# Rigidification of Connective Comodules

Maximilien Péroux  Department of Mathematics, University of Pennsylvania, 209
South 33rd Street, Philadelphia, PA, 19104-6395, USA<EMAIL_ADDRESS>
###### Abstract.

We show that we can rigidify homotopy coherent comodules in connective modules
over the Eilenberg-Mac Lane spectrum of a field, or more generally of a finite
product of fields $\Bbbk$. That is, the $\infty$-category of homotopy coherent
comodules is represented by a model category of strict comodules in non-
negative chain complexes over $\Bbbk$. These comodules are over a coalgebra
that is strictly coassociative and simply connected. The rigidification result
allows us to derive the notion of cotensor product of comodules and endows the
$\infty$-category of comodules with a symmetric monoidal structure via the
two-sided cobar resolution.

###### Key words and phrases:

comodule, derived cotensor product, cobar construction, rigidification,
Postnikov tower, model category, $\infty$-category

###### 2020 Mathematics Subject Classification:

16T15, 18M05, 18G35, 18N40, 18N70, 55P42, 55P43, 55U15, 55T20, 57T30

##  1\. Introduction

_Rigidification_ results, also sometimes called _rectification_ or
_strictification_ results, describe a correspondence between homotopy coherent
associative or commutative algebraic objects with their strict analogue. For
instance, strictly associative (and unital) algebras in the symmetric monoidal
model category of symmetric spectra correspond to
$\mathbb{A}_{\infty}$-algebras in the $\infty$-category of spectra, see
[HSS00] and [Lur17, 4.1.8.4]. We can also rigidify the action of modules over
a ring, see [Lur17, 4.3.3.17].

We are interested here in dual correspondences for _comodules_. It was shown
in [Pér20a] that rigidification results for coalgebras are challenging and do
not occur with our usual model categories. See also [PS19]. We present here,
in Theorem 3.3, a positive result. Let $\Bbbk$ be a finite product of fields
and denote $H\Bbbk$ its Eilenberg-Mac Lane spectrum. We consider
$\mathcal{D}^{\geq 0}(\Bbbk)$ the symmetric monoidal $\infty$-category of
_connective_ modules over $H\Bbbk$. It is represented by the symmetric
monoidal model category of non-negative chain complexes over $\Bbbk$, denoted
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, in which weak equivalences are quasi-
isomorphisms. If $C$ is a (strict) _simply connected_ coalgebra in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, then we show that the $\infty$-category of
(homotopy coherent) comodules in $\mathcal{D}^{\geq 0}(\Bbbk)$ over $C$ is
represented by a model category of (strictly coassociative) differential
graded comodules over $C$ in which weak equivalence are quasi-isomorphisms of
$C$-comodules.

######  Theorem 1.1 (Theorem 3.3).

Let $C$ be a simply connected differential graded coalgebra over $\Bbbk$. Then
right $C$-comodules in the $\infty$-category $\mathcal{D}^{\geq 0}(\Bbbk)$ of
connective $H\Bbbk$-modules correspond precisely to strictly coassociative
differential graded connective comodules over $C$.

The main application of the above result is that it allows us to define a
derived cotensor product of comodules defined via a two-sided cobar
resolution. Simply dualizing the results for modules seen in [Lur17, 4.5.2.1]
is not possible, as it would require totalization to commute with the monoidal
product, which is _not_ the case in $\mathcal{D}^{\geq 0}(\Bbbk)$. We instead
show that the cotensor product is compatible with the model structure of
differential graded $C$-comodules.

######  Theorem 1.2 (Corollary 4.12).

Let $C$ be a simply connected cocommutative differential graded coalgebra over
$\Bbbk$. The $\infty$-category $\mathcal{CoMod}_{C}(\mathcal{D}^{\geq
0}(\Bbbk))$ of $C$-comodules in $\mathcal{D}^{\geq 0}(\Bbbk)$ is endowed with
a symmetric monoidal structure induced by $\widehat{\square}_{C}$, the derived
cotensor of comodules. Moreover:

  1. (1)

Theorem 4.33: the derived cotensor product in
$\mathcal{CoMod}_{C}(\mathcal{D}^{\geq 0}(\Bbbk))$ is given by the two-sided
cobar resolution: $X\widehat{\square}_{C}Y\simeq\Omega(X,C,Y)$;

  2. (2)

Theorem 4.26: there is an Eilenberg-Moore spectral sequence computing
$H_{*}(X\widehat{\square}_{C}Y)$ with an $E^{2}$-page given by
$\mathsf{CoTor}_{H_{*}(C)}(H_{*}(X),H_{*}(Y))$;

  3. (3)

Proposition 4.29: if $C$ and $D$ are quasi-isomorphic simply connected
cocommutative differential graded coalgebras, then
$\mathcal{CoMod}_{C}(\mathcal{D}^{\geq 0}(\Bbbk))$ and
$\mathcal{CoMod}_{D}(\mathcal{D}^{\geq 0}(\Bbbk))$ are equivalent as symmetric
monoidal $\infty$-categories.

The proof of Theorem 1.1 is similar to the case of modules in [Lur17,
4.3.3.17]. However, homotopy limits of comodules are challenging to determine
compared to homotopy colimits of modules. The main obstacle is the fact that
the tensor product does not commute with limits in general. In order to
compute homotopy limits, we construct a Postnikov tower of comodules. Just as
CW approximations represent a cofibrant replacement for spaces, the Postnikov
towers provide an inductive fibrant replacement for differential graded
comodules. In fact, the model structure is fibrantly generated by a certain
class of fibrations, a notion we make precise in [Pér21].

######  Theorem 1.3 (Theorem 2.18).

Let $C$ be a simply connected differential graded coalgebra over $\Bbbk$. Let
$X$ be a right differential graded connective $C$-comodule. There exists a
tower of right $C$-comodules:

${\cdots}$${X(n)}$${X(n-1)}$${\cdots}$${X(2)}$${X(1)}$${0}$

with a sequence of maps $X\hookrightarrow X(n)$ such that:

  1. (1)

The map $X\hookrightarrow X(n)$ induces an isomorphism $H_{i}(X)\rightarrow
H_{i}(X(n))$ for every $i\leq n$;

  2. (2)

Each map $X(n)\rightarrow X(n-1)$ is a fibration of comodules and the fiber is
given by $K(V_{n},n-1)\otimes C$, the cofree comodule on the Eilenberg-Mac
Lane space; where $V_{n}$ is the cofiber $H_{n}(X)\rightarrow H_{n}(X(n))$;

  3. (3)

$X$ is quasi-isomorphic to $\mathsf{lim}^{C}_{n}X(n)$, the limit of the above
tower.

### Simplicial Comodules

The $\infty$-category $\mathcal{D}^{\geq 0}(\Bbbk)$ of connective
$H\Bbbk$-modules is also represented by the symmetric monoidal category of
simplicial $\Bbbk$-modules $\mathsf{sMod}_{\Bbbk}$ in which the weak
equivalences are weak homotopy equivalences. A simplicial coalgebra $C$ over
$\Bbbk$ is said to be simply connected if $C_{0}=\Bbbk$ and there are no non-
degenerate $1$-simplices. All arguments in this paper remain valid if we
replace ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ by $\mathsf{sMod}_{\Bbbk}$. In
particular Theorems 1.1, 1.2 and 1.3 have a simplicial version. However
$\mathsf{sMod}_{\Bbbk}$ and ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ are not
equivalent as monoidal categories, but they are after we derive their tensor
products. We show in [Pér21] there is a Dold-Kan correspondence between
simplicial comodules and differential graded connective comodules which
induces an equivalence of their derived cotensor products. Therefore the
induced derived cotensor product of comodules in $\mathcal{D}^{\geq 0}(\Bbbk)$
in Theorem 1.2 is independent from choosing between $\mathsf{sMod}_{\Bbbk}$ or
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$.

### A Rigidification For Connective Bicomodules

For the sake of simplicity, we focus in this paper on right comodules over a
coalgebra, but the above theorems remain true for left comodules. In fact,
given $C$ and $D$ simply connected differential graded coalgebras, we can
obtain a rigidification result as in Theorem 1.1 for $(C,D)$-bicomodules, as
they are equivalent to left $C\otimes D^{\mathsf{op}}$-comodules. In
particular, we can drop the cocommutativity requirement of Theorem 1.2 and
obtain a derived cotensor product for bicomodules over $C$ which provides a
monoidal (but not symmetric) structure. We provide further details in [KP].

### On the Finite Product of Fields Condition

Throughout this paper, we work exclusively with $\Bbbk$ a finite product of
fields. There are several reasons this condition is imposed. First, it induces
that every object is cofibrant and fibrant in ${\mathsf{Ch}_{\Bbbk}^{\geq
0}}$, and the projective and injective model structures are equal. In
particular, the model structure on comodules is left-induced from a nice
monoidal model category. In [HKRS17], it was shown that the model structures
for comodules are left-induced from injective model structures which are in
general not monoidal model categories. This creates several issues to
understand the induced homotopy theory on comodules. Moreover, as we are
working over a field, every module is flat and thus the tensor product
preserves finite limits. This allows us to understand finite limits of
comodules, which is essential for Theorem 1.3 and thus Theorem 1.1. Finally,
unless working over a finite product of fields, the cotensor product of
comodules is _not_ a comodule, hence Theorem 1.2 cannot be true. Notice though
that in [Pér20b, 6.4.7], we proved an unbounded version of Theorem 1.1, where
$\Bbbk$ is replaced by any commutative ring and $C$ is a differential graded
coalgebra that is perfect as a chain complex.

### On the Simply Connected Condition

The simply connected requirement on the coalgebra is crucial for Theorem 1.3
and thus for Theorem 1.1. It allows us to inductively build our fibrant
replacement, see Lemma A.7. Moreover, it allowed us to identify fibrant
comodules as coflat comodules in Proposition 4.16 and hence was required for
Theorem 1.2.

### Outline

Section 2 introduces the model category of differential graded connective
comodules. In Theorem 2.18 we state the Postnikov tower of Theorem 1.3. We
leave the details of the proof to Appendix A. In Section 3 we state and prove
our main result, which is Theorem 1.1. In Section 3 we show Theorem 1.2.

### 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. I am also grateful to Pete Bousfield who read earlier
versions of this paper and suggested the result Lemma 4.22. Special thanks to
initial conversations with Ben Antieau, Jonathan Beardsley and Kathryn Hess
that sparked some ideas in this paper. Thanks to Sarah Klanderman for her
comments.

### Notation

We begin by setting notation that we use throughout and recalling some
elementary notions of the theory of $\infty$-categories, following [Lur09,
Lur17]. The notions of _symmetric monoidal $\infty$-categories_ and _$\infty$
-operads_ are defined respectively in [Lur17, 2.0.0.7, 2.1.1.10].

  1. (1)

By an _ordinary_ category, we shall refer to a category in the usual sense of
the term. By an _$\infty$ -category_, we mean a quasicategory as in [Lur09].
An ordinary category shall be denoted with bold letters $\mathsf{C}$,
$\mathsf{A}$, etc, while an $\infty$-category shall be denoted with cursive
letters $\mathcal{C}$, $\mathcal{A}$, etc.

  2. (2)

The letter $\Bbbk$ shall always denote a finite product of fields, i.e. a
commutative ring $\Bbbk$ such that it is a product in rings:

$\Bbbk=\Bbbk_{1}\times\cdots\times\Bbbk_{n},$

where each $\Bbbk_{i}$ is a field, for some $1\leq n<\infty$. In the
literature, such rings are referred to as _commutative semisimple Artinian
rings_.

  3. (3)

Let ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ be the category of non-negative chain
complexes of $\Bbbk$-modules (graded homologically). The category is endowed
with a symmetric monoidal structure. The tensor product of two chain complexes
$X$ and $Y$ is defined by:

$(X\otimes Y)_{n}=\bigoplus_{i+j=n}X_{i}\otimes_{\Bbbk}Y_{j},$

with differential given on homogeneous elements by:

$d(x\otimes y)=dx\otimes y+(-1)^{|x|}x\otimes dy.$

We denote the tensor simply as $\otimes$. The monoidal unit is denoted
$\Bbbk$, which is the chain complex $\Bbbk$ concentrated in degree zero.

  4. (4)

Let $V$ be an $\Bbbk$-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 natural chain maps $D^{n}(V)\rightarrow S^{n}(V)$.
When $V=\Bbbk$, we simply write $D^{n}$ and $S^{n}$. Notice that $S^{n}(V)$
corresponds precisely (by the Dold-Kan correspondence) to an Eilenberg-Mac
Lane space $K(V,n)$.

  5. (5)

Let $\mathbb{N}$ denote the poset $\\{0<1<2<\cdots\\}$. Let $\mathsf{C}$ be
any complete category. Objects in $\mathsf{C}^{\mathbb{N}}$ are diagrams of
shape $\mathbb{N}$ and can be represented as (countable) towers in
$\mathsf{C}$:

${\cdots}$${X(2)}$${X(1)}$${X(0).}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$

We denote such an object as $\\{X(n)\\}=(X(n),f_{n})_{n\in\mathbb{N}}$. The
limit of the tower is denoted $\mathsf{lim}_{n}X(n)$.

  6. (6)

Let $\mathsf{M}$ be a model category and denote by $\mathsf{M}_{c}$ its
subcategory spanned by the cofibrant objects. Let $\mathsf{W}$ denote the
class of weak equivalences. The Dwyer-Kan localization, or underlying
$\infty$-category, of $\mathsf{M}$ is defined in [Lur17, 1.3.4.15] and is
denoted $\mathscr{N}(\mathsf{M}_{c})\left[\mathsf{W}^{-1}\right]$.

  7. (7)

Recall from [Lur17, 2.0.0.1] that for any symmetric monoidal category
$\mathsf{C}$, one can define the operator category $\mathsf{C}^{\otimes}$,
such that the nerve $\mathscr{N}(\mathsf{C}^{\otimes})$ is a symmetric
monoidal $\infty$-category whose underlying $\infty$-category is
$\mathscr{N}(\mathsf{C})$, see [Lur17, 2.1.2.21]. Let $\mathsf{M}$ be a
symmetric monoidal model category (see [Hov99, 4.2.6]). By the _symmetric
monoidal Dwyer-Kan localization_ of $\mathsf{M}$, we are referring to the
associated symmetric monoidal structure on the $\infty$-category
$\mathscr{N}(\mathsf{M}_{c})\left[\mathsf{W}^{-1}\right]$ as in [Lur17,
4.1.7.4] and [NS18, A.4, A.5].

##  2\. Postnikov Tower of Comodules

We present here a model structure for comodules over a coalgebra in connective
chain complexes (Proposition 2.12). We first recall the definitions in a
general symmetric monoidal category and then present the differential graded
case. The model structure is combinatorial and simplicial. Moreover, we show
that when working over a simply connected differential graded coalgebra, we
can build an inductive fibrant replacement similar to a Postnikov tower
(Theorem 2.18). This allows us to inductively compute homotopy limits of
comodules and is crucial to the next section. Details of the constructions are
in Appendix A.

###  2.1. Generality

Let us first recall some general notions of coalgebras and comodules in a
symmetric monoidal (ordinary) category.

######  Definition 2.1.

Let $(\mathsf{C},\otimes,\mathbb{I})$ be a symmetric monoidal category. A
_coalgebra_ $(C,\Delta,\varepsilon)$ in $\mathsf{C}$ consists of an object $C$
in $\mathsf{C}$ together with a _coassociative_ comultiplication
$\Delta:C\rightarrow C\otimes C$, such that the following diagram commutes:

${C}$${C\otimes C}$${C\otimes C}$${C\otimes C\otimes
C,}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta}$$\scriptstyle{\textup{id}_{C}\otimes\Delta}$$\scriptstyle{\Delta\otimes\textup{id}_{C}}$

and admits a _counit_ morphism $\varepsilon:C\rightarrow\mathbb{I}$ such that
we have the following commutative diagram:

${C\otimes C}$${C\otimes\mathbb{I}\cong C\cong\mathbb{I}\otimes C}$${C\otimes
C}$${C.}$$\scriptstyle{\textup{id}_{C}\otimes\varepsilon}$$\scriptstyle{\varepsilon\otimes\textup{id}_{C}}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta}$

The coalgebra is _cocommutative_ if the following diagram commutes:

${C\otimes C}$${C\otimes
C}$${C,}$$\scriptstyle{\tau}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta}$

where $\tau$ is the twist isomorphism from the symmetric monoidal structure of
$\mathsf{C}$. A _morphism of coalgebras_
$f:(C,\Delta,\varepsilon)\rightarrow(C^{\prime},\Delta^{\prime},\varepsilon^{\prime})$
is a morphism $f:C\rightarrow C^{\prime}$ in $\mathsf{C}$ such that the
following diagrams commute:

${C}$${C^{\prime}}$${C}$${C^{\prime}}$${C\otimes C}$${C^{\prime}\otimes
C^{\prime},}$${\mathbb{I}.}$$\scriptstyle{f}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varepsilon^{\prime}}$$\scriptstyle{f\otimes
f}$

The coalgebra is _coaugmented_ if there is a map $\eta:\mathbb{I}\rightarrow
C$ of coalgebras.

######  Definition 2.2.

Let $(\mathsf{C},\otimes,\mathbb{I})$ be symmetric monoidal category. Let
$(C,\Delta,\varepsilon)$ be a coalgebra in $\mathsf{C}$. A _right comodule
$(X,\rho)$ over $C$_, or a _right $C$-comodule_, is an object $X$ in
$\mathsf{C}$ together with a _coassociative_ and _counital_ right coaction
morphism $\rho:X\rightarrow X\otimes C$ in $\mathsf{C}$, i.e., the following
diagram commutes:

${X}$${X\otimes C}$${X}$${X\otimes C}$${X\otimes C}$${X\otimes C\otimes
C,}$${X\otimes\mathbb{I}}$${X.}$$\scriptstyle{\rho}$$\scriptstyle{\rho}$$\scriptstyle{\rho\otimes\mathsf{id}_{C}}$$\scriptstyle{\rho}$$\scriptstyle{\mathsf{id}_{X}\otimes\varepsilon}$$\scriptstyle{\mathsf{id}_{X}\otimes\Delta}$$\scriptstyle{\cong}$

The category of right $C$-comodules in $\mathsf{C}$ is denoted
$\mathsf{CoMod}_{C}(\mathsf{C})$. Similarly, we can define the category of
_left $C$-comodules_ where objects are endowed with a left coassociative
counital coaction $X\rightarrow C\otimes X$ and we denote the category by
${}_{C}\mathsf{CoMod}(\mathsf{C})$.

######  Remark 2.3.

If $C$ is a cocommutative comonoid in $\mathsf{C}$, the categories of left and
right comodules over $C$ are naturally isomorphic:
${}_{C}\mathsf{CoMod}(\mathsf{C})\cong\mathsf{CoMod}_{C}(\mathsf{C})$. In this
case, we omit whether the coaction is left or right.

######  Remark 2.4.

Since a coalgebra in $\mathsf{C}$ is an algebra in $\mathsf{C}^{\mathsf{op}}$,
then we can define the category of right comodules as modules in the opposite
category:
$\mathsf{CoMod}_{C}(\mathsf{C})={\left(\mathsf{Mod}_{C}(\mathsf{C}^{\mathsf{op}})\right)}^{\mathsf{op}}$,
and similarly for the left case.

######  Proposition 2.5.

Let $(\mathsf{C},\otimes,\mathbb{I})$ be symmetric monoidal category. Suppose
that $\mathsf{C}$ is presentable and the tensor product $\otimes$ preserves
filtered colimits in each variable. Then for any choice of coalgebra $C$ in
$\mathsf{C}$, the category of right $C$-comodules in $\mathsf{C}$ is
presentable, and we have an adjunction:

${\mathsf{CoMod}_{C}(\mathsf{C})}$${\mathsf{C}.}$$\scriptstyle{U}$$\scriptstyle{\perp}$$\scriptstyle{-\otimes
C}$

###### Proof.

Since $\mathsf{CoMod}_{C}(\mathsf{C})$ is the category of coalgebras over the
comonad $-\otimes C:\mathsf{C}\to\mathsf{C}$, apply [AR94, 2.78, 2.j]. ∎

######  Definition 2.6.

Following Proposition 2.5, for any object $X$ in $\mathsf{C}$, we say that
$X\otimes C$ is the _cofree right $C$-comodule generated by $X$_. Similarly,
we can define the _cofree left $C$-comodule generated by $X$ as $C\otimes X$_.

######  Definition 2.7.

Let $(\mathsf{C},\otimes,\mathsf{Hom}(-,-),\mathbb{I})$ be a closed symmetric
monoidal bicomplete category. Then $\mathsf{C}$ is enriched, tensored and
cotensored over itself. Let $C$ be a monoid in $\mathsf{C}$. Then
$\mathsf{CoMod}_{C}(\mathsf{C})$ is enriched over $\mathsf{C}$. For
$(X,\rho_{X})$ and $(Y,\rho_{Y})$ right $C$-comodules, we define the
enrichment $\mathsf{Hom}_{C}(X,Y)$ as the equalizer in $\mathsf{C}$:

${\mathsf{Hom}_{C}(X,Y)}$${\mathsf{Hom}(X,Y)}$${\mathsf{Hom}(X,Y\otimes C),}$

where we have omitted the forgetful functor $U$ from our notation. The first
parallel morphism is defined by post-composing with $\rho_{Y}:Y\rightarrow
Y\otimes C$. The second morphism is defined by the forgetful-cofree
adjointness. Additionally, the category $\mathsf{CoMod}_{C}(\mathsf{C})$ is
tensored (and cotensored) over $\mathsf{C}$. For any object $M$ in
$\mathsf{C}$, the functor $M\otimes-:\mathsf{C}\rightarrow\mathsf{C}$ is
strong monoidal, and thus, given any comonoid $C$ in $\mathsf{C}$, the functor
lifts to a functor
$M\otimes-:\mathsf{CoMod}_{C}(\mathsf{C})\rightarrow\mathsf{CoMod}_{C}(\mathsf{C})$.

If $\mathsf{M}$ is a symmetric monoidal model category, and if we suppose
$\mathsf{CoMod}_{C}(\mathsf{M})$ is endowed with a model structure left-
induced from $\mathsf{C}$ (in the sense of [HKRS17]), then the Quillen
bifunctor $-\otimes-:\mathsf{M}\times\mathsf{M}\rightarrow\mathsf{M}$ lifts to
a Quillen bifunctor:

$-\otimes-:\mathsf{M}\times\mathsf{CoMod}_{C}(\mathsf{M})\rightarrow\mathsf{CoMod}_{C}(\mathsf{M}),$

as cofibrations, weak equivalences, and colimits in
$\mathsf{CoMod}_{C}(\mathsf{M})$ are determined in $\mathsf{M}$. Thus we have
shown the following.

######  Proposition 2.8.

Let $\mathsf{M}$ be a combinatorial closed symmetric monoidal model category
with cofibrant unit. Let $C$ be a comonoid in $\mathsf{M}$ such that the
forgetful functor left-induces a model structure on
$\mathsf{CoMod}_{C}(\mathsf{M})$. Then $\mathsf{CoMod}_{C}(\mathsf{M})$ is a
$\mathsf{M}$-model category in the sense of [Hov99, 4.2.18].

###  2.2. The Differential Graded Case

We now focus on the case where $\mathsf{C}={\mathsf{Ch}_{\Bbbk}^{\geq 0}}$,
the category of non-negative chain complexes of $\Bbbk$-modules.

######  Definition 2.9.

Let $\Bbbk$ be a finite product of fields. There is a model structure on
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ in which weak equivalences are quasi-
isomorphisms, cofibrations are monomorphisms, and fibrations are positive
levelwise epimorphisms ([Qui67]). It is a combinatorial symmetric monoidal
model category, see [SS03]. It is a simplicial model category via the Dold-Kan
correspondence. Every object is cofibrant and fibrant.

######  Definition 2.10.

A coalgebra $C$ in the symmetric monoidal category ${\mathsf{Ch}_{\Bbbk}^{\geq
0}}$ is said to be a _(connective) dg-coalgebra over $\Bbbk$_. A _right
$C$-dg-comodule_ or _right dg-comodule over $C$_ is a right $C$-comodule in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. A _quasi-isomorphism of (right)
$C$-comodules_ is a map of right $C$-comodules such that it induces an
isomorphism on the homologies as a chain map.

######  Proposition 2.11.

Let $C$ be a dg-coalgebra over $\Bbbk$. The forgetful functor
$U:\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ creates and preserves colimits,
finite limits, monomorphisms and epimorphisms. In particular, a short sequence
is exact in $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ if and only
if it is exact in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ via $U$.

###### Proof.

Since $U$ is a left adjoint, it preserves colimits and epimorphisms. Since we
are working over a finite product of fields, the comonad $-\otimes
C:{\mathsf{Ch}_{\Bbbk}^{\geq 0}}\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq 0}}$
preserves finite limits and monomorphisms. Since
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ is the category of
coalgebras over that comonad, the result follows. ∎

######  Proposition 2.12.

Let $C$ be a dg-coalgebra over $\Bbbk$. Then the category of right $C$-dg-
comodules admits a combinatorial model category left-induced from the
forgetful-cofree adjunction:

${\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})}$${{\mathsf{Ch}_{\Bbbk}^{\geq
0}}.}$$\scriptstyle{U}$$\scriptstyle{\perp}$$\scriptstyle{-\otimes C}$

In particular, $U$ preserves and reflects cofibrations and weak equivalences.
Therefore the weak equivalences are precisely the quasi-isomorphisms of right
$C$-comodules, and the cofibrations are precisely the monomorphisms of right
$C$-comodules. Every object is cofibrant. The model structure is combinatorial
and simplicial.

###### Proof.

One can adapt the arguments from [HKRS17, 6.3.7] for unbounded chain complexes
to ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. The model structure is combinatorial by
[BHK+15, 2.23] combined with [HKRS17, 3.3.4]. It is simplicial by Proposition
2.8 and the Dold-Kan correspondence. ∎

###  2.3. Inductive Fibrant Replacement

We provide here a computationally effective fibrant replacement in the model
structure $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$. We will need
to restrict to simply connected dg-coalgebra (Definition 2.17). In Section 4,
we will also identify fibrant $C$-comodules with _coflat $C$-comodules_
(Proposition 4.16). Just as CW-approximation is a cofibrant replacement for
spaces which provides cell by cell arguments, the fibrant replacement in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ is built as a Postnikov
tower and allows layer by layer arguments. The details of the construction are
left in Appendix A in which we study fibrations of $C$-comodules.

We are interested in homotopy limits of countable towers in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$. In general, unlike
pullbacks, limits of towers in $\mathsf{CoMod}_{C}(\mathsf{M})$ are very
different than limits of the underlying towers in $\mathsf{M}$, for a general
monoidal category $\mathsf{M}$.

######  Notation 2.13.

If $\\{X(n)\\}$ is a tower of right $C$-comodules in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, we denote its limit by
$\mathsf{lim}_{n}^{C}X(n)$, and if we forget the $C$-comodule coactions, we
denote the limit in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ by
$\mathsf{lim}_{n}U(X(n))$.

######  Definition 2.14.

A tower $\\{X(n)\\}$ in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ _stabilizes in each
degree_ if for each degree $i\geq 0$, the tower $\\{X(n)_{i}\\}$ of
$\Bbbk$-modules stabilizes for $n\geq i+1$, i.e., for all $n\geq 0$, and all
$0\leq i\leq n$, we have:

$X(n+1)_{i}\cong X(n+2)_{i}\cong X(n+3)_{i}\cong\cdots.$

Let $C$ be a dg-coalgebra over $\Bbbk$. A tower $\\{X(n)\\}$ in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ _stabilizes in each
degree_ if the underlying tower $\\{U(X(n))\\}$ in ${\mathsf{Ch}_{\Bbbk}^{\geq
0}}$ stabilizes in each degree.

######  Lemma 2.15.

Let $\\{X(n)\\}$ be a tower in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ that
stabilizes in each degree. Let $C$ be any (non-negative) chain complex. Then
the tower $\Big{\\{}X(n)\otimes C\Big{\\}}$ in ${\mathsf{Ch}_{\Bbbk}^{\geq
0}}$ also stabilizes in each degree and we have:

$\Big{(}\mathsf{lim}_{n}X(n)\Big{)}\otimes
C\cong\mathsf{lim}_{n}\Big{(}X(n)\otimes C\Big{)}.$

###### Proof.

For all $n\geq 0$, and all $0\leq i\leq n$, we have:

$\displaystyle\Big{(}X(n+1)\otimes C\Big{)}_{i}$ $\displaystyle=$
$\displaystyle\bigoplus_{a+b=i}X(n+1)_{a}\otimes C_{b}$ $\displaystyle\cong$
$\displaystyle\bigoplus_{a+b=i}X(n+2)_{a}\otimes C_{b}$ $\displaystyle=$
$\displaystyle\Big{(}X(n+2)\otimes C\Big{)}_{i},$

as $0\leq a\leq i\leq n$. This argument generalizes in higher degrees and thus
shows that the desired tower stabilizes in each degree. For all $i\geq 0$,
notice that both $\Big{(}\Big{(}\mathsf{lim}_{n}X(n)\Big{)}\otimes
C\Big{)}_{i}$ and $\Big{(}\mathsf{lim}_{n}\Big{(}X(n)\otimes
C\Big{)}\Big{)}_{i}$ are equal to
$\displaystyle\bigoplus_{a+b=i}X(i+1)_{a}\otimes C_{b}$. ∎

######  Corollary 2.16.

Let $C$ be a dg-coalgebra over $\Bbbk$. Let $\\{X(n)\\}$ be a tower in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ that stabilizes in each
degree. Then the natural map:

$U(\mathsf{lim}^{C}_{n}X(n))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathsf{lim}_{n}U(X(n))$

is an isomorphism in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$.

###### Proof.

This follows directly from Lemma 2.15 as $U$ preserves and reflects a limit
precisely when the comonad $-\otimes C:{\mathsf{Ch}_{\Bbbk}^{\geq
0}}\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ preserves that limit. In detail,
if we denote $X:=\mathsf{lim}_{n}U(X(n))$, then the coaction $X\rightarrow
X\otimes C$ is constructed as follows. For each degree $i\geq 0$, the map
$X_{i}\rightarrow(X\otimes C)_{i}$ is entirely determined by the coaction
$X(i+1)\rightarrow X(i+1)\otimes C$. ∎

######  Definition 2.17.

A dg-coalgebra $C$ over $\Bbbk$ is said to be _simply connected_ if
$C_{0}=\Bbbk$ and $C_{1}=0$.

For instance, given any simply connected space $X$, its singular chain complex
$C_{\ast}(X;\Bbbk)$ is a simply connected dg-coalgebra.

We build a fibrant replacement for any $C$-comodule $X$ inductively as a
Postnikov tower of comodules. Each fiber of the layer is the cofree comodule
of an Eilenberg-Mac Lane space. As the level increases in the tower, the
layers approximate $X$ homologically.

######  Theorem 2.18.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. Let $X$ be a right
$C$-dg-comodule. There exists a fibrant replacement
$X\stackrel{{\scriptstyle\simeq}}{{\hookrightarrow}}\widetilde{X}$ of right
$C$-comodules, constructed inductively as follows. The right $C$-dg-comodule
$\widetilde{X}$ is the limit of a countable tower $\\{X(n)\\}$ of right
$C$-dg-comodules that stabilizes in each degree:

${\widetilde{X}}$${\vdots}$${X(n+1)}$${S^{n}(V_{n+1})\otimes
C}$${X(n)}$${S^{n-1}(V_{n})\otimes C}$${\vdots}$${X(2)}$${S^{1}(V_{2})\otimes
C}$${X}$${X(1)}$${U(X)\otimes C}$${X(0)}$${0}$

where $V_{n}$ is the $\Bbbk$-module defined as the cofiber:

${0}$${H_{n}(X)}$${H_{n}(X(n))}$${V_{n}}$${0.}$

For $n\geq 2$, the right $C$-dg-comodule $X(n)$ is defined as the pullack of
$C$-comodules:

${X({n})}$${D^{n}(V_{n})\otimes C}$${X(n-1)}$${S^{n}(V_{n})\otimes
C.}$${\lrcorner}$

Each map $X(n+1)\rightarrow X(n)$ is a fibration as a right $C$-dg-comodules
and its fiber is $S^{n}(V_{n+1})\otimes C$. Moreover, for $n\geq 1$, the
cofibration $X\hookrightarrow X(n)$ induces an isomorphism $H_{i}(X)\cong
H_{i}(X(n))$ for all $0\leq i\leq n$.

We prove the above theorem at the end of Appendix A.

######  Definition 2.19.

Let $X$ be a right dg-comodule over a simply connected dg-coalgebra $C$ over a
finite product of fields $\Bbbk$. The _Postnikov tower of $X$_ is the tower
$\\{X(n)\\}$ in $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ built in
Theorem 2.18.

######  Remark 2.20.

Our Postnikov towers are not functorials: given a map $X\rightarrow Y$ of
$C$-comodules, and given $\\{X(n)\\}$ and $\\{Y(n)\\}$ their respective
Postnikov towers, we do not obtain maps $X(n)\rightarrow Y(n)$.

We now show that the Postnikov tower is homotopically well behaved. We first
need the following proposition.

######  Proposition 2.21 ([GJ99, VI.1.1]).

Let $\mathsf{C}$ be a model category. Then the category of towers
$\mathsf{C}^{\mathbb{N}}$ can be endowed with the Reedy model structure, where
a map $\\{X(n)\\}\rightarrow\\{Y(n)\\}$ is a weak equivalence (respectively a
cofibration), if each map $X(n)\rightarrow Y(n)$ is a weak equivalence
(respectively a cofibration) in $\mathsf{C}$, for all $n\geq 0$. An object
$\\{X(n)\\}$ is fibrant if and only if $X(0)$ is fibrant and all the maps
$X(n+1)\rightarrow X(n)$ in the tower are fibrations in $\mathsf{C}$.
Moreover, if we denote $\iota:\mathsf{C}\rightarrow\mathsf{C}^{\mathbb{N}}$
the functor induced by the constant diagram, then we obtain a Quillen
adjunction
${\iota:\mathsf{C}}$${\mathsf{C}^{\mathbb{N}}:\mathsf{lim}_{n}.}$$\scriptstyle{\perp}$

######  Definition 2.22.

If $\\{X(n)\\}$ is a tower of right $C$-comodules in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, we denote its homotopy limit by
$\mathsf{holim}_{n}^{C}X(n)$ and is the right derived functor of
$\mathsf{lim}_{n}^{C}$ of Proposition 2.21. If we forget the $C$-comodule
coactions, we denote the homotopy limit in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ by
$\mathsf{holim}_{n}U(X(n))$.

######  Corollary 2.23.

Let $X$ be a right dg-comodule over a simply connected dg-coalgebra $C$ over a
finite product of fields $\Bbbk$. Then the maps $X(n+1)\rightarrow X(n)$ are
epimorphisms of chain complexes and thus in particular fibrations in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. In particular $\widetilde{X}$ is the
homotopy limit of $\\{X(n)\\}$ in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ and in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, i.e. we have:

$U(\widetilde{X})\simeq
U(\mathsf{holim}^{C}_{n}X(n))\simeq\mathsf{holim}_{n}(U(X(n))).$

###### Proof.

We omit the forgetful functor $U$ in our proof. For all $n\geq 1$, we see
$D^{n}(V)\rightarrow S^{n}(V)$ is an epimorphism of chain complexes, for any
$\Bbbk$-module $V$. Since $-\otimes C$ preserves epimorphisms, we see that
$D^{n}(V)\otimes C\rightarrow S^{n}(V)\otimes C$ remains an epimorphism of
chain complexes and thus a fibration in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. As
chain complexes, we have the pullback:

${X({n+1})}$${D^{n}(V_{n+1})\otimes C}$${X(n)}$${S^{n}(V_{n+1})\otimes
C.}$${\lrcorner}$

Since pullback preserves fibrations, we get that $X(n+1)\rightarrow X(n)$ is a
fibration in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. ∎

##  3\. Rigidification of Comodules

We present and prove here our main result in Theorem 3.3. Throughout, let
$\Bbbk$ be a finite product of fields. Let $C$ be a simply connected dg-
coalgebra (Definition 2.17). Let $H\Bbbk$ be the Eilenberg-Mac Lane spectrum
of $\Bbbk$. We show that right $C$-comodules in connective $H\Bbbk$-modules
with a coaction that is coassociative up to higher homotopies are equivalent
to right $C$-dg-comodules over $\Bbbk$ with a strictly coassociative coaction.
Our proof will depend on the Postnikov tower of Definition 2.19.

We first make the notion of homotopy coherent coassociative comodules precise
using an $\infty$-categorical approach. Let $\mathcal{C}$ be a symmetric
monoidal $\infty$-category, as in [Lur17, 2.0.0.7]. Let $\mathbb{A}_{\infty}$
be the associative $\infty$-operad as in [Lur17, 4.1.1.6]. Let $A$ be an
$\mathbb{A}_{\infty}$-algebra in $\mathcal{C}$, as in [Lur17, 4.1.1.6]. We
denote ${}_{A}\mathcal{Mod}(\mathcal{C})$ the $\infty$-category of left
$A$-modules, instead of $\mathcal{LMod}_{A}(\mathcal{C})$ as in [Lur17,
4.2.1.13]. We similarly denote $\mathcal{Mod}_{A}(\mathcal{C})$ the
$\infty$-category of right $A$-modules.

######  Definition 3.1 ([Pér22, 2.1]).

Let $\mathcal{C}$ be a symmetric monoidal $\infty$-category. An
_$\mathbb{A}_{\infty}$ -coalgebra_ in $\mathcal{C}$ is an
$\mathbb{A}_{\infty}$-algebra in the opposite category
$\mathcal{C}^{\mathsf{op}}$.

Given a symmetric monoidal model category $\mathsf{M}$, any coalgebra $C$ in
$\mathsf{M}$ that is cofibrant as an object in $\mathsf{M}$ can be regarded as
an $\mathbb{A}_{\infty}$-coalgebra in the underlying $\infty$-category
$\mathscr{N}(\mathsf{M}_{c})[\mathsf{W}^{-1}]$ of $\mathsf{M}$. See more
details in [Pér20a]. Essentially, a strictly coassociative coalgebra is
naturally coassociative up to coherent homotopies.

######  Definition 3.2.

Let $\mathcal{C}$ be a symmetric monoidal $\infty$-category. Let $C$ be an
$\mathbb{A}_{\infty}$-coalgebra in $\mathcal{C}$. Then define the category of
right $C$-comodules in $\mathcal{C}$ as:

$\mathcal{CoMod}_{C}(\mathcal{C}):={\left(\mathcal{Mod}_{C}(\mathcal{C}^{\mathsf{op}})\right)}^{\mathsf{op}}.$

We define the $\infty$-category of left $C$-comodules
${}_{C}\mathcal{CoMod}(\mathcal{C})$ similarly.

Let $\mathsf{M}$ be a symmetric monoidal category. Let $\mathsf{M}^{\otimes}$
be the operator category of $\mathsf{M}$ as in [Lur17, 2.0.0.1]. Then the
nerve $\mathscr{N}(\mathsf{M}^{\otimes})$ is a symmetric monoidal
$\infty$-category whose underlying $\infty$-category is
$\mathscr{N}(\mathsf{M})$, see [Lur17, 2.1.2.21]. Let $C$ be a coalgebra in
$\mathsf{M}$. It can be regarded as an $\mathbb{A}_{\infty}$-coalgebra in
$\mathscr{N}(\mathsf{M})$. There is an equivalence of $\infty$-categories:

$\mathscr{N}(\mathsf{CoMod}_{C}(\mathsf{M}))\simeq\mathcal{CoMod}_{C}(\mathscr{N}(\mathsf{M})).$

Suppose now that $\mathsf{M}$ is a combinatorial symmetric monoidal model
category in which each object is cofibrant. By [Lur17, 4.1.7.6], the
underlying $\infty$-category
$\mathscr{N}(\mathsf{M})\left[\mathsf{W}^{-1}\right]$ is symmetric monoidal
via the derived tensor product of $\mathsf{M}$. Moreover, the localization
functor
$\mathscr{N}(\mathsf{M}^{\otimes})\rightarrow\mathscr{N}(\mathsf{M})\left[\mathsf{W}^{-1}\right]^{\otimes}$
is symmetric monoidal. In particular, we obtain a functor:

$\mathcal{CoMod}_{C}(\mathscr{N}(\mathsf{M}))\longrightarrow\mathcal{CoMod}_{C}\left(\mathscr{N}(\mathsf{M})\left[\mathsf{W}^{-1}\right]\right).$

By construction, the functor sends weak equivalences to equivalences, and thus
by the universal property of the localization, we obtain a functor:

${\gamma_{C}:\mathscr{N}(\mathsf{CoMod}_{C}(\mathsf{M}))\left[\mathsf{W}_{\mathsf{CoMod}}^{-1}\right]}$${\mathcal{CoMod}_{{C}}\left(\mathscr{N}(\mathsf{M})\left[\mathsf{W}^{-1}\right]\right).}$

Here $\mathsf{W}_{\mathsf{CoMod}}$ denotes the class of weak equivalences
between right $C$-comodules in $\mathsf{M}$.

Objects in
$\mathcal{CoMod}_{{C}}\left(\mathscr{N}(\mathsf{M})\left[\mathsf{W}^{-1}\right]\right)$
are $C$-comodules in $\mathsf{M}$ with a coaction that is coassociative only
up to higher homotopies. Objects in
$\mathscr{N}(\mathsf{CoMod}_{C}(\mathsf{M}))\left[\mathsf{W}_{\mathsf{CoMod}}^{-1}\right]$
are $C$-comodules in $\mathsf{M}$ with a coaction that is strictly
coassociative. We have just shown that comodules with a strictly coassociative
coaction are naturally endowed with a coaction that is coassociative up to
coherent homotopies.

If the functor $\gamma_{C}$ above is an equivalence of $\infty$-categories,
then this shows that any $C$-comodule in $\mathsf{M}$ with a homotopy coherent
coassociative coaction is weakly equivalent to a $C$-comodule with a strictly
coassociative coaction. We say that we can _rigidify the coaction_.

We apply our above discussion to $\mathsf{M}={\mathsf{Ch}_{\Bbbk}^{\geq 0}}$,
the category of non-negative chain complexes over $\Bbbk$, a finite product of
fields. It is a symmetric monoidal model category (with respect to quasi-
isomorphisms) and every object is cofibrant. By [Lur17, 7.1.3.10], the
underlying $\infty$-category $\mathscr{N}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})[\mathsf{W}^{-1}]$ is equivalent as a symmetric monoidal $\infty$-category
to connective $H\Bbbk$-modules in spectra, which we denote $\mathcal{D}^{\geq
0}(\Bbbk)$. Here $H\Bbbk$ denotes the Eilenberg-Mac Lane spectrum of $\Bbbk$.
In this situation, we show we can always rigidify the coaction of a comodule.

######  Theorem 3.3.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. Then there is a
natural equivalence of $\infty$-categories:

${\gamma_{C}:\mathscr{N}\left(\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\right)\left[\mathsf{W}^{-1}_{\mathsf{CoMod}}\right]}$${\mathcal{CoMod}_{C}\left(\mathcal{D}^{\geq
0}(\Bbbk)\right),}$$\scriptstyle{\simeq}$

where $\mathsf{W}_{\mathsf{CoMod}}$ denotes the class of $C$-comodule maps
that are quasi-isomorphisms in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$.

We prove the theorem at the end of this section. The result is dual to case of
modules in [Lur17, 4.3.3.17], and we shall follow the same arguments in our
proof. However, (homotopy) limits in comodules are more challenging to compute
than (homotopy) colimits in modules. In fact, the forgetful functor from
modules to its underlying presentably symmetric monoidal $\infty$-category
$\mathcal{C}$ preserves geometric realizations, but the forgetful functor from
comodules to $\mathcal{C}$ does _not_ preserve totalizations. In order to
compute the desired homotopy limits of comodules, we use our Postnikov towers
of Definition 2.19. Our approach is similar to the proof of [Lur17, 4.1.8.13]
in which homotopy colimits of algebras are computed using the cellular
structure of the model category.

######  Remark 3.4.

We obtained a rigidification for comodules over a _discrete_
$\mathbb{A_{\infty}}$-coalgebra in $\mathcal{D}^{\geq 0}(\Bbbk)$. Unlike the
case of algebras (see [Lur17, 4.1.8.4]), it is unlikely that
$\mathbb{A}_{\infty}$-coalgebras in $\mathcal{D}^{\geq 0}(\Bbbk)$ are
equivalent to (strictly coassociative) dg-coalgebras in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. See [Pér20a, 4.3].

###  3.1. Barr-Beck-Lurie Comonadicity Theorem

We invite the reader to look at the definition in $\infty$-categories of a
_monadic functor_ in [Lur17, 4.7.3.4]. A functor
$\mathcal{C}\rightarrow\mathcal{D}$ of $\infty$-categories will be called
_comonadic_ if its opposite
$\mathcal{C}^{\mathsf{op}}\rightarrow\mathcal{D}^{\mathsf{op}}$ is monadic.
More precisely, a left adjoint functor $\mathcal{C}\rightarrow\mathcal{D}$ in
$\infty$-categories exhibits $\mathcal{C}$ as comonadic over $\mathcal{D}$ if
$\mathcal{C}$ is equivalent to coalgebras over the comonad over $\mathcal{D}$
determined by the adjunction.

We recall a necessary and sufficient condition for a left adjoint functor to
be comonadic. This is analogous to the situation in ordinary categories where
a left adjoint $L$ is comonadic if and only it preserves $L$-split equalizers.
The $\infty$-categorical notion of $L$-split coaugmented cosimplical objects
is entirely dual to the simplicial analogue described in [Lur17, 4.7.2.2].

######  Theorem 3.5 ([Lur17, 4.7.3.5] Barr-Beck-Lurie Comonadicity Theorem).

A functor $F:\mathcal{C}\rightarrow\mathcal{D}$ in $\infty$-categories
exhibits $\mathcal{C}$ as comonadic over $\mathcal{D}$ if and only if it
admits a right adjoint, is conservative, and preserves all limits of $F$-split
coaugmented cosimplicial objects.

######  Theorem 3.6 ([Lur17, 4.7.3.16]).

A functor $V:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ is an equivalence of
$\infty$-categories if there is a left adjoint functor
$F^{\prime}:\mathcal{C}^{\prime}\rightarrow\mathcal{D}$ such that $F^{\prime}$
and $F^{\prime}\circ V$ exhibit both $\mathcal{C}$ and $\mathcal{C}^{\prime}$
as comonadic over $\mathcal{D}$ over the same comonad. More precisely, given
the following diagram of $\infty$-categories where $V$ commutes with the left
adjoints:

${\mathcal{C}}$${\mathcal{D}}$${\mathcal{C}^{\prime}}$$\scriptstyle{V}$$\scriptstyle{F}$$\scriptstyle{\perp}$$\scriptstyle{G}$$\scriptstyle{\perp}$$\scriptstyle{G^{\prime}}$$\scriptstyle{F^{\prime}}$

the functor $V:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ is an equivalence
of $\infty$-categories if:

  * •

the functor $F:\mathcal{C}\rightarrow\mathcal{D}$ exhibits $\mathcal{C}$ as
comonadic over $\mathcal{D}$;

  * •

the functor $F^{\prime}:\mathcal{C}^{\prime}\rightarrow\mathcal{D}$ exhibits
$\mathcal{C}^{\prime}$ as comonadic over $\mathcal{D}$;

  * •

the canonical map $(F\circ G)\rightarrow(F^{\prime}\circ G^{\prime})$ is an
equivalence of functors.

######  Proposition 3.7 ([Lur17, 4.7.2.5]).

Let $\mathcal{C}$ be a monoidal $\infty$-category. Given any
$\mathbb{A}_{\infty}$-coalgebra $C$ in $\mathcal{C}$, the forgetful functor
exhibits the $\infty$-category of (right) $C$-comodule
$\mathcal{CoMod}_{C}(\mathcal{C})$ as comonadic over $\mathcal{C}$.

The following argument appeared in the proof of Theorem 0.3 in [Heu18].

######  Definition 3.8.

Given a pair of adjoint functors
${L:\mathcal{C}}$${\mathcal{D}:R}$$\scriptstyle{\perp}$ of
$\infty$-categories, we define the _canonical $RL$-resolution_ which is the
following $L$-split coaugmented cosimplicial object in $\mathcal{C}$, induced
by the comonad $LR$ on $\mathcal{D}$:

${X}$${RL(X)}$${RLRL(X)}$${RLRLRL(X)\cdots.}$

We shall denote the $L$-split coaugmented cosimplicial object by $X\rightarrow
RL^{\bullet+1}(X)$.

Given a complete $\infty$-category and a small category $S$, we denote the
limit of a functor $F:\mathscr{N}(S)\rightarrow\mathcal{C}$ by
$\mathcal{lim}_{S}^{\mathcal{C}}F$.

######  Proposition 3.9.

Let ${L:\mathcal{C}}$${\mathcal{D}:R}$$\scriptstyle{\perp}$ be a pair of
adjoint functors in $\infty$-categories, such that $L$ is conservative. Then
$L$ is comonadic if and only if the map
$X\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathcal{lim}^{\mathcal{C}}_{\Delta}(RL^{\bullet+1}(X))$
is an equivalence for all objects $X$ in $\mathcal{C}$.

###### Proof.

We show the sufficient condition. Let $X^{-1}\rightarrow X^{\bullet}$ be an
$L$-split cosimplicial object of $\mathcal{C}$. We have the following square:

${X^{-1}}$${\mathcal{lim}_{\Delta}^{\mathcal{C}}(X^{\bullet})}$${\mathcal{lim}_{\Delta}^{\mathcal{C}}\left(RL^{\bullet+1}(X^{-1})\right)}$${\mathcal{lim}_{\Delta\times\Delta}^{\mathcal{C}}\left(RL^{\bullet+1}(X^{\bullet})\right).}$

The vertical maps are equivalences by assumption. The bottom horizontal map is
an equivalence as $X^{-1}\rightarrow X^{\bullet}$ is $L$-split. Indeed, we
have $L(X^{-1})\simeq\mathcal{lim}_{\Delta}^{\mathcal{D}}L(X^{\bullet})$, and
since $R$ preserves limits, we get
$RL(X^{-1})\simeq\mathcal{lim}_{\Delta}^{\mathcal{C}}RL(X^{\bullet})$. Since
the coaugmented cosimplicial object $RL(X^{-1})\rightarrow RL(X^{\bullet})$
remains $L$-split, we can reiterate our argument and thus show that the bottom
horizontal map is an equivalence in $\mathcal{C}$. Therefore the top
horizontal map is an equivalence, as desired. ∎

###  3.2. Two-Sided Cosimplicial Cobar Resolution

Throughout this section, let $C$ be a simply connected dg-coalgebra over
$\Bbbk$. From the Quillen adjunction (Proposition 2.12):

${\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})}$${{\mathsf{Ch}_{\Bbbk}^{\geq
0}},}$$\scriptstyle{U}$$\scriptstyle{\perp}$$\scriptstyle{-\otimes C}$

we obtain an adjunction in $\infty$-categories induced by the derived
functors:

${\mathscr{N}(\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}}))[\mathsf{W}^{-1}_{\mathsf{CoMod}}]}$${\mathscr{N}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})[\mathsf{W}^{-1}]\simeq\mathcal{D}^{\geq
0}(\Bbbk).}$$\scriptstyle{U}$$\scriptstyle{\perp}$$\scriptstyle{-\otimes C}$

Here $\mathsf{W}_{\mathsf{CoMod}}$ denotes the class of $C$-comodule maps that
are quasi-isomorphisms in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, and $\mathsf{W}$
denotes the class of quasi-isomorphisms. Since every object is cofibrant in
$\mathsf{CoMod}_{C}$ and every object is fibrant in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, the derived functors above are identical to
their underived versions. We apply Definition 3.8 to obtain a $U$-split
coaugmented cosimplicial object
$\mathscr{N}(\Delta)\rightarrow\mathscr{N}(\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}}))[\mathsf{W}^{-1}_{\mathsf{CoMod}}]$. By [Lur17, 1.3.4.25], the
cosimplicial object is determined by a cosimplicial object in the model
category $\Delta\rightarrow\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})$:

${X}$${U(X)\otimes C}$${U\Big{(}U(X)\otimes C\Big{)}\otimes C}$${\cdots.}$

Subsequently, we may omit the functor $U$ for convenience. This leads to the
following definition.

######  Definition 3.10.

Let $(C,\Delta,\varepsilon)$ be a simply connected dg-coalgebra over $\Bbbk$.
Let $(X,\rho)$ be a right $C$-dg-comodule. The _two-sided cosimplicial cobar
construction_ $\Omega^{\bullet}(X,C,C)$ of $X$ is the cosimplicial object in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$:

${X}$${X\otimes C}$${X\otimes C\otimes C}$${\cdots,}$

defined as follows.

  * •

For all $n\geq-1$: $\Omega^{n}(X,C,C)=X\otimes C^{\otimes n+1}$.

  * •

The zeroth coface map
$d^{0}=\rho\otimes\mathsf{id}_{C^{n+1}}:\Omega^{n}(X,C,C)\rightarrow\Omega^{n+1}(X,C,C)$.

  * •

For $1\leq i\leq n+1$, the $i$-th coface map is given by:

$d^{i}=\mathsf{id}_{X}\otimes\mathsf{id}_{C^{\otimes
i-1}}\otimes\Delta\otimes\mathsf{id}_{C^{\otimes
n+1-i}}:\Omega^{n}(X,C,C)\rightarrow\Omega^{n+1}(X,C,C)$

  * •

For all $0\leq j\leq n$ the $j$-th codegeneracy map is given by:

$s^{j}=\mathsf{id}_{X}\otimes\mathsf{id}_{C^{\otimes
j}}\otimes\varepsilon\otimes\mathsf{id}_{C^{\otimes
n+1-j}}:\Omega^{n+1}(X,C,C)\rightarrow\Omega^{n}(X,C,C).$

It is $U$-split coaugmented in the sense that there exist extra degeneracies:

$s^{-1}=\mathsf{id}_{X}\otimes\mathsf{id}_{C^{\otimes
n+1}}\otimes\varepsilon:\Omega^{n+1}(X,C,C)\rightarrow\Omega^{n}(X,C,C),$

for all $n\geq-1$, which are dashed in the diagram above. These maps are _not_
maps of $C$-comodules but only chain maps. Since
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ is a simplicial model
category by Proposition 2.12, homotopy limits over cosimplicial diagrams are
computed as in [Hir03, 18.1.8]. We denote the homotopy limit of the
cosimplicial diagram $\Omega^{\bullet}(X,C,C)$ in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ by $\Omega(X,C,C)$ and we
say it is the _two-sided cobar resolution of $X$._ Notice that each object in
the cosimplicial diagram $\Omega^{\bullet}(X,C,C)$ is a right cofree
$C$-comodule, hence fibrant. Thus $\Omega(X,C,C)$ is a fibrant right
$C$-comodule by [Hir03, 18.5.2].

######  Remark 3.11.

We warn the reader that in the literature $\Omega(X,C,C)$ denotes the homotopy
limit in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ (which is weakly equivalent to $X$
since $\Omega^{\bullet}(X,C,C)$ is $U$-split) and not in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$. But as we will show in
Lemma 3.13, this distinction won’t matter.

By construction, the homotopy limit of $\Omega^{\bullet}(X,C,C)$ is quasi-
isomorphic to $X$ in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. We now show that the
homotopy limit of $\Omega^{\bullet}(X,C,C)$ in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ is also equivalent to $X$
as a $C$-comodule. We shall prove it inductively, layer by layer on the
Postnikov tower of $X$ (see Definition 2.19).

######  Lemma 3.12.

Let $M$ be a chain complex in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. Let
$X=M\otimes C$ be a cofree right $C$-comodule. Then the cobar resolution of
$X$ induces a weak equivalence
$X\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega(X,C,C)$ in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$.

###### Proof.

This result follows from the extra codegeneracies being maps of $C$-comodules.
In particular, the splitting lifts to
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$. In more details, regard
$C$ as a right $C$-comodule via its comultiplication $\Delta:C\rightarrow
C\otimes C$. Then the two-sided cobar construction $\Omega^{\bullet}(C,C,C)$
in $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$:

${C}$${C^{\otimes 2}}$${C^{\otimes 3}}$${\cdots,}$

splits in $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ via the map of
$C$-comodules $\varepsilon\otimes\mathsf{id}_{C}:C^{\otimes 2}\rightarrow C$,
where $\varepsilon:C\rightarrow\Bbbk$ is the counit of $C$. Since
$-\otimes-:{\mathsf{Ch}_{\Bbbk}^{\geq
0}}\times\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\rightarrow\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ is a
Quillen bifunctor by Proposition 2.8, then $M\otimes-$ is a left Quillen
functor, and thus induces a functor on the underlying $\infty$-categories:

$M\otimes-:\mathscr{N}\left(\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\right)\left[\mathsf{W}^{-1}_{\mathsf{CoMod}}\right]\rightarrow\mathscr{N}\left(\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\right)\left[\mathsf{W}^{-1}_{\mathsf{CoMod}}\right].$

Split cosimplicial limits are preserved by any functor of $\infty$-categories.
From the isomorphism of cosimplicial diagrams:
$M\otimes\Omega^{\bullet}(C,C,C)\cong\Omega^{\bullet}(M\otimes C,C,C)$, we get
the equivalences: $M\otimes C\simeq M\otimes\Omega(C,C,C)\simeq\Omega(M\otimes
C,C,C)$. ∎

######  Lemma 3.13.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. Let $X$ be a right
$C$-dg-comodule. Then the cobar resolution of $X$ induces a weak equivalence
$X\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega(X,C,C)$ in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$.

###### Proof.

We use Theorem 2.18. Let $\\{X(n)\\}$ be the Postnikov tower of $X$, and
denote by $\widetilde{X}$ the (homotopy) limit of the tower in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$. Then the acyclic
cofibration $X\stackrel{{\scriptstyle\simeq}}{{\hookrightarrow}}\widetilde{X}$
induces an objectwise weak equivalence
$\Omega^{\bullet}(X,C,C)\rightarrow\Omega^{\bullet}(\widetilde{X},C,C)$
between objectwise fibrant cosimplicial diagrams. Thus
$\Omega(X,C,C)\rightarrow\Omega(\widetilde{X},C,C)$ is a weak equivalence by
[Hir03, 18.5.3]. Therefore it suffices to show that the map
$\widetilde{X}\rightarrow\Omega(\widetilde{X},C,C)$ is a weak equivalence.

Since the Postnikov tower $\\{X(n)\\}$ stabilizes in each degree, we have the
quasi-isomorphism:

$U(\mathsf{holim}_{n}^{C}X(n))\simeq\mathsf{holim}_{n}U(X(n)).$

Since the cofree functor $-\otimes C:{\mathsf{Ch}_{\Bbbk}^{\geq
0}}\rightarrow\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ is right
Quillen, we also obtain the weak equivalence:

$(U(\mathsf{holim}_{n}^{C}X(n)))\otimes
C\simeq\mathsf{holim}_{n}^{C}(U(X(n))\otimes C).$

Notice that the tower $\Big{\\{}U(X(n))\otimes C\Big{\\}}$ also stabilizes in
each degree by Lemma 2.15. The maps in that tower are fibrations in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ and in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. Thus the homotopy limit can also be computed
in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. Therefore:

$\Omega(\widetilde{X},C,C)\simeq\mathsf{holim}_{n}^{C}\Big{(}\Omega(X(n),C,C)\Big{)}.$

Hence it is enough to show that the canonical maps
$X(n)\rightarrow\Omega(X(n),C,C)$ are weak equivalences in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ for all $n\geq 0$.

We shall prove it inductively. For $n=0$, we have $X(0)=0$ and the map is
trivial and hence a weak equivalence. For $n=1$, we know that $X(1)$ is a
cofree right $C$-comodule, and hence, by Lemma 3.12, we have
$X(1)\rightarrow\Omega(X(1),C,C)$ is a weak equivalence. Suppose now that we
have shown $X(n)\rightarrow\Omega(X(n),C,C)$ is a weak equivalence for some
$n\geq 1$. Then by construction, the comodule $X({n+1})$ is obtained as a
homotopy pullback in $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$:

${X({n+1})}$${P\otimes C}$${X(n)}$${Q\otimes C,}$${\lrcorner}$

for some chain complexes $P$ and $Q$. By [Hir03, 18.5.2], it induces a
homotopy pullback in $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$:

${\Omega(X({n+1}),C,C)}$${\Omega(P\otimes
C,C,C)}$${\Omega(X(n),C,C)}$${\Omega(Q\otimes C,C,C).}$${\lrcorner}$

Notice $X(n)$, $P\otimes C$ and $Q\otimes C$ are weakly equivalent as
$C$-comodules to their two-sided cobar resolutions, either by induction or by
Lemma 3.12. Therefore $X({n+1})\rightarrow\Omega(X(n+1),C,C)$ is a weak
equivalence since homotopy pullbacks preserve weak equivalences (see also
[Hov99, 5.2.6]):

${X(n+1)}$${P\otimes C}$${\Omega(X({n+1}),C,C)}$${\Omega(P\otimes
C,C,C)}$${X(n)}$${Q\otimes C}$${\Omega(X({n}),C,C)}$${\Omega(Q\otimes
C,C,C)}$${\lrcorner}$$\scriptstyle{\simeq}$${\lrcorner}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

This finishes the proof by induction. ∎

######  Corollary 3.14.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. The two-sided cobar
resolution of $X$ is a fibrant replacement in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$:

${X}$${U(X)\otimes C}$${\Omega(X,C,C)}$$\scriptstyle{\simeq}$

We are now ready to prove Theorem 3.3 using Theorem 3.6, Proposition 3.9 and
our above lemma.

###### Proof of Theorem 3.3.

Since the forgetful functor $U:\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ preserves and reflects weak
equivalences by definition of the model structures, we immediately get that
the left Quillen derived functor
$\mathscr{N}(\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}}))\left[\mathsf{W}^{-1}_{\mathsf{CoMod}}\right]\rightarrow\mathcal{D}^{\geq
0}(\Bbbk)$ is conservative. By [Lur17, 1.3.4.23, 1.3.4.25], homotopy limits
over cosimplicial diagrams in $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})$ correspond exactly to limits over cosimplicial diagrams in the
$\infty$-categorical sense. Hence the left Quillen derived forgetful functor
is comonadic by Lemma 3.13 and Proposition 3.9. The result follows from
Theorem 3.6 since Proposition 3.7 shows that
$\mathcal{CoMod}_{C}(\mathcal{D}^{\geq 0}(\Bbbk))$ is also comonadic over the
same comonad $-\otimes C:\mathcal{D}^{\geq
0}(\Bbbk)\rightarrow\mathcal{D}^{\geq 0}(\Bbbk)$. ∎

##  4\. Derived Cotensor of Comodules

Let $\Bbbk$ be a finite product of fields. Let $C$ be a simply connected
_cocommutative_ dg-coalgebra over $\Bbbk$. Recall that we have defined the
$\infty$-category of (right) $C$-comodules
$\mathcal{CoMod}_{C}(\mathcal{D}^{\geq 0}(\Bbbk))$ in connective
$H\Bbbk$-modules in previous section. In particular, our main result in
Theorem 3.3 showed that this $\infty$-category is equivalent to the underlying
$\infty$-category of the model category of $C$-dg-comodules
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ of Proposition 2.12.
Subsequently, we shall denote $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})$ simply as $\mathsf{CoMod}_{C}$.

We endow a symmetric monoidal structure on the category $\mathsf{CoMod}_{C}$
given by the _cotensor product_ of comodules. We show it behaves well with
respect to the model structure of $\mathsf{CoMod}_{C}$. This provides a
_derived_ cotensor product on the homotopy category of $\mathsf{CoMod}_{C}$
and more importantly, it endows the $\infty$-category
$\mathsf{CoMod}_{C}(\mathcal{D}^{\geq 0}(\Bbbk))$ with a symmetric monoidal
structure.

This was achievable using our rigidification result in Theorem 3.3. We could
not have applied the dual result of [Lur17, 4.4.2.8] as it would have required
the tensor product to commute with totalizations.

######  Remark 4.1.

If $C$ is not cocommutative, most results remain true here. The
$\infty$-category of bicomodules over $C$ is endowed with a monoidal structure
(but not symmetric). We explore this further in [KP].

###  4.1. The Cotensor Product of Comodules

The relative tensor product of modules is obtained by co-equalizing the
actions. It generalizes the classical situation in which, for a commutative
ring $R$ and $R$-modules $A$ and $B$, we have the equality $ar\otimes
b=a\otimes rb$, where $a\in A$, $b\in B$ and $r\in R$. For comodules, we do
the opposite and equalize the coactions.

######  Definition 4.2.

Let $X$ and $Y$ be $C$-comodules in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. Define
their _cotensor product_ $X{{\square}_{C}}Y$ to be the following equalizer in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$:

${X{{\square}_{C}}Y}$${X\otimes Y}$${X\otimes C\otimes Y,}$

where the two parallel morphisms are induced by the coactions $X\rightarrow
X\otimes C$ and $Y\rightarrow C\otimes Y$.

If we defined $X{{\square}_{C}}Y$ in a general monoidal category $\mathsf{C}$
instead of ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, then $X{{\square}_{C}}Y$ is not a
comodule in general. The tensor product generally does not preserve
equalizers. This is not the case when working with ${\mathsf{Ch}_{\Bbbk}^{\geq
0}}$.

######  Lemma 4.3.

The cotensor $X{{\square}_{C}}Y$ is endowed with a $C$-comodule structure.

###### Proof.

Since $-\otimes C:{\mathsf{Ch}_{\Bbbk}^{\geq
0}}\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ preserves equalizers, we obtain
the following dashed map below by universality of equalizers:

${X{{\square}_{C}}Y}$${X\otimes Y}$${X\otimes C\otimes
Y}$${(X{{\square}_{C}}Y)\otimes C}$${X\otimes Y\otimes C}$${X\otimes C\otimes
Y\otimes C.}$

We can check the map is a coassociative and counital coaction of a
$C$-comodule. ∎

######  Lemma 4.4 ([EM66, 2.2]).

For any $C$-comodule $X$, we have a natural isomorphism of $C$-comodules:

$X{{\square}_{C}}C\cong X\cong C{{\square}_{C}}X.$

######  Lemma 4.5 ([EM66, 2.1]).

Let $M$ be an object in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. Then for any cofree
comodule $M\otimes C$ we have natural isomorphisms of $C$-comodules:

$(M\otimes C){{\square}_{C}}X\cong M\otimes X\mbox{ and
}X{{\square}_{C}}(C\otimes M)\cong X\otimes M.$

######  Proposition 4.6.

Let $X$ be a $C$-comodule. Then
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ is a left
exact functor that preserves finite limits and filtered colimits.

###### Proof.

This follows directly from the fact that, when over a finite product of
fields, any object $M$ in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ induces a functor
$M\otimes-:{\mathsf{Ch}_{\Bbbk}^{\geq 0}}\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq
0}}$ that preserves finite limits and all colimits. The cotensor product
preserves filtered colimits as equalizers in presentable categories commute
with filtered colimits. ∎

######  Proposition 4.7.

The cotensor product defines a symmetric monoidal structure on $C$-comodules,
and we shall denote it $\left(\mathsf{CoMod}_{C},{{\square}_{C}},C\right)$.

###### Proof.

Given $C$-comodules $X$, $Y$ and $Z$, we need to provide a natural
associativity isomorphism of $C$-comodules:

$(X{{\square}_{C}}Y){{\square}_{C}}Z\cong
X{{\square}_{C}}(Y{{\square}_{C}}Z).$

First, notice we obtain a natural isomorphism dashed in the diagram below by
naturality of equalizers:

${(X\otimes Y){{\square}_{C}}Z}$${(X\otimes Y)\otimes Z}$${(X\otimes Y)\otimes
C\otimes Z}$${X\otimes(Y{{\square}_{C}}Z)}$${X\otimes(Y\otimes
Z)}$${X\otimes(Y\otimes C\otimes
Z).}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The bottom row is an equalizer as $X\otimes-$ preserves finite limits since we
are working over a finite product of fields. The vertical isomorphisms in the
above diagram are induced by the natural associativity of the tensor product
in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. Therefore, we obtain the desired
isomorphism by naturality of equalizers:

${(X{{\square}_{C}}Y){{\square}_{C}}Z}$${(X\otimes
Y){{\square}_{C}}Z}$${(X\otimes C\otimes
Y){{\square}_{C}}Z}$${X{{\square}_{C}}(Y{{\square}_{C}}Z)}$${X\otimes(Y{{\square}_{C}}Z)}$${X\otimes
C\otimes(Y{{\square}_{C}}Z).}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The top row is an equalizer by Proposition 4.6. The natural isomorphism
obtained is compatible with the $C$-actions.

To conclude, we need to check three coherence diagrams to be commutative, for
instance compatibility of the associativity and the unit. But these follow
from the symmetric monoidal structure on ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ and
naturality of the equalizers. ∎

######  Remark 4.8.

If $M$ is a module over a commutative algebra $A$ in a nice enough monoidal
category $\mathsf{C}$, then in general
$M\otimes-:\mathsf{Mod}_{A}(\mathsf{C})\rightarrow\mathsf{Mod}_{A}(\mathsf{C})$
is a left adjoint. Unfortunately, for a general $C$-comodule $X$, there is no
reason to expect that the functor
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ is a left
nor a right adjoint. In [Tak77], the author introduced the notion of _quasi-
finite_ $C$-comodules. Essentially, a $C$-comodule $X$ is quasi-finite if and
only if $X{{\square}_{C}}-$ is a right adjoint. However, a $C$-comodule need
not be weakly equivalent to a quasi-finite one. For instance, if we choose
$C=\Bbbk$, then $X$ is quasi-finite if and only if $X$ is a perfect chain
complex.

###  4.2. How To Derive The Cotensor Product

Given a symmetric monoidal model category $(\mathsf{C},\otimes)$ (see [Hov99,
4.2.6]), we can (left) derive the tensor product $\otimes^{\mathbb{L}}$ to
obtain a symmetric monoidal structure on the homotopy category of $\mathsf{C}$
(see [Hov99, 4.3.2]). It also endows its underlying $\infty$-category
$\mathscr{N}(\mathsf{C}_{c})\left[\mathsf{W}^{-1}\right]$ with a symmetric
monoidal structure (see [Lur17, 4.1.7.6]). In the case of modules over a
commutative algebra $A$, the derived tensor product $M\otimes^{\mathbb{L}}N$
of $A$-modules $M$ and $N$ is given by a two-sided bar construction $B(M,A,N)$
(see [Lur17, 4.4.2.8]). Here if $\mathsf{C}$ is a model category, we denoted
$\mathsf{C}_{c}$ and $\mathsf{C}_{f}$ the full subcategory of cofibrant and
fibrant objects in $\mathsf{C}$ respectively.

We provide here a formal way to derive the cotensor product of comodules.
Given $C$-dg-comodules $X$ and $Y$, the derived cotensor product
$X\widehat{\square}_{C}Y$ is given by the two-sided cobar construction
$\Omega(X,C,Y)$. However, $(\mathsf{CoMod}_{C},{{\square}_{C}},C)$ is not a
symmetric monoidal model category. Instead, we show the cotensor product
behaves well with respect to fibrant objects. We right derive the cotensor
product. However the functor
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ is not a
right adjoint, even for $X$ a fibrant $C$-comodule. Nevertheless, we provide
sufficient conditions for a model category $\mathsf{C}$ with a symmetric
monoidal structure that guarantees its underlying $\infty$-category
$\mathscr{N}(\mathsf{C}_{f})\left[\mathsf{W}^{-1}\right]$ is also endowed with
a derived tensor product.

######  Definition 4.9.

A _(symmetric) comonoidal model category_ $\mathsf{C}$ is a category endowed
with both a model structure and a (symmetric) monoidal structure
$(\mathsf{C},\otimes,\mathbb{I})$, such that:

  1. (i)

for any fibrant object $X$ in $\mathsf{C}$, the functors
$X\otimes-:\mathsf{C}\rightarrow\mathsf{C}$ and $-\otimes
X:\mathsf{C}\rightarrow\mathsf{C}$ preserve fibrant objects and weak
equivalences between fibrant objects;

  2. (ii)

for any fibrant replacement $\mathbb{I}\rightarrow f\mathbb{I}$ of the unit,
the induced morphism $X\cong\mathbb{I}\otimes X\rightarrow f\mathbb{I}\otimes
X$ is a weak equivalence, for any fibrant object $X$.

The requirement (ii) is automatic if $\mathbb{I}$ is already fibrant

######  Proposition 4.10 ([Lur17, 4.1.7.6], [NS18, A.7]).

Let $(\mathsf{C},\otimes,\mathbb{I})$ be a symmetric comonoidal model
category. Suppose that $\mathbb{I}$ is fibrant. Then the Dwyer-Kan
localization $\mathscr{N}(\mathsf{C}_{f})[{\mathsf{W}}^{-1}]$ of $\mathsf{C}$
can be given the structure of a symmetric monoidal $\infty$-category via the
symmetric monoidal Dwyer-Kan localization of
$\mathscr{N}(\mathsf{C}_{f}^{\otimes})$:

${\mathscr{N}(\mathsf{C}_{f}^{\otimes})}$${\mathscr{N}(\mathsf{C}_{f})[{\mathsf{W}}^{-1}]^{\otimes},}$

where ${\mathsf{W}}$ is the class of weak equivalences restricted to fibrant
objects in $\mathsf{C}$.

######  Theorem 4.11.

Let $C$ be a simply connected cocommutative dg-coalgebra over $\Bbbk$. Then
the symmetric monoidal category
$(\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}}),{{\square}_{C}},C)$ is a
symmetric comonoidal model category with fibrant unit.

###### Proof.

This follows from Corollary 4.17 and Corollary 4.27 below. ∎

######  Corollary 4.12.

Let $C$ be a simply connected cocommutative dg-coalgebra over $\Bbbk$. Then
the $\infty$-category $\mathcal{CoMod}_{C}(\mathcal{D}^{\geq 0}(\Bbbk))$ is
symmetric monoidal when endowed with the derived cotensor product of
$C$-comodules.

###### Proof.

By Theorem 3.3, we have:

${\mathscr{N}\left(\mathsf{CoMod}_{C}\right)\left[\mathsf{W}^{-1}_{\mathsf{CoMod}}\right]}$${\mathcal{CoMod}_{C}\left(\mathcal{D}^{\geq
0}(\Bbbk)\right),}$$\scriptstyle{\simeq}$

where $\mathsf{W}_{\mathsf{CoMod}}$ denotes the class of quasi-isomorphisms of
$C$-comodules. Since the model category of $\mathsf{CoMod}_{C}$ is
combinatorial by Proposition 2.12, we get:

$\mathscr{N}\left(\mathsf{CoMod}_{C}\right)\left[\mathsf{W}^{-1}_{\mathsf{CoMod}}\right]\simeq\mathscr{N}\left((\mathsf{CoMod}_{C})_{f}\right)\left[\mathsf{W}^{-1}_{\mathsf{CoMod}}\right],$

by [Lur17, 1.3.4.16]. Then the result follows from Proposition 4.10 and
Theorem 4.11. ∎

######  Remark 4.13.

The bifunctor
$-{{\square}_{C}}-:\mathsf{CoMod}_{C}\times\mathsf{CoMod}_{C}\longrightarrow\mathsf{CoMod}_{C}$
is _not_ a co-Quillen bifunctor (dual definition of [Hov99, 4.2.1]). Indeed,
in general $X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$
does not preserve fibrations, even if $X$ is fibrant. An example is given by
applying the functor to the generating fibration $0\rightarrow S^{0}(V)\otimes
C$. If we choose $V=\Bbbk$, then we obtain a map $0\rightarrow X$ which is
clearly not a fibration (consider the case $C=\Bbbk$). Moreover the functor
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ is not a
right adjoint, even when $X$ is fibrant $C$-comodule.

###  4.3. Coflat Comodules

We show here, in Proposition 4.16, that fibrant $C$-comodules in the model
category $\mathsf{CoMod}_{C}$ are precisely the _coflat $C$-comodules_. In
particular, this will show that the cotensor product of fibrant $C$-comodules
remains fibrant (see Corollary 4.17), a key argument in the proof of Theorem
4.11.

######  Definition 4.14.

Let $X$ be a $C$-dg-comodule. The functor
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ is left
exact by Proposition 4.6. We say the $C$-comodule $X$ is _coflat_ if
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ is (right)
exact.

Recall that exactness in $\mathsf{CoMod}_{C}$ is entirely determined in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ by Proposition 2.11.

######  Proposition 4.15.

Let $X$ and $Y$ be coflat $C$-comodules. Then $X{{\square}_{C}}Y$ is a coflat
$C$-comodule.

###### Proof.

We consider the following exact sequence in $\mathsf{CoMod}_{C}$:

${0}$${Z^{\prime}}$${Z}$${Z^{\prime\prime}}$${0.}$

Since $Y$ is coflat, we obtain the following exact sequence:

${0}$${Y{{\square}_{C}}Z^{\prime}}$${Y{{\square}_{C}}Z}$${Y{{\square}_{C}}Z^{\prime\prime}}$${0.}$

Since $X$ is coflat, we then obtain the following exact sequence:

${0}$${X{{\square}_{C}}(Y{{\square}_{C}}Z^{\prime})}$${X{{\square}_{C}}(Y{{\square}_{C}}Z)}$${X{{\square}_{C}}(Y{{\square}_{C}}Z^{\prime\prime})}$${0.}$

By associativity of cotensor product, this exact sequence is equivalent to the
following one:

${0}$${(X{{\square}_{C}}Y){{\square}_{C}}Z^{\prime}}$${(X{{\square}_{C}}Y){{\square}_{C}}Z}$${(X{{\square}_{C}}Y){{\square}_{C}}Z^{\prime\prime}}$${0.}$

Thus $X{{\square}_{C}}Y$ is coflat by definition. ∎

A key observation is the following.

######  Proposition 4.16.

A $C$-comodule is coflat if and only if it is fibrant in the model category
$\mathsf{CoMod}_{C}$.

We shall prove the proposition above at the end of this section. An immediate
consequence is the following.

######  Corollary 4.17.

Let $X$ and $Y$ be fibrant $C$-comodules. Then $X{{\square}_{C}}Y$ is fibrant.

Before we prove Proposition 4.16, we introduce several definitions from
homological algebra. Notice first that the category of $C$-comodules
$\mathsf{CoMod}_{C}$ is Abelian and has enough injective objects. More
specifically, any injective object $M$ in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$
induces an injective $C$-comodule $M\otimes C$, and thus we see that any
$C$-comodule $X$ can be embedded into an injective $C$-comodule via its
coaction. Thus we can derive the cotensor product in the sense of Abelian
categories.

######  Definition 4.18.

Let $X$ be a $C$-comodule. Define:

$\mathsf{CoTor}_{C}^{i}(X,-):\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C},$

to be the $i$-th right derived functor of $X{{\square}_{C}}-$, for $i\geq 0$.
More specifically, given an injective resolution of a $C$-comodule $Y$:

${0}$${Y}$${I^{0}}$${I^{1}}$${\cdots,}$

then $\mathsf{CoTor}_{C}^{i}(X,Y)$ is given by the $i$-th cohomology
$H^{i}(X{{\square}_{C}}I^{\bullet})$.

We have that $\mathsf{CoTor}_{C}^{0}(X,Y)=X{{\square}_{C}}Y$ for any comodules
$X$ and $Y$. If $Y$ is an injective $C$-comodule, then
$\mathsf{CoTor}_{C}^{i}(X,Y)=0$ for any comodule $X$ and $i>0$.

######  Remark 4.19.

Notice that a $C$-comodule is injective if and only if it is acyclic fibrant
in the model category $\mathsf{CoMod}_{C}$.

The following proposition is an immediate consequence of Proposition 4.6.

######  Proposition 4.20.

Let $X$ be a $C$-comodule. The following are equivalent:

  1. (i)

the $C$-comodule $X$ is coflat;

  2. (ii)

the functor
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ preserves
all colimits;

  3. (iii)

the functor
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ is a left
adjoint;

  4. (iv)

for any $C$-comodule $Y$, we have $\mathsf{CoTor}_{C}^{i}(X,Y)=0$ for all
$i\geq 1$;

  5. (v)

for any $C$-comodule $Y$, we have $\mathsf{CoTor}_{C}^{1}(X,Y)=0$.

We see in particular from the above result that every injective $C$-comodule
is automatically coflat.

######  Definition 4.21.

Let $X$ be a $C$-comodule. The functor
$\mathsf{Hom}_{C}(X,-):\mathsf{CoMod}_{C}\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq
0}}$ from Definition 2.7 is left exact. Define
$\mathsf{Ext}_{C}^{i}(X,-):\mathsf{CoMod}_{C}\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq
0}}$ to be the $i$-th right derived functor of $\mathsf{Hom}_{C}(X,-)$, for
$i\geq 0$. More specifically, given an injective resolution of a $C$-comodule
$Y$:

${0}$${Y}$${I^{0}}$${I^{1}}$${\cdots,}$

then $\mathsf{Ext}_{C}^{i}(X,Y)$ is given by the $i$-th cohomology
$H^{i}(\mathsf{Hom}_{C}(X,I^{\bullet}))$.

Before proving Proposition 4.16, we need the following result due to
Bousfield, who shared this result after reading an earlier version of this
paper.

######  Lemma 4.22 (Bousfield).

Let $\mathsf{A}$ be an Abelian category endowed with a model structure where
acyclic cofibrations are precisely monomorphisms with acyclic cokernels. Let
$f:X\rightarrow Y$ be an epimorphism in $\mathsf{A}$. Let $F$ be its kernel.
Then $f$ is a fibration if and only if $F$ is fibrant.

###### Proof.

A fibration always has fibrant kernel, regardless of being an epimorphism.
This is because pullbacks preserve fibrations and the kernel $F$ is given by
the pullback:

${F}$${X}$${0}$${Y.}$${\lrcorner}$$\scriptstyle{f}$

Now suppose $F$ is fibrant, let us show that $f$ is a fibration. Since
$\mathsf{A}$ is a model category, we can factor $f$ as follows:

${X}$${Y}$${X^{\prime},}$$\scriptstyle{f}$$\scriptstyle{\simeq}$$\scriptstyle{i}$$\scriptstyle{f^{\prime}}$

where $i$ is an acylic cofibration and $f^{\prime}$ is a fibration. Denote
$F^{\prime}$ the kernel of $f^{\prime}$. We obtain the following morphism of
short exact sequences in $\mathsf{A}$:

${0}$${F}$${X}$${Y}$${0}$${0}$${F^{\prime}}$${X^{\prime}}$${Y}$${0.}$$\scriptstyle{f}$$\scriptstyle{i}$$\scriptstyle{\simeq}$$\scriptstyle{f^{\prime}}$

We have used the fact that since $f$ is an epimorphism and $f=f^{\prime}\circ
i$, then $f^{\prime}$ must also be an epimorphism. Since $i$ is a
monomorphism, the snake lemma guarantees that the induced map $F\rightarrow
F^{\prime}$ is also a monomorphism. Therefore we can take the cokernels of the
vertical maps:

${0}$${0}$${0}$${0}$${F}$${X}$${Y}$${0}$${0}$${F^{\prime}}$${X^{\prime}}$${Y}$${0}$${0}$${K}$${K^{\prime}}$${0}$${0.}$${0}$${0}$${0}$$\scriptstyle{f}$$\scriptstyle{i}$$\scriptstyle{\simeq}$$\scriptstyle{f^{\prime}}$

The $9$-lemma guarantees that the third row is exact, and thus $K$ is acylic.
Therefore $F\rightarrow F^{\prime}$ is an acylic cofibration. Since $F$ is
fibrant, then we obtain the following section of $F\rightarrow F^{\prime}$:

${F}$${F}$${F^{\prime}}$${0.}$$\scriptstyle{\simeq}$$\scriptstyle{\ell}$

We define then $P$ to be the following pushout in $\mathsf{A}$:

${F^{\prime}}$${X^{\prime}}$${F}$${P.}$$\scriptstyle{\ell}$${\ulcorner}$

In an Abelian category, pushouts preserve monomorphisms so $F\rightarrow P$ is
a monomorphism. Pushouts also preserve cokernels, thus $Y$ is the cokernel of
$F\rightarrow P$. Therefore we obtain the following composite of short exact
sequences:

${0}$${F}$${X}$${Y}$${0}$${0}$${F^{\prime}}$${X^{\prime}}$${Y}$${0}$${0}$${F}$${P}$${Y}$${0.}$$\scriptstyle{\simeq}$$\scriptstyle{f}$$\scriptstyle{i}$$\scriptstyle{\simeq}$$\scriptstyle{\ell}$$\scriptstyle{f^{\prime}}$

The composite of the left vertical arrows is the identity on $F$ by
construction of $\ell$. By the $5$-lemma, we get that $P$ is isomorphic to
$Y$. Therefore, we have just shown that $f$ is a retract of $f^{\prime}$ which
is a fibration. Hence $f$ is also a fibration. ∎

###### Proof of Proposition 4.16.

As $C$ is simply connected, it is also _coaugmented_ , i.e. there is a map of
coalgebras $\eta:\Bbbk\rightarrow C$ which is trivial in every non-zero
degree, and in degree zero is the identity $\mathsf{id}_{\Bbbk}$. Since $C$ is
coaugmented, any object $X$ of ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ admits a
trivial $C$-comodule induced by $X\cong
X\otimes\Bbbk\stackrel{{\scriptstyle\mathsf{id}\otimes\eta}}{{\longrightarrow}}X\otimes
C$. See more details in [Hes09, 2.1]. We shall endow the spheres and disks
below with their trivial $C$-comodule structures.

Let $F$ be a $C$-comodule. We show that the following statements are
equivalent.

  1. (i)

$F$ is a coflat $C$-comodule.

  2. (ii)

$\mathsf{CoTor}_{C}^{1}(S^{n},F)=0$, for all $n\geq 0$.

  3. (iii)

$\mathsf{CoTor}_{C}^{1}(S^{0},F)=0$.

  4. (iv)

$\mathsf{Ext}_{C}^{1}(D^{n},F)=0$, for all $n\geq 0$.

  5. (v)

$\mathsf{Ext}^{1}_{C}(X,F)=0$ for any acyclic $C$-comodule $X$.

  6. (vi)

$F$ is a fibrant $C$-comodule.

(i) $\Leftrightarrow$ (ii):

We have (i) $\Rightarrow$ (ii) from Proposition 4.20. So let us show (ii)
$\Rightarrow$ (i). Suppose $\mathsf{CoTor}_{C}^{1}(S^{n},F)=0$ for all $n\geq
0$. Notice that it implies $\mathsf{CoTor}_{C}^{1}(S^{n}(V),F)=0$ for all
$\Bbbk$-modules $V$ and all $n\geq 0$. Indeed, since $\Bbbk$ is a finite
product of fields, $V$ is a retract of a free $\Bbbk$-module, and thus it is
enough to show $\mathsf{CoTor}_{C}^{1}(\bigoplus_{\lambda}S^{n},F)=0$ for all
$n\geq 0$ and any ordinal $\lambda$. But this follows directly from the fact
that the cotensor product preserves coproducts by Proposition 4.6.

We now introduce a notation. For any non-negative chain complex $X$, and for
$n\geq 0$, define $X_{\leq n}$ as the subchain complex of $X$ where, for all
$i\geq 0$:

$(X_{\leq n})_{i}=\left\\{\begin{tabular}[]{ll}$X_{i}$,&$0\leq i\leq n$,\\\
$0$,&$i>n$.\end{tabular}\right.$

As $(X_{\leq n}\otimes C)_{i}=\left((X\otimes C)_{\leq n}\right)_{i}$ for
$0\leq i\leq n$, we obtain an unique $C$-comodule structure on $X_{\leq n}$
from a $C$-comodule structure on $X$ such that the inclusion $X_{\leq
n}\hookrightarrow X$ is a map of $C$-comodules. In particular, any
$C$-comodule $X$ is the filtered colimit in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ of:

${X_{\leq 0}}$${X_{\leq 1}}$${X_{\leq 2}}$${\cdots.}$

Let us prove that $\mathsf{CoTor}_{C}^{1}(X,F)=0$ for any $C$-comodule $X$. We
first prove by induction on $n\geq 0$ that $\mathsf{CoTor}_{C}^{1}(X_{\leq
n},F)=0$. For the initial case, notice that $X_{\leq 0}\cong S^{0}(V)$ for
some $\Bbbk$-module $V$, and thus $\mathsf{CoTor}_{C}^{1}(X_{\leq 0},F)=0$ by
our above argument. Now suppose $\mathsf{CoTor}_{C}^{1}(X_{\leq n},F)=0$ for
some $n\geq 0$. Then we obtain a short exact sequence of $C$-comodules:

${0}$${X_{\leq n}}$${X_{\leq n+1}}$${S^{n+1}(V)}$${0,}$

for some $\Bbbk$-module $V$. The induced long exact sequence from right
deriving the cotensor $-{{\square}_{C}}F$ in particular gives the exact
sequence:

${\mathsf{CoTor}_{C}^{1}(X_{\leq n},F)}$${\mathsf{CoTor}_{C}^{1}(X_{\leq
n+1},F)}$${\mathsf{CoTor}_{C}^{1}(S^{n+1}(V),F).}$

Thus by induction, and our above argument, we get
$\mathsf{CoTor}_{C}^{1}(X_{\leq n+1},F)=0$. Now by Proposition 4.6, we get:

$\mathsf{CoTor}_{C}^{1}(X,F)\cong\mathsf{colim}_{n}\mathsf{CoTor}_{C}^{1}(X_{\leq
n},F),$

and hence we can conclude $\mathsf{CoTor}_{C}^{1}(X,F)=0$. Thus $F$ is coflat.

(ii) $\Leftrightarrow$ (iii):

We only need to show that (iii) $\Rightarrow$ (ii). For all $n\geq 0$, notice
that:

$S^{n}{{\square}_{C}}F\cong S^{n}\otimes(S^{0}{{\square}_{C}}F).$

Since the functor
$S^{n}\otimes-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ is exact, the
result follows.

(iii) $\Leftrightarrow$ (iv):

First notice that, for any map of $C$-comodules $I\rightarrow K$, the induced
chain map $\mathsf{Hom}_{C}(D^{n},I)\rightarrow\mathsf{Hom}_{C}(D^{n},K)$ is
an epimorphism for all $n\geq 0$ if and only if
$S^{0}{{\square}_{C}}I\rightarrow S^{0}{{\square}_{C}}K$ is an epimorphism of
$C$-comodules. Indeed, this follows from:

$\mathsf{Hom}(D^{n},X)_{n}=X_{n}=(S^{0}\otimes X)_{n},$

for any chain complex $X$ and any $n\geq 0$.

Now, let $I$ be an injective $C$-comodule such that $F$ maps into $I$. Then we
can form the short exact sequence of $C$-comodules:

${0}$${F}$${I}$${K}$${0.}$

Thus $\mathsf{CoTor}_{C}^{1}(S^{0},F)=0$ if and only if
$S^{0}{{\square}_{C}}I\rightarrow S^{0}{{\square}_{C}}K$ is an epimorphism,
and $\mathsf{Ext}_{C}^{1}(D^{n},F)=0$ if and only if
$\mathsf{Hom}_{C}(D^{n},I)\rightarrow\mathsf{Hom}_{C}(D^{n},K)$ is an
epimorphism for all $n\geq 0$. We can conclude by our above argument.

(iv) $\Leftrightarrow$ (v):

We only need to show that (iv) $\Rightarrow$ (v). Notice that
$\mathsf{Ext}_{C}^{1}(D^{n},F)=0$ for all $n\geq 1$, implies that
$\mathsf{Ext}_{C}^{1}(D^{n}(W),F)=0$ for any $\Bbbk$-module $W$ and all $n\geq
1$. Indeed, since $W$ is a retract of a free $\Bbbk$-module, we only need to
show that $\mathsf{Ext}_{C}^{1}(\bigoplus_{\lambda}D^{n},F)=0$ for some
ordinal $\lambda$. But this follows from:

$\mathsf{Hom}_{C}\left(\bigoplus_{\lambda}D^{n},F\right)\cong\prod_{\lambda}\mathsf{Hom}_{C}(D^{n},F).$

Let $X$ be an acylic non-negative chain complex. By Proposition A.5, we have
that:

$X\cong\prod_{i\geq 1}D^{i}(W_{i}),$

for some $\Bbbk$-modules $W_{i}$. Define for $n\geq 1$ the subchain complex
$X_{<n}$ of $X$ by:

$X_{<n}=\prod_{0<i\leq n}D^{i}(W_{i}).$

Notice that $(X_{<n})_{i}=X_{i}$ for $0\leq i<n$. Given a $C$-comodule
structure on $X$, there is a unique $C$-comodule structure on $X_{<n}$ such
that $X_{<n}\hookrightarrow X$ is a map of $C$-comodules. In particular, any
acylic $C$-comodule $X$ is the filtered colimit in
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ of:

${X_{<1}}$${X_{<2}}$${X_{<3}}$${\cdots.}$

Let us now show that $\mathsf{Ext}_{C}^{1}(X,F)=0$ for any acyclic
$C$-comodule $X$. We first prove by induction on $n\geq 1$ that
$\mathsf{Ext}_{C}^{1}(X_{<n},F)=0$. For the initial case, notice that
$X_{<1}\cong D^{1}(W_{1})$ and thus $\mathsf{Ext}_{C}^{1}(X_{<1},F)=0$ by our
above argument. Now suppose $\mathsf{Ext}_{C}^{1}(X_{<n},F)=0$ for some $n\geq
1$. Then we obtain a short exact sequence of $C$-comodules:

${0}$${X_{<n}}$${X_{<n+1}}$${D^{n+1}(W_{n+1})}$${0.}$

The induced long exact sequence from left deriving the functor
$\mathsf{Hom}_{C}(-,F)$ in particular gives the exact sequence:

${\mathsf{Ext}_{C}^{1}(D^{n+1}(W_{n+1}),F))}$${\mathsf{Ext}_{C}^{1}(X_{<n+1},F)}$${\mathsf{Ext}_{C}^{1}(X_{<n},F)}$

Thus by induction, and our above argument, we get
$\mathsf{Ext}_{C}^{1}(X_{<n+1},F)=0$. Thus we get:

$\mathsf{Ext}_{C}^{1}(X,F)\cong\mathsf{lim}_{n}\mathsf{Ext}_{C}^{1}(X_{<n},F),$

and hence we can conclude $\mathsf{Ext}_{C}^{1}(X,F)=0$ for any acyclic
$C$-comodule $X$.

(v) $\Leftrightarrow$ (vi):

Let us first show (v) $\Rightarrow$ (vi). Factor the map $F\rightarrow 0$ in
$\mathsf{CoMod}_{C}$ as:

${F}$${0}$${E,}$$\scriptstyle{\simeq}$

such that $E$ is fibrant. Let $X$ be the cokernel of $F\hookrightarrow E$.
Then we obtain a short exact sequence in $\mathsf{CoMod}_{C}$:

${0}$${F}$${E}$${X}$${0.}$$\scriptstyle{\simeq}$

Thus $X$ is an acyclic $C$-comodule. But since $\mathsf{Ext}_{C}^{1}(X,F)=0$,
then the above short exact sequence must split. Thus $F$ is a retract of $E$,
and hence $F$ is fibrant.

Let us show now (vi) $\Rightarrow$ (v). Suppose we are given any extension of
$F$ with an acyclic $C$-comodule $X$:

${0}$${F}$${E}$${X}$${0.}$

Then by Lemma 4.22, we get that the map $E\rightarrow X$ must be a fibration
of $C$-comodules. But the lifting property provides the dashed map in the
following commutative diagram in $\mathsf{CoMod}_{C}$:

${0}$${E}$${X}$${X.}$$\scriptstyle{\simeq}$

Thus the extension must split. Hence $\mathsf{Ext}_{C}^{1}(X,F)=0$. ∎

###  4.4. An Eilenberg-Moore Spectral Sequence

We show here that if $X$ is a fibrant $C$-comodule (i.e. coflat), then the
functor $X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$
preserves quasi-isomorphisms. We shall use classical methods from [EM66].

######  Definition 4.23.

Given any Abelian category $\mathsf{A}$, recall that the conormalization
functor provides an equivalence of categories
$\mathsf{N}^{\bullet}:\mathsf{A}^{\Delta}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathsf{CoCh}^{\geq
0}(\mathsf{A})$, between cosimplicial objects in $\mathsf{A}$ and non-negative
cochain complexes of $\mathsf{A}$. See [Wei94, 8.4.3]. Given $\Phi$ a
cosimplicial object in $\mathsf{A}$, we have that $\mathsf{N}^{i}(\Phi)$ is
given by $\Phi^{0}$ if $i=0$, and by the kernel of the codegeneracies:

$\bigcap_{j=0}^{i-1}\mathsf{ker}(\Phi^{j}\rightarrow\Phi^{j-1}),$

for $i\geq 1$. The differentials are given by the alternating sum of the
coface maps of $\Phi$. Apply this to the Abelian category
$\mathsf{A}=\mathsf{CoMod}_{C}$. Let $X$ be a $C$-comodule and consider
$\Omega^{\bullet}(X,C,C)$ as in Definition 3.10. The _two-sided conormalized
cobar construction of $X$_, denoted $\underline{\Omega^{\bullet}}(X,C,C)$ is
the cochain complex of $C$-comodules obtained from the conormalization of
$\Omega^{\bullet}(X,C,C)$.

######  Definition 4.24.

Let $X$ and $Y$ be $C$-comodules. Define the _conormalized cobar construction_
of $X$ and $Y$ to be the cochain complex
$\underline{\Omega^{\bullet}}(X,C,C){{\square}_{C}}Y$ of $C$-comodules, which
is denoted $\underline{\Omega^{\bullet}}(X,C,Y)$. If we denote $\underline{C}$
the unit coideal, i.e. the kernel of the counit
$\varepsilon:C\rightarrow\Bbbk$, then $\underline{\Omega^{n}}(X,C,Y)$ is given
by $X\otimes\underline{C}^{\otimes n}\otimes Y$.

######  Remark 4.25.

Let $X$ and $Y$ be $C$-comodules. As noticed in [Rav86, A1.2.12], for all
$i\geq 0$ we have an isomorphism of $C$-comodules:

$\mathsf{CoTor}_{C}^{i}(X,Y)\cong H^{i}(\underline{\Omega^{\bullet}}(X,C,Y)).$

Recall that to any chain complex $M$, we can regard its homology $H_{*}(M)$ as
a chain complex with trivial differentials. If $C$ is a simply connected dg-
coalgebra over $\Bbbk$, then $H_{*}(C)$ is also a simply connected
cocommutative differential graded coalgebra. Moreover, for any $C$-comodule
$X$, we can check that $H_{*}(X)$ is a $H_{*}(C)$-comodule.

######  Theorem 4.26 (Eilenberg-Moore Spectral Sequence).

Let $C$ be a simply connected cocommutative dg-coalgebra over $\Bbbk$. Let $X$
be a fibrant $C$-comodule, i.e. a coflat $C$-comodule. Let $Y$ be any
$C$-comodule. Then there is a convergent spectral sequence:

$E^{2}_{\bullet,q}=\mathsf{CoTor}^{q}_{H_{*}(C)}(H_{*}(X),H_{*}(Y))\Rightarrow
E^{\infty}_{\bullet,0}=H_{*}(X{{\square}_{C}}Y).$

###### Proof.

The conormalized cobar resolution of $X$ and $Y$, denoted
$\underline{\Omega^{\bullet}}(X,C,Y)$, is a cochain complex of a chain complex
and thus defines a second quadrant double chain complex
$(\underline{\Omega^{\bullet}}(X,C,Y))_{\bullet}$, where we grade the row
cohomologically, but the columns homologically. For any $p,q\geq 0$, we have:

$(\underline{\Omega^{q}}(X,C,Y))_{p}=(X\otimes\underline{C}^{\otimes q}\otimes
Y)_{p}.$

As $C$ is simply connected, its unit coideal $\underline{C}$ is trivial in
degrees $0$ and $1$. Hence we obtain $(\underline{\Omega^{q}}(X,C,Y))_{p}=0,$
for $0\leq p\leq 2q-1$. Thus the two associated spectral sequences to the
double complex converge, see [McC01, 2.15].

The first spectral sequence has its $E^{1}$-page induced by the cohomology of
the rows, and therefore:

$E^{1}_{\bullet,q}=H^{q}(\underline{\Omega^{\bullet}}(X,C,Y)\cong\mathsf{CoTor}_{C}^{q}(X,Y).$

Since $X$ is a coflat $C$-comodule, then $E^{1}_{\bullet,q}=0$ for all $q\geq
1$, and we have $E^{1}_{\bullet,0}=X{{\square}_{C}}Y$. Thus the spectral
sequence collapses onto its second page
$E^{2}_{\bullet,0}=H_{*}(X{{\square}_{C}}Y)$. The second spectral sequence has
its $E^{1}$-page induced by the homology of the columns, and therefore:

$E^{1}_{\bullet,q}=H_{*}(\underline{\Omega^{q}}(X,C,Y))=\underline{\Omega^{q}}(H_{*}(X),H_{*}(C),H_{*}(Y)).$

Thus, as its $E^{2}$-page is given by the cohomology of the induced cochain
complex, we obtain:

$E^{2}_{\bullet,q}=\mathsf{CoTor}^{q}_{H_{*}(C)}(H_{*}(X),H_{*}(Y)).$

It converges to the page with trivial columns except its $0$-th column which
is given by the cohomology $H_{*}(X{{\square}_{C}}Y)$. ∎

######  Corollary 4.27.

Let $C$ be a simply connected cocommutative dg-coalgebra over $\Bbbk$. Let $X$
be a fibrant $C$-comodule. (i.e. a coflat $C$-comodule). Then the functor
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ sends weak
equivalences to weak equivalences.

###### Proof.

Let $Y\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Y^{\prime}$ be a
quasi-isomorphism of $C$-comodules. It induces an isomorphism $H_{*}(Y)\cong
H_{*}(Y^{\prime})$ of $H_{*}(C)$-comodules. Therefore we obtain:

$\mathsf{CoTor}^{q}_{H_{*}(C)}(H_{*}(X),H_{*}(Y))\cong\mathsf{CoTor}^{q}_{H_{*}(C)}(H_{*}(X),H_{*}(Y^{\prime})),$

for all $q\geq 0$. By Theorem 4.26, we obtain $H_{*}(X{{\square}_{C}}Y)\cong
H_{*}(X{{\square}_{C}}Y^{\prime})$ via the map $Y\rightarrow Y^{\prime}$. Then
$X{{\square}_{C}}Y\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}X{{\square}_{C}}Y^{\prime}$
is a quasi-isomorphism of $C$-comodules. ∎

######  Remark 4.28.

In an earlier version of this paper, we have shown the following. Let $X$ be
_any_ $C$-dg-comodule. Then the functor
$X{{\square}_{C}}-:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{C}$ sends
acyclic fibrations to weak equivalences. In particular, it preserves weak
equivalences between fibrant objects. This follows by generating acyclic
fibrations from a class of maps (as for fibrations in Appendix A) see [Pér21,
4.14].

###  4.5. Change of Coalgebras

We observe here a direct consequence from Corollary 4.27. Let $f:C\rightarrow
D$ be a map of simply connected cocommutative coalgebras in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. The map endows the coalgebra $C$ with a
right $D$-comodule structure:

${C}$${C\otimes C}$${C\otimes
D,}$$\scriptstyle{\Delta_{C}}$$\scriptstyle{\mathsf{id}_{C}\otimes f}$

such that $f:C\rightarrow D$ is a map of $D$-comodules. We obtain a functor:

$f^{*}:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{D},$

where each $C$-comodule $(X,\rho)$ is sent to the $D$-comodule
$(X,(\mathsf{id}_{X}\otimes f)\circ\rho)$. We shall often write $f^{*}(X)$
simply as $X$.

Given any $D$-comodule $X$, we can form the cotensor of $D$-comodules
$X\square_{D}C$, which can be endowed with the structure of $C$-comodule as
follows. The $C$-coaction is induced by the natural map of equalizers:

${X\square_{D}C}$${X\otimes C}$${X\otimes D\otimes C}$${(X\square_{D}C)\otimes
C}$${X\otimes C\otimes C}$${X\otimes D\otimes C\otimes C,}$

where the vertical arrows are induced by the comultiplication on $C$.
Therefore, we obtain a functor
$-\square_{D}C:\mathsf{CoMod}_{D}\rightarrow\mathsf{CoMod}_{C}$ which is right
adjoint to $f^{*}$.

######  Proposition 4.29.

Let $f:C\rightarrow D$ be a map of simply connected cocommutative dg-
coalgebras over $\Bbbk$. Then the adjunction:

${\mathsf{CoMod}_{C}}$${\mathsf{CoMod}_{D},}$$\scriptstyle{f^{*}}$$\scriptstyle{\perp}$$\scriptstyle{-\square_{D}C}$

is a Quillen pair. The adjunction is a Quillen equivalence if and only if the
map $f$ is a quasi-isomorphism. Moreover, when $f$ is a quasi-isomorphism, we
obtain an equivalence of symmetric monoidal $\infty$-categories:

$\mathcal{CoMod}_{C}(\mathcal{D}^{\geq
0}(\Bbbk))\simeq\mathcal{CoMod}_{D}(\mathcal{D}^{\geq 0}(\Bbbk)),$

with respect to their derived cotensor product.

###### Proof.

The first statement follows directly from the fact that the functor $f^{*}$
preserves monomorphisms and weak equivalences.

For the second statement, we shall apply [Hov99, 1.3.16]. Notice that $f^{*}$
reflects weak equivalences. Suppose first that $f$ is a weak equivalence. Now
let $X$ be any fibrant $D$-comodule, the counit of the adjunction:

${X\square_{D}C}$${X\square_{D}D\cong X,}$$\scriptstyle{\simeq}$

is a weak equivalence by Corollary 4.27. Conversely, if we suppose the
adjunction to be a Quillen equivalence, then the map:

$f:C\cong D\square_{D}C\longrightarrow D\square_{D}D\cong D,$

must be a weak equivalence, as $D$ is always fibrant as a $D$-comodule.

The third statement, it follows from the universal property of the symmetric
Dwyer-Kan localization [Lur17, 4.1.7.4]. The functor
$f^{*}:\mathsf{CoMod}_{C}\rightarrow\mathsf{CoMod}_{D}$ is colax monoidal with
respect to the cotensor products. The induced lax monoidal map on the right
adjoint:

$(X\square_{D}C){{\square}_{C}}(Y\square_{D}C)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(X\square_{D}Y)\square_{D}C,$

is actually strong monoidal by Lemma 4.5, for any $D$-comodules $X$ and $Y$.
Therefore we obtain a symmetric monoidal functor of $\infty$-categories:

$-\square_{D}C:\mathscr{N}((\mathsf{CoMod}_{D})_{f}^{\otimes})\longrightarrow\mathscr{N}((\mathsf{CoMod}_{C})_{f}^{\otimes}).$

If we suppose $f$ to be a weak equivalence, then if we post-compose the
functor with the symmetric monoidal Dwyer-Kan localization:

$\mathscr{N}((\mathsf{CoMod}_{C})_{f}^{\otimes})\longrightarrow\mathscr{N}((\mathsf{CoMod}_{C})_{f}^{\otimes})[\mathsf{W}^{-1}_{\mathsf{CoMod}}]\simeq\mathcal{CoMod}_{C}(\mathcal{D}^{\geq
0}(\Bbbk)),$

which is strong monoidal, it sends weak equivalence in $\mathsf{CoMod}_{D}$ to
equivalences in $\mathcal{CoMod}_{C}(\mathcal{D}^{\geq 0}(\Bbbk))$. Thus we
obtain the desired equivalence of symmetric monoidal $\infty$-categories. ∎

###  4.6. The Two-Sided Cobar Resolution

We conclude by providing an explicit formula for the derived cotensor product
$X\widehat{\square}_{C}Y$ of $C$-comodules. We show here that for $Y$ a
fibrant $C$-comodule, we obtain the equivalence
$X\widehat{\square}_{C}Y\simeq\Omega(X,C,C){{\square}_{C}}Y\simeq\Omega(X,C,Y)$.

######  Definition 4.30.

Let $C$ be a simply connected (cocommutative) dg-coalgebra. Let $X$ and $Y$ be
$C$-comodules. We define the _two-sided cosimplicial cobar construction of $X$
and $Y$_ to be the (coaugmented) cosimplicial object $\Omega^{\bullet}(X,C,Y)$
in $\mathsf{CoMod}_{C}$ by $\Omega^{\bullet}(X,C,C){{\square}_{C}}Y$. We
denote $\Omega(X,C,Y)$ the homotopy limit of $\Omega^{\bullet}(X,C,Y)$ in
$\mathsf{CoMod}_{C}$ computed as in [Hir03, 18.1.8].

######  Lemma 4.31.

Let $C$ be a simply connected cocommutative dg-coalgebra over $\Bbbk$. Let $X$
be a $C$-comodule. Let $M$ be a (non-negative) chain complex over $\Bbbk$.
Then there is a natural quasi-isomorphism of $C$-comodules:

${\Omega(X,C,C)\otimes M}$${\Omega(X,C,C\otimes M).}$$\scriptstyle{\simeq}$

###### Proof.

The natural map fits in the following commutative diagram in
$\mathsf{CoMod}_{C}$:

${X\otimes M}$${\Omega(X,C,C)\otimes M}$${\Omega(X,C,C\otimes M).}$

The vertical map is a quasi-isomorphism by Lemma 3.13. We now show that the
diagonal map is a quasi-isomorphism of $C$-comodules. The proof is similar as
of Lemma 3.13. Let $\widetilde{X}$ be the (homotopy) limit of Postnikov tower
$\\{X(n)\\}$ of $X$ as in Definition 2.19. From the fibrant replacement
$X\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\widetilde{X}$, we obtain the
commutative diagram in $\mathsf{CoMod}_{C}$:

${X\otimes M}$${\Omega(X,C,C\otimes M)}$${\widetilde{X}\otimes
M}$${\Omega(\widetilde{X},C,C\otimes M).}$

The left vertical map is a quasi-isomorphism as $M$ is a chain complex over a
finite product of fields. For each $n\geq 0$, notice that the natural maps
$\Omega^{n}(X,C,C\otimes M)\rightarrow\Omega^{n}(\widetilde{X},C,C\otimes M)$
are quasi-isomorphisms between coflat $C$-comodules (i.e. fibrant
$C$-comodules). By [Hir03, 18.5.3], the right vertical map is also a quasi-
isomorphism of $C$-comodules.

Thus we need to show that we have a quasi-isomorphism of $C$-comodules:

$\widetilde{X}\otimes M\simeq\Omega(\widetilde{X},C,C\otimes M).$

For $n=1$, then $X(1)$ is a right $C$-cofree comodule $N\otimes C$. The
coaugmented cosimplicial object $\Omega^{\bullet}(N\otimes C,C,C\otimes M)$:

${N\otimes C\otimes M}$${N\otimes C^{\otimes 2}\otimes M}$${N\otimes
C^{\otimes 3}\otimes M}$${\cdots,}$

splits in $\mathsf{CoMod}_{C}$, just as in the proof of Lemma 3.12. Thus we
have shown that $X(1)\otimes M\simeq\Omega(X(1),C,C\otimes M)$. Suppose we
have shown $X(n)\otimes M\simeq\Omega(X(n),C,C\otimes M)$ for some $n>0$.
Recall that $X(n+1)$ is the (homotopy) pullback in $\mathsf{CoMod}_{C}$ of:

${X(n+1)}$${P\otimes C}$${X(n)}$${Q\otimes C.}$${\lrcorner}$

Notice that $-\otimes M$ preserves finite limits, epimorphisms and thus
homotopy pullbacks, as we are working over $\Bbbk$ a finite product of fields.
Notice that the functor $\Omega(-,C,C\otimes M)$ preserves fibrations and
(homotopy) pullbacks. Thus we obtain the quasi-isomorphism given by the dashed
map in the diagram below:

${X(n+1)\otimes M}$${P\otimes C\otimes M}$${\Omega(X({n+1}),C,C\otimes
M)}$${\Omega(P\otimes C,C,C\otimes M)}$${X(n)\otimes M}$${Q\otimes C\otimes
M}$${\Omega(X({n}),C,C\otimes M)}$${\Omega(Q\otimes C,C,C\otimes
M)}$${\lrcorner}$$\scriptstyle{\simeq}$${\lrcorner}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Therefore, we can conclude that $\widetilde{X}\otimes
M\simeq\Omega(\widetilde{X},C,C\otimes M)$ by Lemma 2.15 and Proposition 2.21.
∎

In general, the cotensor product does not behave well with non-finite limits.
However, we have the following result.

######  Lemma 4.32.

Let $\\{X(n)\\}$ be a Postnikov tower of a $C$-comodule $X$. Let $Y$ be any
$C$-comodule. Then $\left\\{X(n){{\square}_{C}}Y\right\\}$ stabilizes in each
degree and:

$(\mathsf{lim}_{n}^{C}X(n)){{\square}_{C}}Y\cong\mathsf{lim}_{n}^{C}(X(n){{\square}_{C}}Y).$

In particular, if we denote $\widetilde{X}$ the limit of $\\{X(n)\\}$, then
the Postnikov tower of $\widetilde{X}\square_{C}Y$ is
$\\{X(n){{\square}_{C}}Y\\}$.

###### Proof.

Equalizers of towers that stabilize in each degree also stabilize in each
degree. Then the result follows from Lemma 2.15. ∎

######  Theorem 4.33.

Let $\Bbbk$ be a finite product of fields. Let $C$ be a simply connected
cocommutative dg-coalgebra over $\Bbbk$. Let $X$ be a $C$-dg-comodule. Let $Y$
be a fibrant $C$-dg-comodule (i.e. a coflat $C$-comodule). Then there is a
natural quasi-isomorphism of $C$-comodules:

${\Omega(X,C,C){{\square}_{C}}Y}$${\Omega(X,C,Y).}$$\scriptstyle{\simeq}$

###### Proof.

Let $\widetilde{Y}$ denote the (homotopy) limit of the Postnikov tower
$\\{Y(n)\\}$ of $Y$ (as a left $C$-comodule), as in Definition 2.19. Since $Y$
is fibrant, then $Y$ is a retract of $\widetilde{Y}$ and thus it is sufficient
to prove the quasi-isomorphism:

$\Omega(X,C,C){{\square}_{C}}\widetilde{Y}\simeq\Omega(X,C,\widetilde{Y}).$

We first show that for all $n\geq 0$:

$\Omega(X,C,C){{\square}_{C}}Y(n)\simeq\Omega(X,C,Y(n)).$

For $n=0$, the statement is vacuous. For $n=1$, then we can regard $Y$ as a
(left) cofree $C$-comodule $C\otimes M$ and we can conclude by the previous
lemma. Suppose we have shown the equivalence for some $n\geq 1$. Recall that
$Y(n+1)$ is obtained as a pullback of (left) $C$-comodules:

${Y(n+1)}$${C\otimes P}$${Y(n)}$${C\otimes Q,}$${\lrcorner}$

for some chain complexes $P$ and $Q$. On one hand, since
$\Omega(X,C,C){{\square}_{C}}-$ preserves finite limits we obtain the pullback
of (left) $C$-comodules:

${\Omega(X,C,C){{\square}_{C}}Y(n+1)}$${\Omega(X,C,C)\otimes
P}$${\Omega(X,C,C){{\square}_{C}}Y(n)}$${\Omega(X,C,C)\otimes
Q.}$${\lrcorner}$

On the other hand, we also obtain the pullback of cosimplicial left
$C$-comodules:

${\Omega^{\bullet}(X,C,Y(n+1))}$${\Omega^{\bullet}(X,C,C\otimes
P)}$${\Omega^{\bullet}(X,C,Y(n))}$${\Omega^{\bullet}(X,C,C\otimes
Q)}$${\lrcorner}$

The right vertical map is an objectwise fibration (both as $C$-comodules and
as chain complexes), thus we get the (homotopy) pullback in (left)
$C$-comodules:

${\Omega(X,C,Y(n+1))}$${\Omega(X,C,C\otimes
P)}$${\Omega(X,C,Y(n))}$${\Omega(X,C,C\otimes Q)}$${\lrcorner}$

Therefore, by induction and Lemma 4.31, we get:

$\Omega(X,C,C){{\square}_{C}}Y(n)\simeq\Omega(X,C,Y(n)).$

for all $n\geq 0$. We can conclude by Lemma 4.32, Corollary 2.23 and
Proposition 2.21. ∎

##  Appendix A Fibrations of Comodules

Throughout this appendix, we let $C$ be a simply connected dg-coalgebra over
$\Bbbk$. The goal is to prove Theorem 2.18. It states, by means of a Postnikov
tower, that we can inductively fibrant replace right $C$-comodules in the
model structure $\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ defined
in Proposition 2.12. Subsequently, let us write
$\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ simply by
$\mathsf{CoMod}_{C}$.

We prove in Theorem A.4 the result in greater generality. We show that we can
generate fibrations in $\mathsf{CoMod}_{C}$ as retract of fibrations built as
_Moore-Postnikov towers_. Formally, we define a class of generating fibrations
but unlike for cofibrantly generated model categories, we cannot apply a
_cosmall object argument_ (dual of [Hov99, 2.1.14]). Instead we build an ad-
hoc factorization that is not functorial. In fact, we can show that the model
structure $\mathsf{CoMod}_{C}$ admits an interesting _Postnikov presentation_
(in the sense of [BHK+15]). More of this explored in [Pér21] and we shall not
need it here.

We begin with some terminology. We dualize the notion of relative cell complex
[Hov99, 2.2.9].

######  Definition A.1 ([Hes09, 5.12]).

Let $\mathsf{P}$ be a class of morphisms in a category $\mathsf{C}$ closed
under pullbacks. Let $\lambda$ be an ordinal. Given a functor
$Y:\lambda^{\mathsf{op}}\rightarrow\mathsf{C}$ such that for all
$\beta<\lambda$, the morphism $Y_{\beta+1}\rightarrow Y_{\beta}$ fits into the
pullback diagram:

${Y_{\beta+1}}$${X^{\prime}_{\beta+1}}$${Y_{\beta}}$${X_{\beta+1},}$${\lrcorner}$

where $X^{\prime}_{\beta+1}\rightarrow X_{\beta+1}$ is some morphism in
$\mathsf{P}$, and $Y_{\beta}\rightarrow X_{\beta+1}$ is a morphism in
$\mathsf{C}$, and we denote:

$Y_{\gamma}:=\underset{\beta<\gamma}{\mathsf{lim}}Y_{\beta},$

for any limit ordinal $\gamma<\lambda$. We say that the composition of the
tower $Y$:

$\underset{\lambda^{\mathsf{op}}}{\mathsf{lim}}Y_{\beta}\longrightarrow
Y_{0},$

if it exists, is a _$\mathsf{P}$ -Postnikov tower_. The class of all
$\mathsf{P}$-Postnikov towers is denoted $\mathsf{Post}_{\mathsf{P}}$.

######  Proposition A.2 ([BHK+15, 2.10]).

If $\mathsf{C}$ is a complete category, the class $\mathsf{Post}_{\mathsf{P}}$
is the smallest class of morphisms in $\mathsf{C}$ containing $\mathsf{P}$
closed under composition, pullbacks and limits indexed by ordinals.

###### Proof.

See dual statements in [Hov99, 2.1.12, 2.1.13]. ∎

######  Definition A.3.

Let $C$ be a dg-coalgebra over $\Bbbk$. Define a _class_ of morphisms in
$\mathsf{CoMod}_{C}$:

$\mathsf{P}_{C}:=\Big{\\{}0\longrightarrow S^{0}(V)\otimes C\mid V\text{ any
$\Bbbk$-module}\Big{\\}}\bigcup\Big{\\{}D^{n}(V)\otimes C\longrightarrow
S^{n}(V)\otimes C\mid V\text{ any $\Bbbk$-module}\Big{\\}}_{n\geq 1}.$

Notice that the maps $D^{n}(V)\rightarrow S^{n}(V)$ and $0\rightarrow
S^{0}(V)$ are fibrations in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. Since $-\otimes
C:{\mathsf{Ch}_{\Bbbk}^{\geq 0}}\rightarrow\mathsf{CoMod}_{C}$ is a right
Quillen functor by Proposition 2.12, we see that the morphisms in
$\mathsf{P}_{C}$ are fibrations in $\mathsf{CoMod}_{C}$. We call the class
$\mathsf{P}_{C}$ the _generating fibrations of $\mathsf{CoMod}_{C}$_.

The following result is a generalization of [Hes09, 2.10] in which only finite
dimensional comodules were considered.

######  Theorem A.4.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. Every fibration in
$\mathsf{CoMod}_{C}$ is a retract of a $\mathsf{P}_{C}$-Postnikov tower. Any
morphism in $\mathsf{CoMod}_{C}$ factors as a cofibration followed by a
$\mathsf{P}_{C}$-Postnikov tower.

We spend the rest of the appendix proving the above 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 A.5.

Let $X$ be a (possibly unbounded) chain complex over $\Bbbk$. Then $X$ is
split as a chain complex and we have a non-canonical decomposition:

$X_{n}\cong H_{n}(X)\oplus B_{n}(X)\oplus B_{n-1}(X).$

In particular, any chain complex $X$ can be decomposed non-canonically as
product of disks and spheres:

$X\cong\prod_{n\in\mathbb{Z}}S^{n}(V_{n})\oplus D^{n}(W_{n}),$

where $V_{n}=H_{n}(X)$ and $W_{n}=B_{n-1}(X)$.

###### Proof.

We have the following short exact exact sequences of $\Bbbk$-modules:

${0}$${B_{n}(X)}$${Z_{n}(X)}$${H_{n}(X)}$${0,}$
${0}$${Z_{n}(X)}$${X_{n}}$${B_{n-1}(X)}$${0.}$$\scriptstyle{d_{n}}$

Since any short exact sequence splits, we can choose sections (the dashed maps
denoted above), such that we obtain the following isomorphism of
$\Bbbk$-modules:

$\displaystyle X_{n}$ $\displaystyle\cong$ $\displaystyle Z_{n}(X)\oplus
B_{n-1}(X)$ $\displaystyle\cong$ $\displaystyle H_{n}(X)\oplus B_{n}(X)\oplus
B_{n-1}(X).$

An investigation of the differentials $X_{n}\rightarrow X_{n-1}$ give the
desired result. ∎

Recall that every object is fibrant in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, and
thus every cofree $C$-comodule is fibrant in $\mathsf{CoMod}_{C}$ by
Proposition 2.12.

######  Lemma A.6.

Let $C$ be a dg-coalgebra over $\Bbbk$. Let $X$ be a right $C$-dg-comodule.
Then the fibration $U(X)\otimes C\rightarrow 0$ is a
$\mathsf{P}_{C}$-Postnikov tower.

###### Proof.

Given any $\Bbbk$-module $V$, notice that we have the following pullback in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$:

${S^{n}(V)}$${D^{n+1}(V)}$${0}$${S^{n+1}(V).}$${\lrcorner}$

Thus $S^{n}(V)\otimes C\rightarrow 0$ is a $\mathsf{P}_{C}$-Postnikov tower by
Proposition A.2. Moreover, since $D^{n}(V)\rightarrow 0$ is the composite
$D^{n}(V)\rightarrow S^{n}(V)\rightarrow 0$, we get that $D^{n}(V)\otimes
C\rightarrow 0$ is also a $\mathsf{P}_{C}$-Postnikov tower.

By Proposition A.5, we see that the underlying chain complex $U(X)$ splits as:

$U(X)\cong S^{0}(V_{0})\oplus\prod_{n\geq 1}\left(S^{n}(V_{n})\oplus
D^{n}(W_{n})\right).$

Since $-\otimes C:{\mathsf{Ch}_{\Bbbk}^{\geq 0}}\rightarrow\mathsf{CoMod}_{C}$
is a right adjoint, we obtain an isomorphism of $C$-comodules:

$U(X)\otimes C\cong(S^{0}(V_{0})\otimes C)\oplus\prod_{n\geq
1}^{C}\Big{(}(S^{n}(V_{n})\otimes C)\oplus(D^{n}(W_{n})\otimes C)\Big{)}.$

Here $\displaystyle\prod^{C}$ denotes the product in $\mathsf{CoMod}_{C}$. We
can conclude the desired result by Proposition A.2 ∎

For any chain complex $C$ and any $\Bbbk$-module $V$, we see that the $i$-th
term of the chain complex $S^{n}(V)\otimes C$ is the $\Bbbk$-module $V\otimes
C_{i-n}$. If we choose $C$ to be a simply connected dg-coalgebra, we get:

$\left(S^{n}(V)\otimes C\right)_{i}=\left\\{\begin{tabular}[]{ll}$0$&$i<n$,\\\
$V$&$i=n$,\\\ $0$&$i=n+1$,\\\ $V\otimes C_{i-n}$&$i\geq
n+2$.\end{tabular}\right.$

Thus, around the $n$-th term, the chain complex $S^{n}(V)\otimes C$ 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.

######  Lemma A.7.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. Let $X$ be a right
$C$-dg-comodule. Let $V$ be a $\Bbbk$-module. Let $n\geq 1$ be an integer.
Given a surjective linear map $f_{n}:X_{n}\rightarrow V$ that is non-trivial
only on $n$-cycles, there is a $C$-comodule map $f:X\rightarrow
S^{n}(V)\otimes C$, and the pullback comodule $P$ in the following diagram in
$\mathsf{CoMod}_{C}$:

${P}$${D^{n}(V)\otimes C}$${X}$${S^{n}(V)\otimes C,}$${\lrcorner}$

has homology:

$H_{i}(P)\cong\left\\{\begin{tabular}[]{ll}$\mathsf{ker}\left(H_{n}(f)\right)$&$i=n$,\\\
$H_{i}(X)$&$i<n$,\end{tabular}\right.$

and we have $P_{i}=X_{i}$ for $0\leq i<n-1$ and $i=n$, and
$P_{n-1}=X_{n-1}\oplus V$.

###### Proof.

First observe the following:

$\left(D^{n}(V)\otimes
C\right)_{i}=\left\\{\begin{tabular}[]{ll}$0$&$i<n-1$,\\\ $V$&$i=n-1,n$,\\\
$(V\otimes C_{i-n})\oplus(V\otimes C_{i-(n-1)})$&$i\geq
n+1$.\end{tabular}\right.$

Notice that the differential $\left(D^{n}(V)\otimes
C\right)_{n+1}\rightarrow\left(D^{n}(V)\otimes C\right)_{n}$ is trivial.

The properties of the map $U(X)_{n}\rightarrow V$ determine a chain map
$U(X)\rightarrow S^{n}(V)$, and thus a map of $C$-comodules $X\rightarrow
S^{n}(V)\otimes C$. By construction, since pullbacks in $\mathsf{CoMod}_{C}$
are determined in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, which are computed
levelwise, for $0\leq i<n-1$ and $j\in\\{n-1,n+1\\}$, we have the pullbacks of
$\Bbbk$-modules:

${P_{j}}$${V}$${P_{n}}$${V}$${P_{i}}$${0\,}$${X_{j}}$${0,}$${X_{n}}$${V,}$${X_{i}}$${0.}$${\lrcorner}$${\lrcorner}$${\lrcorner}$$\scriptstyle{f_{n}}$

Thus $P_{j}\cong X_{j}\oplus V$ and $P_{i}=X_{i}$ for any $i<{n-1}$ and $i=n$.
The differential $P_{n}\rightarrow P_{n-1}$ is the linear map
${X_{n}}$${X_{n-1}\oplus V,}$$\scriptstyle{d_{n}\oplus f}$ and the
differential $P_{j}\rightarrow P_{j-1}$ is the linear map:

${X_{j}\oplus V}$${X_{j}}$${X_{j-1},}$$\scriptstyle{d_{j}}$

where the unlabeled map is the natural projection, for $j\in\\{n-1,n+1\\}$.
All the differentials $P_{i}\rightarrow P_{i-1}$ for $0\leq i<n-1$ are the
differentials $X_{i}\rightarrow X_{i-1}$ of the chain complex $X$. Clearly, we
get $H_{i}(P)=H_{i}(X)$ for $0\leq i<n-1$. For $i=n-1$, by Proposition A.5, we
can choose a decomposition:

$X_{n}\cong H_{n}(X)\oplus B_{n-1}(X)\oplus B_{n}(X).$

The differential $d_{n}:X_{n}\rightarrow X_{n-1}$ sends the factor
$B_{n-1}(X)$ in $X_{n}$ to itself, and the factor $H_{n}(X)\oplus B_{n}(X)$ to
zero. By definition, the map $f_{n}:X_{n}\rightarrow V$ 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 $B_{n-1}(X)\oplus B_{n}(X)$ to zero. Thus the image
of the differential $P_{n}\longrightarrow P_{n-1}$, is precisely
$B_{n-1}(X)\oplus V$. Therefore, we obtain:

$\displaystyle H_{n-1}(P)$ $\displaystyle=$
$\displaystyle\frac{\mathsf{ker}(P_{n-1}\rightarrow
P_{n-2})}{\mathsf{im}(P_{n}\rightarrow P_{n-1})}$ $\displaystyle\cong$
$\displaystyle\frac{Z_{n-1}(X)\oplus V}{B_{n-1}(X)\oplus V}$
$\displaystyle\cong$ $\displaystyle\frac{Z_{n-1}(X)}{B_{n-1}(X)}$
$\displaystyle=$ $\displaystyle H_{n-1}(X).$

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 $f_{n}(x)=0$. Since $f_{n}:X_{n}\rightarrow V$ is entirely defined
on the copy $H_{n}(X)$ in $X_{n}$, we get from the commutative diagram:

${Z_{n}(X)}$${X_{n}}$${V}$${H_{n}(X),}$$\scriptstyle{f_{n}}$$\scriptstyle{H_{n}(f)}$

that $H_{n}(P)\cong\mathsf{ker}(H_{n}(f)).$ ∎

######  Lemma A.8.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. Let $j:X\rightarrow
Y$ be a monomorphism in $\mathsf{CoMod}_{C}$ (i.e. a cofibration), such that
it induces a monomorphism in homology in each degree. Let $n\geq 1$ be a fixed
integer. Then the map $j$ factors in $\mathsf{CoMod}_{C}$ as:

${X}$${Y}$${F_{n}(Y)}$$\scriptstyle{j}$$\scriptstyle{F_{n}(j)}$$\scriptstyle{F_{n}(p_{j})}$

where $F_{n}(Y)$ is a right $C$-comodule built with the following properties.

  * •

The map $F_{n}(p_{j}):F_{n}(Y)\rightarrow Y$ is a $\mathsf{P}_{C}$-Postnikov
tower.

  * •

The map $F_{n}(j):X\rightarrow F_{n}(Y)$ is a monomorphism (i.e. a cofibration
in $\mathsf{CoMod}_{C}$).

  * •

The $\Bbbk$-module $(F_{n}(Y))_{i}$ differs from $Y_{i}$ only in degrees
$i=n-1$ and $i\geq n+1$.

  * •

In degrees $i<n$ in homology, the map
$H_{i}(F_{n}(p_{j})):H_{i}(F_{n}(Y))\rightarrow H_{i}(Y)$ is an isomorphism.
For all degrees $i\geq 0$, the maps $H_{i}(F_{n}(j))$ are monomorphisms, such
that, if the maps $H_{i}(j)$ are isomorphisms, then so are the maps
$H_{i}(F_{n}(j))$.

  * •

In degree $n$ in homology, we have $H_{n}(F_{n}(Y))\cong H_{n}(X)$ and the
map:

$H_{n}(F_{n}(j)):H_{n}(X)\longrightarrow H_{n}(F_{n}(Y))\cong H_{n}(X),$

is an isomorphism.

###### Proof.

We construct below the chain complex $F_{n}(Y)$ explicitly using Lemma A.7.
For simplicity, we drop the forgetful functor $U$ from the notation. By
Proposition A.5, we can decompose $Y_{n}$ as:

$Y_{n}\cong
H_{n}(Y)\oplus\overline{Y_{n}}\cong\mathsf{im}\Big{(}H_{n}(j)\Big{)}\oplus\mathsf{coker}\Big{(}H_{n}(j)\Big{)}\oplus\overline{Y_{n}},$

where $\overline{Y_{n}}$ is the direct sum of the copies of the boundaries.
Denote the $\Bbbk$-module $V=\mathsf{coker}(H_{n}(j))$ and define the linear
map $f_{n}:Y_{n}\rightarrow V$ to be the natural projection. In particular,
the map $f_{n}$ sends $n$-boundaries of $Y$ to zero. This defines a chain map:
$f:Y\longrightarrow S^{n}(V)$. By adjointness, obtain a $C$-comodule map
$Y\rightarrow S^{n}(V)\otimes C$. Notice that since $j:X\rightarrow Y$ is a
monomorphism, we get $j(\overline{X_{n}})\subseteq\overline{Y_{n}}$, and so,
by construction of $f$, we get that the composite
${X}$${Y}$${S^{n}(V)}$$\scriptstyle{j}$$\scriptstyle{f}$ is the zero chain
map. We obtain $F_{n}(Y)$ as the following pullback in $\mathsf{CoMod}_{C}$,
with a chain map $F_{n}(j)$ induced by the universality of pullbacks:

${X}$${F_{n}(Y)}$${D^{n}(V)\otimes C}$${Y}$${S^{n}(V)\otimes
C.}$$\scriptstyle{F_{n}(j)}$$\scriptstyle{\exists!}$$\scriptstyle{0}$$\scriptstyle{j}$${\lrcorner}$$\scriptstyle{F_{n}(p_{j})}$$\scriptstyle{f}$

By construction, the induced map of $C$-comodules
$F_{n}(p_{j}):F_{n}(Y)\rightarrow Y$ is a $\mathsf{P}_{C}$-Postnikov tower.
From the commutativity of the diagram:

${X}$${F_{n}(Y)}$${Y,}$$\scriptstyle{F_{n}(j)}$$\scriptstyle{j}$$\scriptstyle{F_{n}(p_{j})}$

since $j$ is a monomorphism, so is $F_{n}(j)$. Since $H_{i}(j)$ is a
monomorphism for all $i$, then so is $H_{i}(F_{n}(j))$. By Lemma A.7, we get
$H_{i}(F_{n}(Y))\cong H_{i}(Y)$ for all $i<n$. For $i=n$, we get:

${H_{n}(F_{n}(Y))\cong\mathsf{ker}\Big{(}H_{n}(f)\Big{)}\cong H_{n}(X),}$

as we have the short exact sequence of $\Bbbk$-vector spaces:

${0}$${H_{n}(X)}$${H_{n}(Y)}$${V}$${0,}$$\scriptstyle{H_{n}(j)}$$\scriptstyle{H_{n}(f)}$

since $V=\mathsf{coker}(H_{n}(j))$. Thus $H_{n}(F_{n}(j))$ is an isomorphism
as desired. ∎

We state the case $n=0$.

######  Lemma A.9.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. Let $j:X\rightarrow
Y$ be a monomorphism in $\mathsf{CoMod}_{C}$, such that it induces a
monomorphism in homology in each degree. Then the map $j$ factors in
$\mathsf{CoMod}_{C}$ as:

${X}$${Y}$${F_{0}(Y)}$$\scriptstyle{j}$$\scriptstyle{F_{0}(j)}$$\scriptstyle{F_{0}(p_{j})}$

where $F_{0}(Y)$ is a right $C$-comodule built with the following properties.

  * •

The map $F_{0}(p_{j}):F_{0}(Y)\rightarrow Y$ is a $\mathsf{P}_{C}$-Postnikov
tower.

  * •

The map $F_{0}(j):X\rightarrow F_{0}(Y)$ is a monomorphism and a monomorphism
in homology.

  * •

In degree zero, the map $H_{0}(F_{0}(j)):H_{0}(X)\rightarrow H_{0}(F_{0}(Y))$
is an isomorphism of $\Bbbk$-modules.

###### Proof.

Let $V=\mathsf{coker}(H_{0}(j))$ which defines a map $f:Y\rightarrow
S^{0}(V)\otimes C$ of right $C$-comodules, such that, if we precompose with
$j:X\rightarrow Y$, it is the zero map. Define the right $C$-comodule
$F_{0}(Y)$ as follows:

${F_{0}(Y)}$${0}$${Y}$${S^{0}(V)\otimes
C.}$${\lrcorner}$$\scriptstyle{F_{0}(p_{0})}$$\scriptstyle{f}$

A computation shows that $H_{0}(F_{0}(Y))$ is given by the kernel of
$H_{0}(f)$ which is $H_{0}(X)$ by construction. The other properties follow
from similar arguments as previous lemma. ∎

###### Proof of Theorem A.4.

The first statement follows from the second using the retract argument (see
[Hov99, 1.1.9]). Given a map $f:X\rightarrow Y$ of right $C$-comodules, we
build below a right $C$-comodule $\widetilde{W}$ as a tower in
$\mathsf{CoMod}_{C}$ using Lemma A.8 repeatedly so that $f$ factors as:

${\widetilde{W}}$${\vdots}$${G_{1}(W)}$${G_{0}(W)}$${X}$${W}$${Y}$$\scriptstyle{F_{1}(p)}$$\scriptstyle{F_{0}(p)}$$\scriptstyle{f}$$\scriptstyle{j}$$\scriptstyle{G_{0}(j)}$$\scriptstyle{G_{1}(j)}$$\scriptstyle{\widetilde{j}}$$\scriptstyle{p}$

where $\widetilde{j}$ is a monomorphism and a quasi-isomorphism, and all the
vertical maps and $p$ are in $\mathsf{Post}_{\mathsf{P}_{C}}$. The composition
of all the vertical maps and $p$ is a map of $C$-comodules
$\widetilde{W}\rightarrow Y$, which is a $\mathsf{P}_{C}$-Postnikov tower by
Proposition A.2.

First we define $W$ as the $C$-comodule $(U(X)\otimes C)\oplus Y$. Then $f$
factors through $W$ via the following pullback in $\mathsf{CoMod}_{C}$:

${X}$${W}$${U(X)\otimes
C}$${Y}$${0,}$$\scriptstyle{\rho}$$\scriptstyle{f}$${\lrcorner}$$\scriptstyle{p}$

where $\rho$ is the right $C$-coaction of $X$. The induced map $p$ is a
$\mathsf{P}_{C}$-Postnikov tower by Lemma A.6. By commutativity of the upper
triangle, we see that the monomorphism $\rho$ induces a monomorphism
$j:X\hookrightarrow W$ which is also a monomorphism in homology (as $\rho$ is
a monomorphism in homology by counitality).

We construct the $C$-comodule $\widetilde{W}$ below as the limit
$\mathsf{lim}_{n}^{C}(G_{n}(W))$ in $\mathsf{CoMod}_{C}$ of the tower of maps:

${\cdots}$${G_{2}(W)}$${G_{1}(W)}$${G_{0}(W)}$${W,}$$\scriptstyle{G_{2}(p)}$$\scriptstyle{G_{1}(p)}$$\scriptstyle{G_{0}}$

where each $G_{n}(p)$ is a $\mathsf{P}_{C}$-Postnikov tower. Moreover, the
tower stabilizes in each degree, in the sense of Definition 2.14. The map
$\widetilde{j}:X\rightarrow\widetilde{W}$ is induced by the monomorphisms
$G_{n}(j):X\rightarrow G_{n}(W)$ which are compatible with the tower:

${G_{n}(W)}$${X}$${G_{n-1}(W),}$$\scriptstyle{G_{n}(p)}$$\scriptstyle{G_{n}(j)}$$\scriptstyle{G_{n-1}(j)}$

and $G_{n}(j)$ induces an isomorphism in homology in degrees $i$, for $0\leq
i\leq n$, and a monomorphism otherwise. We construct the $C$-comodule
$G_{n}(W)$ in the tower inductively as follows.

  * •

For the initial step, apply Lemma A.9 to the monomorphism $j:X\rightarrow W$.
Denote $G_{0}(W):=F_{0}(W)$. The cofibration $G_{0}(j):=F_{0}(j):X\rightarrow
G_{0}(X)$ is an isomorphism in homology in degree $0$, and a monomorphism in
positive degrees. The map $G_{0}(p):=F_{0}(p_{j}):G_{0}(W)\rightarrow Y$ is a
$\mathsf{P}_{C}$-Postnikov tower.

  * •

For the inductive step, for a fixed integer $n\geq 0$, suppose the
$C$-comodule $G_{n}(W)$ is defined, together with a cofibration
$G_{n}(j):X\rightarrow G_{n}(W)$ inducing an isomorphism in homology for
degrees $i$, where $0\leq i\leq n$, and a monomorphism in homology for higher
degrees. Apply Lemma A.8 to the monomorphism $G_{n}(j)$ for the degree $n+1$.
Denote:

$G_{n+1}(W):=F_{n+1}(G_{n}(W)).$

The cofibration $G_{n+1}(j)$ defined as the $C$-comodule map:

$F_{n+1}(G_{n}(j)):X\longrightarrow F_{n+1}(G_{n}(W))=G_{n+1}(W),$

is an isomorphism in homology in degrees $i$ where $0\leq i\leq n+1$, and a
monomorphism in other degrees. We obtain a $\mathsf{P}_{C}$-Postnikov tower
$G_{n+1}(p)$ defined as the $C$-comodule map:

$F_{n+1}\left(p_{G_{n}(j)}\right):G_{n+1}(W)=F_{n+1}(G_{n}(W))\longrightarrow
G_{n}(W),$

such that the following diagram commutes:

${X}$${G_{n}(W)}$${G_{n+1}(W).}$$\scriptstyle{G_{n}(j)}$$\scriptstyle{G_{n+1}(j)}$$\scriptstyle{G_{n+1}(p)}$

The induced map $\widetilde{j}:X\rightarrow\widetilde{W}$ is a monomorphism of
$C$-comodules. Indeed, notice that the tower stabilizes in each degree and
thus, for each $i\geq 0$:

$\widetilde{W}_{i}=(\mathsf{lim}_{n}^{C}G_{n}(W))_{i}\cong(\mathsf{lim}_{n}G_{n}(W))_{i}\cong(G_{i+1}(W))_{i}\cong(G_{i+2}(W))_{i}=\cdots,$

by Corollary 2.16. Therefore the map
$\widetilde{j}_{i}:X_{i}\rightarrow\widetilde{W}_{i}$ is the map:

${\left(G_{n+1}(j)\right)}_{i}:X_{i}\longrightarrow{\left(G_{i+1}(W)\right)}_{i},$

which is a monomorphism. Similarly, we get: $H_{i}(\widetilde{W})\cong
H_{i}(G_{i+1}(W))\cong H_{i}(X),$ for all $i\geq 0$, and so $\widetilde{j}$ is
an acyclic cofibration as desired. ∎

We can now prove Theorem 2.18 as it is a particular case of Theorem A.4 where
instead of factorizing a general map of $C$-comodules $X\rightarrow Y$, we
take $Y=0$.

###### Proof of Theorem 2.18.

Following the proof of Theorem A.4, we first consider the $C$-comodule
$X(1)=W=U(X)\otimes C$. Notice that as $C$ is simply connected, then $X(1)$
already has the correct homology at level $0$ and $1$. The rest of the proof
follows. ∎

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