (Chk ) and Chk . Moreover, the tower {N (X(n))} stabilizes in each degree (as it is 25 the equalizer of towers that stabilizes in each degree by Lemma 4.19). Thus the homopy limits of {NC(X(n))} ≥0 ≥0 C in CoModC (Chk ) and in Chk are equivalent to N (X). ≥0 ≥0 Consider the above pullback in Chk , then as Γ : Chk → sModk is a right Quillen functor, we obtain the following homotopy pullback in sModk: e

ΓNC(X(n + 1)) Γ(N(P ) ⊗ C) y

ΓNC(X(n)) Γ(N(Q) ⊗ C).

Applying the weak homotopy equivalences:

Γ(N(Q) ⊗ C) −→≃ Q ⊗ Γ(C), Γ(N(P ) ⊗ C) −→≃ P ⊗ Γ(C)

≥0 induced again by Γ : Chk → sModk being part of a weak monoidal Quillen equivalence, we obtain by [Hov99, 5.2.6] the homotopy pullback in sModk:

ΓNC(X(n + 1)) P ⊗ Γ(C) y

ΓNC(X(n)) Q ⊗ Γ(C).

By induction, since ΓN(X(n)) → X(n) is a weak homotopy equivalence, we obtain that ΓN(X(n + 1)) → X(n + 1) is a weak homotopy equivalence. We conclude that: e e ΓNC (X) −→ X is a weak homotopy equivalence by noticing that (we omit the underlying functor): e e C C Γ(C) ΓN (X) ≃ ΓN (holimn X(n)) C C ≃ Γ(holimn N (X(n))) e C ≃ Γ(holimnN (X(n))) C ≃ holimnΓN (X(n))

≃ holimnX(n) ≃ X. We have used that the homotopy limits of {X(n)} and {NC (X(n))} can be computed in their underlying e categories and that NC is a right Quillen functor. 

≥0 5.2. A Weak Comonoidal Quillen Equivalence. Let C be a cocommutative coalgebra in Chk . Con- sider 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:

Γ(XCY ) Γ(X ⊗ Y ) Γ(X ⊗ C ⊗ Y )

Γ(X)Γ(C)Γ(Y ) Γ(X) ⊗ Γ(Y ) Γ(X) ⊗ Γ(C) ⊗ Γ(Y ).

≥0 Thus Γ lifts to a functor CoModC(Chk ) → CoModΓ(C)(sModk) 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 ≥0 spectrum Hk. It is equivalent to the underlying ∞-categories of sModk and Chk . 26 Theorem 5.3. Let k be a finite product of fields. Let C be a simply connected cocommutative coalgebra in ≥0 Chk . Then the induced adjunction:

Γ ≥0 (CoModC (Chk ), C, C)⊥ (CoModΓ(C)(sModk), Γ(C), Γ(C)), NC is a weak comonoidal Quillen equivalence. In particular, we obtain an equivalence of symmetric monoidal ∞-categories: ≥0 ≥0 CoModC (D (k)) ≃ CoModΓ(C)(D (k)), 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 sModk, then there is no ≥0 guarantee that N(D) is also cocommutative. Thus CoModN(D)(Chk ) 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 NC (Γ(C)) =∼ C, thus we only need to show that the lax monoidal map: C C C N (X)CN (Y ) −→ N (XΓ(C)Y ), 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: NC (X) NC(Y ) −→ NC(X Y ), e C Γ(C) is a weak homotopy equivalence. C e C C e We show inductively that N (X(n))C N (Y ) → N (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), C for some M in sModk. Then from the formula N (M ⊗ Γ(C)) =∼ N(M) ⊗ C, we get: C C C N (M ⊗ Γ(C))C N (Y ) =∼ N(M) ⊗ N (Y ). Thus we need to show that the natural map N(M) ⊗ NC (Y ) → NC(M ⊗ Y ) is a weak homotopy equivalence. We show this by using a Postnikov argument on Y . Let {Y (n)} be its Postnikov tower with (homotopy) limit Y . We need to show that N(M) ⊗ NC(Y ) → NC(M ⊗ Y ) is a weak homotopy equivalence. Let us first show that: e N(M) ⊗ NeC (Y (n)) → NCe(M ⊗ Y (n)), by induction. For n = 0 the case is vacuous. For n = 1, we have that Y (1) is equivalent to M ′ ⊗ Γ(C) for some M ′ in sModk. Then the map: N(M) ⊗ NC (M ′ ⊗ Γ(C)) → NC(M ⊗ M ′ ⊗ Γ(C)), becomes simply the map N(M) ⊗ N(M ′) ⊗ C → N(M ⊗ M ′) ⊗ C. It is a weak homotopy equivalence as N(M) ⊗ N(M ′) → N(M ⊗ M ′) is a weak homotopy equivalence as N is part of a weak monoidal Quillen adjunction. Now suppose that we have shown N(M) ⊗ NC(Y (n)) → NC (M ⊗ Y (n)) is a weak homotopy equivalence for some n ≥ 1. Recall that Y (n + 1) can be described as a homotopy pullback both in CoModΓ(C)(sModk) and in sModk: Y (n + 1) P ⊗ Γ(C) y

Y (n) Q ⊗ Γ(C),

≥0 for some epimorphism P → Q in sModk. Then as the functors N : sModk → Chk , M ⊗− : sModk → sModk, ≥0 ≥0 C ≥0 N(M) ⊗− : Chk → Chk and N : CoModΓ(C)(sModk) → CoModC (Chk ) preserve homotopy pullbacks, we 27 obtain the following two homotopy pullbacks in sModk:

N(M) ⊗ NC (Y (n + 1)) N(M) ⊗ N(P ) ⊗ C y

N(M) ⊗ NC (Y (n)) N(M) ⊗ N(Q) ⊗ C,

and: NC (M ⊗ Y (n + 1)) N(M ⊗ P ) ⊗ C y

NC (M ⊗ Y (n)) N(M ⊗ Q) ⊗ C.

Then by induction and our above argument on cofree objects, we obtain by [Hov99, 5.2.6] that N(M) ⊗ NC (Y (n + 1)) → NC (M ⊗ Y (n + 1)) is a weak homotopy equivalence. Now we can conclude as:

C C Γ(C) N(M) ⊗ N (Y ) ≃ N(M) ⊗ N (holimn Y (n)) C C ≃ N(M) ⊗ holimn N (Y (n)) e C C ≃ holimn (N(M) ⊗ N (Y (n)) C C ≃ holimn (N (M ⊗ Y (n)) C Γ(C) ≃ N (holimn (M ⊗ Y (n))) ≃ NC(M ⊗ Y ).

C We have used the fact that N and M ⊗− and N(M) ⊗− preservee towers that stabilize in each degree and they remain fibrant in the sense of Proposition 2.14. C C C We have just shown N (X(1))CN (Y ) → N (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 CoModΓ(C)(sModk) and in sModk:

X(n + 1) P ′ ⊗ Γ(C) y

X(n) Q′ ⊗ Γ(C),

′ ′ C C for some epimorphism P → Q in sModk. Since N and −CN (Y ) preserves fibrations that are epimor- phisms and pullbacks, we obtain the following two homotopy pullbacks in CoModΓ(C)(sModk):

C C ′ C N (X(n + 1))CN (Y ) N(P ) ⊗ N (Y ) y

C C ′ C N (X(n))C N (Y ) N(Q ) ⊗ N (Y ),

and: C C ′ N (X(n + 1)Γ(C)Y ) N (P ⊗ Y ) y

C C ′ N (X(n)Γ(C)Y ) N (Q ⊗ Y ).

By induction and our previous argument, this shows:

C C C N (X(n + 1))CN (Y ) → N (X(n + 1)Γ(C)Y ), is a weak homotopy equivalence. 28 Since NC preserves towers that stabilize in each degree and by Lemma 4.28, we can conclude by the following string of weak homotopy equivalences: C  C C Γ(C)  C N (X) CN (Y ) ≃ N (holimn X(n)) C N (Y ) C C  C ≃ (holimn N (X(n))) CN (Y ) e C C  C ≃ holimn (N (X(n)) CN (Y )) C C  ≃ holimn N (X(n) Γ(C)Y ) C Γ(C)  ≃ N (holimn (X(n) Γ(C)Y ) C Γ(C)  ≃ N ((holimn X(n)) Γ(C)Y ) C ≃ N (XΓ(C)Y ). NC  NC NC   Thus (X) C (Y ) −→ (X Γ(C)Y ) is a weak homotopye equivalence. Remark 5.5.e There is an alternativee 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:

CoModLR(D)(D) ⊥ CoModD(D). −DLR(D) If we combine with the previous construction:

L CoModR(D)(C) ⊥ CoModLR(D)(D), RR(D) and apply to the adjunction from the Dold-Kan correspondence (1.1), we obtain:

≥0   (CoModN(D)(Chk ), N(D), N(D))⊥ (CoModD(sModk), D,D),

for any simply connected cocommutative coalgebra D in sModk. 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 ≥0 cases. In particular the functor N : CoCAlg(sModk) → CoCAlg(Chk ) does not represent the correct right adjoint of the left derived functor of Γ on the ∞-categories of E∞-coalgebras in the underlying ∞-categories ≥0 of sModk and Chk . 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.

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Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, 19104-6395, USA Email address: [email protected]

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