The Dold–Kan correspondence and coalgebra structures

Abstract

By using the Dold–Kan correspondence we construct a Quillen adjunction between the model categories of non-cocommutative coassociative simplicial and differential graded coalgebras over a field. We restrict to categories of connected coalgebras and prove a Quillen equivalence between them.

1 Introduction

Schwede and Shipley [21] give sufficient conditions for lifting the model category structures of closed monoidal model categories to their categories of monoids along the free-forgetful adjoint pairs. Then in [22], they consider a Quillen equivalence between the closed monoidal model categories and state conditions for obtaining a Quillen equivalence at the level of categories of monoids as illustrated in the following diagram

In this paper we restrict to the categories of simplicial and differential non-negatively graded vector spaces. Both categories have closed monoidal model structures and are equivalent via the classical Dold–Kan correspondence. We consider their respective categories of comonoids, namely, simplicial and differential non-negatively graded coalgebras and we try to mimic dual methods from [22] for providing the categories of comonoids with a Quillen equivalence. We prove that Dold–Kan’s normalization functor descends to the level of categories of coalgebras and gives rise to an adjoint pair. Moreover, we produce model categories structures for simplicial and differential non-negatively graded coalgebras (we refer to [7] and to the unpublished [6] where these model category structures have been studied at first) and observe that the adjunction obtained turns out to be a Quillen pair. We then restrict to the categories of simplicial and differential non-negatively graded connected coalgebras. We derive corresponding model structures for these categories and establish a Quillen equivalence between them.

Section 2 is devoted to some generalities about the monoidal categories of differential non-negatively graded vector spaces, simplicial vector spaces and their corresponding categories of comonoids.

Section 3 revisits the model structure on the category of non-cocommutative coassociative differential graded coalgebras over a field. This model category structure is due to Getzler and Goerss in their unpublished paper [6]. We reproduce with more details most of their results and proofs. However we slightly modify their arguments for the cofibration-(acyclic fibration) axiom.

Section 4 shows that the normalization functor $N$ defines a functor from the category of simplicial coalgebras to the category of differential non-negatively graded coalgebras. Similar observations hold for the inverse functor $\Gamma$, however, as in [22], we point out that both functors are not adjoint on the level of comonoids. We then construct a functor that is right adjoint to the normalization functor $N$ on the level of comonoids. The adjoint pair of functors obtained in this way turns out to be an adjoint Quillen pair. We refer to [18] where similar constructions are also considered.

Section 5 restricts to the categories of connected simplicial coalgebras and connected differential graded coalgebras. In this section, we first investigate the completeness and cocompleteness properties of connected differential graded coalgebras. We then lift along a suitable adjunction the model structure on the category of differential graded coalgebras to the category of connected differential graded coalgebras by means of the transfer principle. In order to obtain the (acyclic cofibration)-fibration factorization axiom in a functorial way we adapt and reprove some useful techniques from [19] in our setting. We end this section by proving a Quillen equivalence between the categories of connected differential graded and simplicial coalgebras.

Section 6 closes this paper with an appendix on the category of connected differential graded algebras. The results therein are intended as a step for constructing limits for connected differential graded coalgebras.

2 Preliminaries

2.1 Differential graded vector spaces

In this section we review some basic results from homological algebra. For a thorough treatment, we refer to [13]. Let Vct be the category of vector spaces over a fixed field $K$. We denote by DGVct the category of differential graded $K$-vectors spaces which are concentrated in non-negative degrees and have differentials of degree $-1$. We denote by $\mathbb {S}^n$ the $n$-sphere chain complex. This is the object of DGVct which has the field $K$ in degree $n$ and $0$ in other degrees. All differentials in $\mathbb {S}^n$ are trivial. The $n$-disk, denoted by $\mathbb {D}^n$, is the object of DGVct which has the field $K$ in degrees $n$ and $n-1$ and $0$ elsewhere. The identity on $K$ is its only non-trivial differential and we set $\mathbb D^0 = \mathbb S^0$.

Recall that if $X$ and $Y$ are two objects in the category DGVct, a symmetric monoidal product $\otimes$ is given by $(X\otimes Y)_n = \bigoplus _{p+q=n} X_p\otimes _K Y_q$ with differential $d(x\otimes y) = dx\otimes y + (-1)^{\mid x\mid }x\otimes dy$. The unit of this monoidal product is $\mathbb {S}^0$, the differential graded vector space concentred in degree $0$, which we sometimes denote by $K[0]$. The category DGVct endowed with the monoidal product $\otimes$ is closed. Given differential graded vector spaces $(X,d_X)$ and $(Y,d_Y)$, let $\mathrm {Hom}(X,Y)$ be the object of DGVct with:

$$\begin{aligned} \mathrm {Hom}(X,Y)_0&= \left\{ f \in \prod _{p \ge 0} \mathbf Vct (X_p, Y_p) \mid d_Y(f_px)-f_{p-1}(d_Xx)=0, x \in X_p\right\} \\ \mathrm {Hom}(X,Y)_n&= \prod _{p \ge 0} \mathbf Vct (X_p,Y_{p+n}), \text{ for } n\ge 1 \end{aligned}$$

and differential $d_H$ for any map $f = \{f_p :X_p \rightarrow Y_{p+n} \}_{p \ge 0}$ given by

$$\begin{aligned} (d_Hf)_p(x) = d_Y (f_p x) + (-1)^{n+1}f_{p-1}(d_X x), \quad x \in X_p. \end{aligned}$$

The specified right adjoint of the functor $-\otimes Y :\text{ DGVct } \rightarrow \text{ DGVct }$ is then given by the functor $\mathrm {Hom}(Y,-) :\text{ DGVct } \rightarrow \text{ DGVct }$. This right adjoint functor is given by applying the good truncation below $0$ (see [24, 1.2.7]) to the unbounded version of $\mathrm {Hom}(X,Y)$.

2.2 Simplicial vector spaces

We denote by SVct the category of simplicial vector spaces, that is, the category of functors $X :\Delta ^{\mathrm {op}} \rightarrow \mathbf Vct$, where $\Delta$ is the category of finite ordered sets $\left[ n\right] = \left\{ 0<1<\cdots <n\right\}$ for $n \in \mathbb N$ and whose morphisms are non-decreasing monotone functions (see for instance [24, 8.1]).

Let $X$ and $Y$ be two objects in SVct. A monoidal product $\widehat{\otimes }$ is given by $(X \widehat{\otimes }Y)_n = X_n \otimes _K Y_n$ with coordinatewise structure maps. The unit of the monoidal product $\widehat{\otimes }$ is the simplicial vector space having $K$ in each degree and identity maps on $K$ as face and degeneracy operators. We denote this unit by $I(K)$. This monoidal product $\widehat{\otimes }$ is symmetric and SVct is closed.

2.3 Differential graded coalgebras

We denote by DGcoAlg the category of counital coassociative differential graded $K$-coalgebras. In other words DGcoAlg is the category of comonoids in the monoidal category (DGVct, $\otimes$, $K[0]$).

2.4 Simplicial coalgebras

The category of simplicial coalgebras, denoted by ScoAlg, is the category of comonoids in the monoidal category (SVct, $\widehat{\otimes }$, $I(K)$).

In the coming sections the symbols $\sqcap$ and $\sqcup$ will stand respectively for the categorical product and coproduct in the appropriate categories.

3 Model category structures on categories of coalgebras

3.1 A model category structure on DGcoAlg

In this section, we consider the category of non-cocommutative coassociative, counital differential non-negatively graded coalgebras over a fixed field $K$, denoted here by DGcoAlg. We review the different arguments for proving the following result due to Getzler and Goerss in their unpublished paper [6]:

Theorem 3.1

[6, Definition 2.3, Theorem 2.8]. Define $f :C \rightarrow D \in$ DGcoAlg to be

• 1. a weak equivalence if $H_*f$ is an isomorphism.

• 2. a cofibration if $f$ is a degreewise injection of graded vector spaces.

• 3. a fibration if $f$ has the right lifting property with respect to acyclic cofibrations.

With these definitions, DGcoAlg becomes a closed model category.

Before proving the theorem stated above, we need to establish an analogue of the Fundamental Theorem on Coalgebras (see [23, Theorem 2.2.1]) for the objects of DGcoAlg. Namely if $C$ is an object of DGcoAlg and $c \in C$ is an homogeneous element then the sub-coalgebra generated by $c$ is finite dimensional. We recall that given a graded coalgebra $C$, a graded right $C$-comodule $M$ is a graded vector space with a structure map $\omega :M\rightarrow M\otimes C$. For a homogeneous basis $\left\{ c_i\right\}$ of $C$ the structure map is given by $\omega (m)=\sum _i m_i \otimes c_i$ under Sweedler’s notation with all but finitely many of the $m_i=0$. Moreover the graded linear dual $C^*=\left\{ \mathbf Vct (C_n, K)\right\}$ becomes a non-positively graded algebra and $M$ inherits a left $C^*$-module structure by setting $f.m = \sum _i \left\langle f,c_i\right\rangle m_i$ for $f\in C^*$. These facts are well-known and may be found in [23, Chapter 2] for the ungraded case or in [10, Section 2.5] for comodule over differential graded Hopf algebras.

Lemma 3.2

[6, Lemma 1.1] Let $\left( C, \Delta _C, \epsilon _C\right)$ be a graded coalgebra and $(M, \omega : M \rightarrow M \otimes C)$ be a right graded $C$-comodule. If $x\in M$ is a homogeneous element, then the subcomodule generated by $x$ is finite-dimensional.

Proof

Let $\left\{ c_i\right\}$ be a homogeneous basis for $C$. Then $\omega (x)=\sum _i x_i\otimes c_i$ with all but finitely many $x_i$ being zero. Now let $N$ denote the vector space spanned by the $x_i$. The identity $\mathrm {id}_M = (\mathrm {id}_M\otimes \epsilon _C)\circ \omega$ yields $x=\sum _i x_i.\epsilon _C(c_i)$ and hence we conclude that $x\in N$. Since $N$ is finite-dimensional it remains to check that $N$ is a subcomodule of $M$, that is, $\omega (N)\subseteq N\otimes C$. This is obtained by performing the following computations

$$\begin{aligned} \sum _i \omega (x_i) \otimes c_i&= (\omega \otimes \mathrm {id}_C)\circ \omega (x) \\&= (\mathrm {id}_M \otimes \Delta _C)\circ \omega (x)\\&= (\mathrm {id}_M \otimes \Delta _C)\left( \sum _j x_j\otimes c_j \right) \\&= \sum _j x_j \otimes \Delta _C(c_j) \\&= \sum _{j,i} x_j \otimes c_{ij} \otimes c_i \end{aligned}$$

for some $c_{ij} \in C$. This yields $\omega (x_i)=\sum _j x_j\otimes c_{ij} \in N\otimes C$. $\square$

Lemma 3.3

[6, Lemma 1.2] Let $C$ be a graded coalgebra and $x\in C$ a homogeneous element. Then there is a finite dimensional sub-coalgebra $D\subseteq C$ such that $x\in D$. Moreover it can be assumed that $D_n=0$ for $n$ larger that the degree of $x$.

Proof

Since the coalgebra $C$ is a right $C$-comodule with structure map given by $\omega =\Delta _C$, the Lemma 3.2 above supplies a finite-dimensional subcomodule with $x\in N \subseteq C$. Note that there is a comodule structure map $\omega :N \rightarrow N\otimes C$ defined by $\omega (n) = \sum _i n_i\otimes c_i$ for any homogeneous $n\in N$. This comodule structure induces a left $C^*$-module structure on $N$ by setting $f.n=\sum _i\left\langle f,c_i\right\rangle n_i$ for all $f\in C^*$ and homogeneous $n\in N$. Now let $\varphi :C^*\rightarrow \mathrm {End}_K(N)$ be the morphism induced by the above left $C^*$-module structure on $N$ and consider the orthogonal

$$\begin{aligned} \left( \ker \varphi \right) ^\bot =\left\{ y \in C \mid \left\langle f,y\right\rangle =0 \forall f\in \ker \varphi \right\} . \end{aligned}$$

Firstly $\left( \ker \varphi \right) ^\bot$ is a sub-coalgebra of $C$ since $\ker \varphi$ is an ideal of $C^*$. Secondly

$$\begin{aligned} \dim _K\left( \ker \varphi \right) ^\bot = \dim _K \left( C^*/ \left( \ker \varphi \right) ^{\bot \bot }\right) \le \dim _K \left( C^*/ \ker \varphi \right) . \end{aligned}$$

But $\dim _K \left( C^*/ \ker \varphi \right)$ is finite by the First Isomorphism Theorem on $\varphi$ and the fact that $N$ is finite dimensional. Hence we conclude that $\left( \ker \varphi \right) ^\bot$ is finite dimensional. Since $x\in N \subseteq \left( \ker \varphi \right) ^\bot$ we may set $D= \left( \ker \varphi \right) ^\bot$. Finally in order to have $D_n=0$ for $n$ greater than the degree of $x$, we just consider the sub-coalgebra of $D$ generated by the homogeneous elements of degree less than or equal to that of $x$. $\square$

Proposition 3.4

[6, Proposition 1.5] Let $(C, \partial )$ be a differential graded coalgebra and $x\in C$ a homogeneous element. Then there is a finite dimensional differential graded coalgebra $D\subseteq C$ such that $x\in D$.

Proof

Suppose $x \in C$ is a homogeneous element degree of degree $n$. By Lemma 3.3 there is a finite dimensional graded sub-coalgebra denoted $D(n)$ such that $x\in D(n) \subseteq C$ and $D(n)_k =0$ for $k>n$. Then choose a basis $\left\{ y_i\right\}$ for $D(n)_n$ and use again Lemma 3.3 to produce finite dimensional sub-coalgebras $D(y_i) \subseteq C$ for each $y_i$ such that $\partial _n y_i \in D(y_i)_{n-1}$. Then set

$$\begin{aligned} D(n-1)=D(n)+ \sum _i D(y_i). \end{aligned}$$

Thus $D(n-1)$ is a sub-coalgebra since the sum of coalgebras is again a coalgebra. Moreover $D(n-1)$ is finite dimensional. The process for obtaining $D(n-1)$ may be repeated to form an ascending sequence of sub-coalgebras

$$\begin{aligned} D(n)\subseteq D(n-1) \subseteq \cdots \subseteq D(0) \subseteq C \end{aligned}$$

with the properties that for $0\le k\le n-1$

• (1) each $D(k)$ is finite dimensional,

• (2) $D(k-1)_l = D(k)_l$ for $l\ge k$,

• (3) $\partial _k \left( D(k)_k\right) \subseteq D(k-1)_{k-1}$.

Finally setting $D=D(0)$ gives the required result. $\square$

With this key result in hand we can now proceed to the proof of the model category axioms. We refer to [5, Definition 3.3] for the numbering of model category axioms. The coming sections will essentially focus on the (co)completeness and factorizations axioms.

3.1.1 Axiom MC1

This section aims at proving the completeness of the category of coalgebras. The category DGcoAlg turns out to be anti-equivalent to a category in which colimits are easier to describe. The completeness of DGcoAlg is then derived from the cocompleteness of that category.

We denote by ProDGAlg $_{\le 0}$ the category of profinite non-positively graded differential algebras. An object $A$ in this category is an inverse limit $\lim _\alpha A_\alpha$ of degreewise finite dimensional graded differential algebras $A_\alpha$. By endowing the finite dimensional algebras $A_\alpha$ with the discrete topology, the object $A \in$ ProDGAlg $_{\le 0}$ inherits a topology where two-sided ideals of finite codimension form a neighborhood basis of $0$. Moreover, for an object $A \in$ ProDGAlg $_{\le 0}$, the continuous linear dual $A'$ defined degreewise by

$$\begin{aligned} A_n' = \left\{ f :A_{-n} \longrightarrow K \mid \ker f \text{ is } \text{ open }\right\} \end{aligned}$$

becomes an object of DGcoAlg, since by [12, Proposition 6, Page 290] the continuous linear dual takes limits to colimits.

Lemma 3.5

[6, Proposition 1.7] The algebraic linear dual and the continuous linear dual define an anti-equivalence of categories between DGcoAlg and ProDGAlg $_{\le 0}$.

Proof

By Proposition 3.4 any $C \in$ DGcoAlg can be written as a filtered colimit of its finite dimensional subcoalgebras. It follows that

$$\begin{aligned} \left( C^*\right) ' = \left( (\mathrm {colim} C_\alpha )^*\right) ' \cong (\lim C_\alpha ^*)' \cong \mathrm {colim} \left( C_\alpha ^*\right) ' \cong \mathrm {colim C_\alpha } = C. \end{aligned}$$

For $A\in$ ProDGAlg $_{\le 0}$, a similar argument yields $\left( A^*\right) ' \cong A.$ $\square$

Proposition 3.6

[6, Proposition 1.8] The category DGcoAlg is complete and cocomplete.

Proof

The forgetful functor from DGcoAlg to DGVct creates colimits and therefore the cocompleteness of DGcoAlg follows immediately. Then, we note that ProDGAlg $_{\le 0}$ is cocomplete. Indeed, let $A :I \rightarrow$ ProDGAlg $_{\le 0}$ be a diagram of profinite differential graded algebras. Here are the steps for defining its colimit in ProDGAlg $_{\le 0}$:

• (1) form the colimit $B = \mathrm {colim}_{i \in I} A_i$ of this diagram in the category of non-positively graded differential algebras.

• (2) endow $B$ with the topology where a neighborhood basis of $0$ is given by the set $\mathcal {J}$ of two-sided ideals $J$ which can be realized as the kernel of maps of algebras $B\rightarrow C$ such that $C$ is a finite dimensional differential graded algebra and such that for each $i \in I$, the composite $A_i \rightarrow B \rightarrow C$ is continuous.

• (3) define the required colimits as the profinite completion $\lim _{J \in \mathcal {J}} B/J$ of $B$ with respect to this topology.

By Proposition 3.5, the continuous linear dual carries colimits in ProDGAlg $_{\le 0}$ to limits in DGcoAlg and therefore DGcoAlg is complete. $\square$

3.1.2 Axiom MC5(i)

In this section, we prove the cofibration-(acyclic fibration) axiom. First, we construct a functor from DGVct to DGcoAlg that is right adjoint to the forgetful functor. Then, this functor is used to provide the required factorization axiom in a functorial way. The arguments used here differ from that of [6, Lemma 1.12, Theorem 2.1] and refer rather to that used in the proof of [19, Corollary 4.15].

Lemma 3.7

[6, Lemma 1.9]. Let $\left\{ C_\alpha \right\} _\alpha$ be a right filtered diagram in DGcoAlg and $D$ be a finite dimensional object in DGcoAlg, then the natural map

$$\begin{aligned} \mathrm {colim}_\alpha {\varvec{DGcoAlg}}\left( D, C_\alpha \right) \longrightarrow {\varvec{DGcoAlg}}\left( D, \mathrm {colim}_\alpha C_\alpha \right) \end{aligned}$$

is a bijection.

Proposition 3.8

[6, Proposition 1.10] Let $V$ be an object in DGVct. Then, there is a functor denoted $S_d$ from DGVct to DGcoAlg that is right adjoint to the functor $U_d$ that forgets coalgebra structure:

• (1) if $V$ is degreewise finite dimensional,

  $$\begin{aligned} S_d(V) = \left( \widehat{T_d(V^*)}\right) ' \end{aligned}$$

• (2) for any $V \in$ DGVct,

  $$\begin{aligned} S_d(V) =\mathrm {colim}_\alpha (S_d(V_\alpha )) \end{aligned}$$

  with $V_\alpha$ running over finite dimensional subvector spaces of $V$.

Proof

If $V$ is finite-dimensional using the following bijections with any object $D \in$ DGcoAlg

$$\begin{aligned} {\mathbf {DGcoAlg}} (D, (\widehat{T_d(V^*)})')&\cong {\mathbf {ProDGcoAlg}}_{\le 0} ( \widehat{T_d(V^*)}, D^*) \\&\cong {\mathbf {DGAlg}}_{\le 0} ( T_d(V^{*}), D^*) \\&\cong {\mathbf {ProDGVct}}_{\le 0} ( V^*, D^*) \\&\cong {\mathbf {DGVct}} ( \left( D^*\right) ', \left( V^*\right) ') \end{aligned}$$

give the desired result.

Now consider a general $V \in$ DGVct. Any object $D \in$ DGcoAlg can be written as the colimit $D = \mathrm {co}\!\lim _{\beta } D_\beta$ of its finite-dimensional subcoalgebras. Then, by Lemma 3.7 the following bijections

$$\begin{aligned} {\mathbf {DGcoAlg}}(D,\mathrm{colim}_\alpha (S_d(V_\alpha )))&\cong \lim _{\beta } {\mathbf {DGcoAlg}}(D_\beta ,\mathrm {colim}_\alpha (S_d(V_\alpha ))) \\&\cong \lim _{\beta } {\mathrm {co}}\!\lim _{\alpha } {\mathbf {DGcoAlg}} (D_\beta ,S_d(V_\alpha )) \\&\cong \lim _{\beta } \mathrm {co}\!\lim _{\alpha } {\mathbf {DGVct}}(D_\beta ,V_\alpha ) \\&\cong {\mathbf {DGVct}}(D,V) \end{aligned}$$

complete the proof. $\square$

Definition 3.9

Let $K$ be a field and $K\left\langle x\right\rangle$ denote the vector space with basis x. We denote by $(I,d_I)$ the unit interval, i.e., the following differential graded vector space concentrated in degrees $1$ and $0$:

$$\begin{aligned} \cdots 0 \rightarrow 0 \rightarrow K\left\langle a \right\rangle \mathop {\longrightarrow }\limits ^{d} K\left\langle b,c\right\rangle \rightarrow 0 \cdots \end{aligned}$$

with $d(a) = c-b$. Setting $\Delta (b) = b \otimes b$, $\Delta (c) = c \otimes c$ and $\Delta (a) = b \otimes a + a \otimes c$ yields a coassociative counital coalgebra structure on the differential graded vector space $(I,d_I)$.

Lemma 3.10

Let $f, g :V \rightarrow W$ be morphisms of DGVct. A chain homotopy between $f$ and $g$ is a chain map $H :V \otimes I \rightarrow W$ such that $H(v \otimes b) =f(v)$ and $H(v \otimes c) =g(v)$.

Proof

For $v \in V_n$, it suffices to set $s_n(v) = (-1)^n H_{n+1}(v\otimes a)$ to recover the classical chain homotopy definition. $\square$

Lemma 3.11

[19, Proposition 4.10] Let $f, g :V \rightarrow W$ be chain homotopic morphisms in DGVct.

Then, $S_d(f), S_d(g) :S_d(V) \rightarrow S_d(W)$ in DGcoAlg are chain homotopic in DGcoAlg.

Proof

There is a homotopy $H :V \otimes I \rightarrow W$. Applying the cofree functor to this homotopy yields a morphism of coalgebras $S_d(H) :S_d(V \otimes I) \rightarrow S_d(W)$. We note that $S_d(V) \otimes I$ inherits a coalgebra structure from that of $S_d(V)$ and $I$ with a comultiplication given by the composite

$$\begin{aligned} \tau \circ \Delta _{S_d(V)} \otimes \Delta _{I} :S_d(V) \otimes I&\rightarrow S_d(V) \otimes S_d(V) \otimes I \otimes I\\&\rightarrow S_d(V) \otimes I \otimes S_d(V) \otimes I \end{aligned}$$

where $\tau$ is the switch map. The morphism $S_d(V) \otimes I \rightarrow V \otimes I$, together with the universal property of the cofree coalgebra functor $S_d$ yields a morphism $\pi$

Finally, the composite $S_d(H) \circ \pi :S_d(V) \otimes I \rightarrow S_d(V \otimes I) \rightarrow S_d(W)$ gives the required homotopy. $\square$

Lemma 3.12

[19, Proposition B.17] Let $f :A \rightarrow B$ and $g :C \rightarrow D$ be morphisms in DGcoAlg. If $f$ and $g$ are homotopy equivalences, then $f \sqcap g :A \sqcap C \rightarrow B \sqcap D$ is a homotopy equivalence.

Proposition 3.13

[6, Lemma 2.4 2.] Let $f :C \rightarrow D$ be a morphism in DGcoAlg. Choose an acyclic differential graded vector space $V$ containing $C$. Then the morphism $f$ can be factored

$$\begin{aligned} C \mathop {\longrightarrow }\limits ^{i} D \sqcap S_d(V) \mathop {\longrightarrow }\limits ^{p} D \end{aligned}$$

with $i$ a cofibration and $p$ an acyclic fibration.

Proof

First forget the coalgebra structure on $C$ by considering $U_d(C) \in$ DGVct. Then define $V$ to be cone($U_d(C)$). The object $\mathrm {cone}(U_d(C)) \in$ DGVct is acyclic and comes with a canonical embedding $j :U_d(C) \rightarrow \mathrm {cone}(U_d(C))$.

We define $p$ to be the projection map $D \sqcap S_d(V) \rightarrow D$. The homotopy equivalence between $\mathrm {cone} U_d(C)$ and $0$ yields a homotopy equivalence between $S_d(\mathrm {cone} U_d(C))$ and $S_d(0) = K[0]$ by Lemma 3.11. Then, the morphism $p :D \sqcap S_d(\mathrm {cone} U_d(C)) \rightarrow D \sqcap K[0] \cong D$ is a homotopy equivalence by Lemma 3.12. Note that we are working with bounded chain complexes over a fixed field, and hence, homotopy equivalences correspond to quasi-isomorphisms. It follows that the morphism $p$ is acyclic.

Then, to give a morphism in DGcoAlg $\left( C, D\sqcap S_d(V)\right)$ amouts to give a pair of morphisms in

$$\begin{aligned} {\varvec{DGcoAlg}}(C,D) \times {\varvec{DGcoAlg}}(C,S_d(V)). \end{aligned}$$

But using the adjunction between the categories of coalgebras and vector spaces, the previous product is equivalent to

$$\begin{aligned} {\varvec{DGcoAlg}}(C,D) \times {\varvec{DGVct}}(U_d(C),V). \end{aligned}$$

In this way, the pair $(f,j)$ yields a coalgebra morphism $i$. Now consider $\pi :D \sqcap S_d(V) \rightarrow S_d(V)$ the projection onto $S_d(V)$ and the map $\epsilon :S_d(V)\rightarrow V$ coming from the counit of the adjunction between coalgebras and vector spaces. Then the following composite

$$\begin{aligned} C \mathop {\longrightarrow }\limits ^{i} D \sqcap S_d(V) \mathop {\longrightarrow }\limits ^{\pi } S_d(V) \mathop {\longrightarrow }\limits ^{\epsilon } V =\mathrm {cone}(U_d(C)) \end{aligned}$$

is the embedding $j$ and therefore insures that the coalgebra morphism $i$ is a cofibration as required. $\square$

3.1.3 Axiom MC5(ii)

In this section, we prove the (acyclic cofibration)-fibration axiom. We exhibit a class of morphisms in DGcoAlg and use the small object argument to perform the required factorization axiom with respect to this class.

Lemma 3.14

[6, Lemma 2.5] Let $j :C \rightarrow D$ be an acyclic cofibration in DGcoAlg and $x \in D$ be a homogeneous element. Then there exists a subcoalgebra $B$ of $D$ such that

• 1. $x \in B$

• 2. $B$ has a countable homogeneous basis,

• 3. $C \cap B \rightarrow B$ is an acyclic cofibration in DGcoAlg.

Proof

The idea of the proof is to construct a sequence $\left( B(n)\right) _{n\ge 1}$ of subcoalgebras of $D$ having the following properties:

• (1) $B(1) \subseteq B(2) \subseteq \cdots \subseteq B(n) \cdots$,

• (2) each $B(n) \in$ DGcoAlg is finite dimensional,

• (3) the induced map $B(n-1)/\left[ C \cap B(n-1)\right] \rightarrow B(n)/\left[ C \cap B(n)\right]$ of differential graded vector spaces is zero in homology.

First, there is a finite dimensional subcoalgebra $B(1)$ of $D$ that contains $x$. Then, suppose that $B(n-1)$ has been constructed. Since $B(n-1)$ is finite dimensional, choose a finite set of homogeneous cycles $z_i + C \cap B(n-1) \in B(n-1)/\left[ C\cap B(n-1)\right]$ so that the resulting homology classes span $H_*\big (B(n-1)/\left[ C\cap B(n-1)\right] \big )$. For each index $i$, there is a homogeneous element $x_i \in D$ which is a boundary of $z_i$ since $H_*\left( D/C\right)$ is zero. Then choose a finite dimensional subcoalgebra $x_i \in A(x_i) \subseteq D$ and set $B(n) = B(n-1) + \sum _i A(x_i)$. Finally, setting $B = \bigcup _n B(n)$ satisfies the statements of the lemma. Indeed, we have

$$\begin{aligned} H_*(C \cap B) = H_*\left( \bigcup _n C \cap B(n)\right) = \bigcup _n H_*(C \cap B(n)) = H_*C. \end{aligned}$$

This comes from the facts that homology commutes with filtered colimits and that $H_*(C \cap B(n)) \cong H_*(B(n))$ due to the long exact sequence on homology resulting from the exact sequence $0 \rightarrow C \cap B(n) \rightarrow B(n) \rightarrow B(n)/\left[ C \cap B(n)\right] \rightarrow 0.$ $\square$

Lemma 3.15

[6, Lemma 2.6] A morphism $q :X \rightarrow Y$ in DGcoAlg is a fibration if and only if it has the right lifting property with respect to all acyclic cofibrations $A \rightarrow B$ so that $B$ has a countable homogeneous basis.

Proof

The first implication follows immediately from the definition of fibration given in Definition 3.1. For the second implication, suppose that a morphism $q :X \rightarrow Y \in$ DGcoAlg has the right lifting property with respect to all acyclic cofibrations $A \rightarrow B$ with $B$ having a countable homogeneous basis. We have to prove that $q$ is a fibration, that is the lifting problem in

where the morphism $i$ is an arbitrary acyclic cofibration, has a solution. This problem is solved by using Zorn’s lemma. Let $\Omega$ be the set of pairs $(\bar{D}, g)$ where $\bar{D}$ fits into a sequence of acyclic cofibrations $C \mathop {\longrightarrow }\limits ^{\subseteq } \bar{D} \mathop {\longrightarrow }\limits ^{\subseteq } D$ in DGcoAlg and $g :\bar{D} \rightarrow X$ is a solution to the restricted lifting problem. We order the set $\Omega$ by setting $(\bar{D}_1, g_1) \preceq (\bar{D}_2, g_2)$ if $\bar{D}_1 \subseteq \bar{D}_2$ and ${g_2}_{\mid _{\bar{D}_1}} = g_1$. The poset $(\Omega , \preceq )$ is non empty since it contains $(C, f)$. Moreover, any chain in $(\Omega , \preceq )$ has an upper bound given by the union. Hence, by Zorn’s lemma $(\Omega , \preceq )$ posses a maximal element $(E, g)$. We now prove that $E=D$. Let $x$ be a homogeneous element of $D$. Since $E \mathop {\longrightarrow }\limits ^{\subseteq } D$ is an acyclic cofibration, Lemma 3.14 guarantees the existence of a subcoalgebra $B$ such that $x \in B$, $B$ has a countable homogeneous basis and $E \cap B \rightarrow B$ is an acyclic cofibration. Thus, the induced lifting problem in

has a solution by hypothesis. Note that the object $E+B$ is a pushout of the diagram $E \leftarrow E \cap B \rightarrow B$ in DGcoAlg because the forgetful functor $U_d$ creates colimits by Proposition 5.16. Hence, the map $g$ can be extended as $\bar{g} :E + B \rightarrow X$ with $\bar{g}(e+b) = g(e)+l(b)$. Moreover, the resulting Mayer–Vietoris sequence

$$\begin{aligned} \cdots \rightarrow H_n(E \cap B) \rightarrow H_n(E) \oplus H_n(B) \rightarrow H_n(E+B) \rightarrow H_{n-1}(E \cap B) \rightarrow \cdots \end{aligned}$$

shows that $E \rightarrow E+B$ is an acyclic cofibration. Both facts imply that $(E+B, \bar{g}) \in (\Omega , \preceq )$. Since $(E, g)$ is a maximal element, we deduce that $E = E+B$ and therefore $x \in E$ as required. $\square$

Proposition 3.16

[6, Lemma 2.7] A morphism $C \rightarrow D$ of differential graded coalgebras can be factored as $C \mathop {\longrightarrow }\limits ^{i} X \mathop {\longrightarrow }\limits ^{p} D$ where $i$ is an acyclic cofibration and $p$ a fibration. Furthermore $i$ is in the class of morphisms generated by the acyclic cofibrations $A \rightarrow B$ such that $B$ has a countable homogeneous basis.

Proof

By Lemma 3.7, any object in DGcoAlg is small relative to the whole category. Moreover the category DGcoAlg is cocomplete. Hence, in light of Lemma 3.15 the small object argument applies by using Bousfield–Smith cardinality argument. We refer to [3, Section 11] where this latter argument first appeared and to the proof of [10, Theorem 2.3.13] where this technique is applied to an example similar to ours. $\square$

We can now sum up the various ingredients for providing DGcoAlg with a model category structure.

Proof of Theorem 3.1 The axiom MC1 is given by Proposition 5.16. The axioms MC2 and MC3 follow by inspection. Parts (i) and (ii) of the axiom MC5 are proven respectively in Proposition 3.13 and in Proposition 3.16. Part (ii) of the axiom MC4 is the definition of fibrations as stated in Theorem 3.1. It remains to prove the part (i) of the axiom MC4. Let $p :C \rightarrow D$ be an acyclic fibration. By Proposition 3.13, factor $p$ as $C \mathop {\longrightarrow }\limits ^{j} X \mathop {\longrightarrow }\limits ^{q} D$ where $j$ is a cofibration and $q$ is an acyclic fibration with the right lifting property with respect to all cofibrations. Note that $j$ is also a weak equivalence by the axiom MC2. Thus the lifting problem

has a solution since $p$ has the right lifting property with respect to all acyclic cofibrations. Hence, the map $p$ is a retract of $q$ and any lifting problem

with a cofibration $f$ has a solution given by the composite of the curved arrows. $\square$

We end this section with the following useful result that is a consequence of Lemma 3.15 and Bousfield-Smith cardinality argument.

Proposition 3.17

[6, Lemma 2.9] The category of differential graded coalgebras DGcoAlg is cofibrantly generated:

• (1) The set of generating cofibrations is given by cofibrations $A\rightarrow B$ in DGcoAlg with $B$ finite dimensional.

• (2) The set of generating acyclic cofibrations is given by acyclic cofibrations $A \rightarrow B$ in DGcoAlg with $B$ having a countable homogeneous basis.

3.2 A model category structure on ScoAlg

The category of cocommutative simplicial coalgebras has a model structure due to Goerss in [7, Section 3]. But one can adapt the arguments therein to the non-cocommutative case as well. A map $f$ in ScoAlg is a weak equivalence if $\pi _{*}f \cong H_*Nf$ is an isomorphism, a cofibration if it is a levelwise inclusion, and a fibration if it has the right lifting property with respect to trivial cofibrations. Goerss’ main line of argumentation remains unchanged for the case of non-cocommutative simplicial coalgebras. The only slight difference is concerned with Lemma 3.19 recalled below.

Lemma 3.18

The forgetful functor $U_s$ from the category of simplicial coalgebras to the category of simplicial vector spaces has a right adjoint $S_s$.

Proof

The functor $S_s$ is obtained by extending degreewise the cofree coalgebra functor $S$ from the category of vector spaces to the category of non-cocommutative coalgebras as constructed in [23, Theorem 6.4.1]. $\square$

Lemma 3.19

[7, Lemma 3.5] Let $f :C \rightarrow D$ be a morphism of coalgebras. Then, $f$ can be factored as $f= p \circ i$

$$\begin{aligned} C \mathop {\longrightarrow }\limits ^{i} X \mathop {\longrightarrow }\limits ^{p} D \end{aligned}$$

where $i$ is a cofibration and $p$ is an acyclic fibration.

In the proof of this lemma we may replace the cocommutative cofree functor by its non-cocommutative version $S_s :\mathbf SVct \longrightarrow \mathbf ScoAlg$. In this way, Goerss’ arguments transfer to the non-cocommutative setting since only the cofreeness property is required.

4 A comparison of coalgebra categories

4.1 Dold–Kan functors for coalgebras

The Dold–Kan correspondence asserts that SVct and DGVct are equivalent. This equivalence of categories is achieved by the normalization functor $N$ and its inverse $\Gamma$. For a fuller description of these functors, we refer to [24, 8.8.4]. Moreover, it is well-known that the normalized version of the shuffle map $\nabla :NA \otimes NB \rightarrow N(A \widehat{\otimes } B)$ makes the normalization functor $N$ lax monoidal while the Alexander-Whitney map $AW :N(A \widehat{\otimes } B) \rightarrow NA \otimes NB$ makes it oplax monoidal. We refer to [13, Chapter VIII, Section 8, Corollaries 8.6, 8.9] for a detailed description of the Alexander-Whitney and shuffle maps and their normalized versions. From these observations we derive that the normalization functor $N$ and its inverse $\Gamma$ pass to the level of comonoids as stated in the propositions below.

Proposition 4.1

If $(A,\Delta _A,\varepsilon _A)$ is a simplicial coalgebra, then $(NA,\Delta _{NA},\varepsilon _{NA})$ is a differential graded coalgebra with a comultiplication given by the composition

and counit given by $N (\varepsilon _A)$.

Proposition 4.2

If $(B,\Delta _B ,\varepsilon _B)$ is a differential graded coalgebra, then $(\Gamma B,\Delta _{\Gamma B},\varepsilon _{\Gamma B})$ is a simplicial coalgebra with a comultiplication $\Delta _{\Gamma B}$ given by the following composition

and counit given by $\Gamma (\varepsilon _B).$

Notice that both propositions and their proofs are dual to results in [22, Section 2.3]. For detailed proofs, we refer to [20, Section 5.1].

Dually to [22, Section 2.4], we point out that the coalgebra-valued functors $N$ and $\Gamma$ are not adjoint. This failure is made more precise in Remark 1 below.

Lemma 4.3

The adjunction counit $\varepsilon :N \Gamma \longrightarrow \mathrm {Id}$ is a comonoidal transformation. Let $\psi _{X,Y}$ be the composition of natural maps

$$\begin{aligned} \psi _{X,Y} = \eta ^{-1}_{\Gamma X \widehat{\otimes }\Gamma Y} \circ \Gamma (\nabla _{\Gamma X,\Gamma Y}) \circ \Gamma (\varepsilon ^{-1}_{X} \otimes \varepsilon ^{-1}_{Y}). \end{aligned}$$

Then the diagram

commutes for every $X$,$Y$ in DGcoAlg.

Proposition 4.4

The functor $\Gamma :\text{ DGcoAlg } \longrightarrow \text{ ScoAlg }$ is full and faithful and respects coalgebra structures. Moreover, the composite endofunctor $N\Gamma$ is naturally isomorphic to the identity functor on the level of comonoids categories.

Remark 1

The unit $\eta :\mathrm {Id} \longrightarrow \Gamma N$ does not have good comonoidal properties. More precisely, there are objects $X$ and $Y$ in the category ScoAlg so that the diagram

does not commute. Indeed, consider for example $X = Y = \Gamma (\mathbf Z [1])$ as in [22, Remark 2.14]. Since $N$ is left inverse to $\Gamma$ by the previous proposition, one has

$$\begin{aligned} NX = NY = N \Gamma (\mathbf Z [1]) \cong \mathbf Z [1] \mathrm {and} NX \otimes NY \cong \mathbf Z [1] \otimes \mathbf Z [1] = \mathbf Z [2]. \end{aligned}$$

Therefore the lower composite map in the previous diagram vanishes in degree 1 since

$$\begin{aligned}{}[\Gamma (NX \otimes NY)]_1 = [\Gamma (\mathbf Z [2])]_1 = 0. \end{aligned}$$

But in degree 1, the right map $\eta _X \widehat{\otimes }\eta _Y$ is an isomorphism between free abelian groups of rank one since

$$\begin{aligned}{}[\Gamma (NY)]_1 \cong [Y]_1 \cong \mathbf Z \cong [X]_1 \cong [\Gamma (NX)]_1. \end{aligned}$$

4.2 Quillen adjunctions for coalgebras

In this section we consider Quillen’s setting of model categories and we investigate whether the coalgebra-valued functors $N$ and $\Gamma$ fit into this framework. In addition to the model structures on the categories of coalgebras proven in Sect. 3 we recall the model category structures of the categories of vector spaces involved in this work.

• (1) The category DGVct has a model category structure (see [16, Chapter I, Example B]). A map $f$ in the category DGVct is a weak equivalence if $H_{*}f$ is an isomorphism, a cofibration if for each $n \ge 0$, $f_n$ is injective, and a fibration if for each $n \ge 1$, $f_n$ is surjective. In [5, Section 7.18] it is proven that DGVct is cofibrantly generated. The generating acyclic cofibrations are given by $\big \{0 \rightarrow \mathbb {D}^n \mid n \ge 1 \big \}$ and the generating cofibrations by $\big \{\mathbb {S}^{n-1} \rightarrow \mathbb {D}^n \mid n \ge 0 \big \}$ with $\mathbb S^{-1}$ being the zero chain complex.

• (2) The category SVct has a model category structure (see [16, II.4, II.6]). A map $f$ in the category SVct is a weak equivalence if $\pi _{*}f$ is an isomorphism and a fibration if it is a fibration in the underlying category of simplicial sets. Since DGVct is cofibrantly generated, one can deduce that SVct is cofibrantly generated by applying the transfer result by Crans in [4, Section 3] to the adjoint pair $(\Gamma , N)$ provided by the Dold–Kan correspondence. Hence in the category SVct, the generating acyclic cofibrations are given by $\big \{0 \rightarrow \Gamma (\mathbb {D}^n) \mid n \ge 1 \big \}$ and the generating cofibrations by $\big \{ \Gamma (\mathbb {S}^{n-1}) \rightarrow \Gamma (\mathbb {D}^n) \mid n\ge 0 \big \}$.

Our aim now is to compare ScoAlg and DGcoAlg in terms of Quillen adjunctions. Recall that the forgetful functor $U_d$ from the category of differential graded coalgebras to the category of differential graded vector spaces has a right adjoint $S_d$ as proven in Proposition 3.8. The counterpart result for the categories of simplicial coalgebras and vector spaces is given by Lemma 3.18. In this way, the situation to be studied may be illustrated in the diagram

where $\widetilde{N}$ stands for the coalgebra-valued normalization functor.

Proposition 4.5

In the above situation the functor $\widetilde{N}$ has a right adjoint $R$. Moreover the adjoint pair $(\widetilde{N}, R)$ is a Quillen pair.

Proof

First, let $V$ be a differential graded vector space and $S_d(V)$ its differential graded cofree coalgebra. We set

$$\begin{aligned} R(S_d(V)) = S_s \Gamma (V). \end{aligned}$$

Indeed, considering the various adjoint pairs $(U_s,S_s)$, $(N, \Gamma )$, $(U_d, S_d)$ and the identity $NU_s = U_d\widetilde{N}$ successively, yields the following bijection

$$\begin{aligned} {\varvec{ScoAlg}} \Big (X, RS_dV \Big ) \cong {\varvec{DGcoAlg}} \Big (\widetilde{N}X, S_dV \Big ). \end{aligned}$$

This means that the functor $R$ is right adjoint to $\widetilde{N}$ for cofree coalgebras. We now notice that the adjunction $(U_d,S_d)$ defines a monad $S_dU_d$ on the category DGcoAlg. Thus, if $C$ is a differential graded coalgebra, it can be written as the equalizer of the diagram

Since the functor $R$ should be a right adjoint it has to preserve limits. Therefore, defining $R(C)$ as the equalizer of the maps $R(d^0)$ and $R(d^1)$ yields the desired right adjoint.

Finally we observe that the cofibrations and acyclic cofibrations in SVct and DGVct match with those of their respective categories of comonoids ScoAlg and DGcoAlg. Since the functor $N$ is a left Quillen functor the identity $NU_s = U_d\widetilde{N}$ ensures that the functor $\widetilde{N}$ is a left Quillen functor. $\square$

Remark 2

We mention that instead of $\widetilde{N}$, we could consider the coalgebra-valued functor $\widetilde{\Gamma }$. By similar techniques, it is possible to construct a right adjoint functor $R$ to $\widetilde{\Gamma }$ and show that $(\widetilde{\Gamma }, R)$ is a Quillen pair.

5 Categories of connected coalgebras

5.1 Connected differential graded coalgebras

In this section we restrict to connected differential graded objects. An object $V$ in DGVct is connected if $V_0 =0$ and we denote by DGVct $_\mathrm {c}$ the full category of connected differential vector spaces. With mild changes, the category DGVct $_\mathrm {c}$ inherits a model category structure from DGVct. Indeed, DGVct $_\mathrm {c}$ has limits and colimits constructed degreewise as in DGVct. Moreover, the model category factorization axioms may be performed as in [8, Section 1.3], but discarding this time objects such as the $0$-sphere $\mathbb {S}^0$, $\mathbb D^{0}$ and $\mathbb {D}^1$. In this way, a map $f$ in the category DGVct $_\mathrm {c}$ is a weak equivalence if $H_{*}f$ is an isomorphism, a cofibration if for each $n \ge 1$, $f_n$ is injective, and a fibration if for each $n \ge 2$, $f_n$ is surjective.

An object $C$ in DGcoAlg is connected if $C_0 =K$. We denote by DGcoAlg $_\mathrm {c}$ the category of connected differential graded coalgebras and in the rest of this section we discuss its model category structure.

Let $V$ be an object in the category DGVct $_\mathrm {c}$. The tensor coalgebra on $V$ is defined by $T'_d(V) = \bigoplus _{n\ge 0} V^{\otimes n}$. Since $V_0 =0$, it follows that $T'_d(V) \cong \prod _{n\ge 0} V^{\otimes n}$. We mention that the structure maps on $T'_d(V)$ are given by

$$\begin{aligned} \Delta _{T'_d(V)}(v_1 \otimes \cdots \otimes v_n)&= \sum _{r=0}^{n} (v_1 \otimes \cdots \otimes v_r) \otimes (v_{r+1} \otimes \cdots \otimes v_n)\\ \Delta _{T'_d(V)}(1)&= 1\otimes 1 \\ \epsilon _{T'_d(V)}(v_1 \otimes \cdots \otimes v_n)&= 0 \text{ for } n \ge 1 \text{ and } \epsilon _{T'_d(V)}(1)= 1. \end{aligned}$$

Definition 5.1

[9, Section II.2]. Let $C$ be a connected differential graded coalgebra. Then, the functor $I'_d :{\varvec{DGcoAlg}}_\mathrm {c} \rightarrow {\varvec{DGVct}}_\mathrm {c}$ is defined by

$$\begin{aligned} I'_d(C) = C / K[0]. \end{aligned}$$

In other words, the differential graded vector space $I'_d(C)_n$ is $C_n$ for $n \ge 1$, and $0$ for $n=0$.

Proposition 5.2

[9, Section II.2]. The tensor coalgebra functor

$$\begin{aligned} T'_d :{\varvec{DGVct}}_\mathrm {c}\rightarrow {\varvec{DGcoAlg}}_\mathrm {c} \end{aligned}$$

is right adjoint to the functor $I'_d$.

5.2 A model category structure for ${\varvec{DGcoAlg}}_\mathrm {c}$

This section aims at transferring the model category structure of DGcoAlg to DGcoalg $_c$ by means of the transfer principle as stated for instance in [2, Sections 2.5, 2.6]. To this end, Proposition 5.3 below exhibits an adjunction between the categories DGcoAlg and DGcoalg $_c$. Then Proposition 5.4 establishes the completeness and cocompleteness of the category DGcoalg $_c$.

The rest of the section is intended to construct a functorial path-object for fibrant objects in DGcoalg $_c$. But more is done here through a construction due to [19, Definition 4.17, Lemma 4.19]: we explain how to achieve the acyclic cofibration-fibration factorization axiom in a functorial way for any morphism in DGcoalg $_c$. Indeed, given a morphism $f:C\rightarrow D$ in DGcoAlg $_c$, the Lemma 5.5 below provides a functorial way to factorize $f$ as $C\mathop {\longrightarrow }\limits ^{i} X \mathop {\longrightarrow }\limits ^{p} D$ with $i$ a cofibration and $p$ a fibration. Then Smith’s construction in Sect. 5.2 builds out of $X$ an object $G$ that is weak equivalent to $C$ proving this way the required axiom MC5(ii). We close this section with an important Lemma 5.12, useful for proving our main result Theorem 5.20.

Proposition 5.3

There is an adjunction between the categories ${\varvec{DGcoAlg}}$ and ${\varvec{DGcoAlg}}_\mathrm {c}$.

Proof

Let $C \in$ DGcoAlg with counit $\epsilon _C :C \rightarrow K[0]$. Since $(\epsilon _C)_0 :C_0 \rightarrow K$ is a non-zero linear form, it follows that $C_0 \cong \ker (\epsilon _C)_0 \oplus K$. Moreover, the vector space $\ker (\epsilon _C)_0$ is a coideal of the coalgebra $C_0$ by [23, Theorem 1.4.7], hence $C_0 / \ker (\epsilon _C)_0$ has a coalgebra structure and we may define a functor

$$\begin{aligned} F :{\varvec{DGcoAlg}} \rightarrow {\varvec{DGcoAlg}}_\mathrm {c} \text{ by } F(C) = C/\left( \ker (\epsilon _C)_0\right) [0] \end{aligned}$$

where $\left( \ker (\epsilon _C)_0\right) [0]$ denotes the object of DGVct with $\ker (\epsilon _C)_0$ concentrated in degree $0$. Let $(C, D) \in \mathbf DGcoAlg \times \mathbf DGcoAlg _\mathrm {c}$ and $f :C \rightarrow D$ a morphism in DGcoAlg. Since $(\epsilon _D)_0 :K \rightarrow K$ is an isomorphism and $(\epsilon _D)_0 f_0 = (\epsilon _C)_0$, the following universal problem

has a solution. In other words the functor $F$ is left adjoint to $U :\mathbf DGcoAlg _\mathrm {c} \rightarrow \mathbf DGcoAlg$ that forgets connectedness. $\square$

Proposition 5.4

The category of connected differential graded coalgebras is complete and cocomplete.

Proof

We first start with limits. A terminal object in DGcoAlg $_\mathrm {c}$ is given by $K[0]$. For other limits, we recall that there is an anti-equivalence between the category of differential graded coalgebras and the category of profinite differential graded algebras. Since by the appendix in Sect. 6, the category of connected differential graded algebras is complete and cocomplete, we derive limits for DGcoAlg $_\mathrm {c}$ by applying the steps given in Proposition 3.6. We mention that the usual tensor product $\otimes$ of differential graded coalgebras is not the categorical product in DGcoAlg $_\mathrm {c}$ since we do not assume cocommutativity.

Regarding colimits, we refer to [14, Section 1] where similar constructions appear for cocommutative coalgebras. Since as a left adjoint, the functor $I'_d$ must preserve initial objects, we deduce that $K[0]$ is initial in DGcoAlg $_\mathrm {c}$. Let $f,g :C \rightarrow D$ be two maps in DGcoAlg $_\mathrm {c}$. Then, their coequalizer is given by $D / \mathrm im (f-g).$ This quotient is constructed degreewise and each of its homogeneous parts is in fact a coalgebra by [23, Proposition 1.4.8]. Finally, if $C$ and $D$ are two objects in DGcoAlg $_\mathrm {c}$, we may form the maps

$$\begin{aligned} K[0] \mathop {\longrightarrow }\limits ^{\varphi _C} C \mathop {\longrightarrow }\limits ^{i_C} C \oplus D \mathrm and K[0] \mathop {\longrightarrow }\limits ^{\varphi _D} D \mathop {\longrightarrow }\limits ^{i_D} C \oplus D. \end{aligned}$$

The coproduct of $C$ and $D$ in DGcoAlg $_\mathrm {c}$ is thus given by

$$\begin{aligned} C \sqcup D = \left( C \oplus D \right) / \mathrm im \left( i_C \circ \varphi _C - i_D \circ \varphi _D \right) . \end{aligned}$$

Notice that the direct sum is the coproduct of the underlying differential graded vector spaces and that the quotient guarantees the required connectedness condition. $\square$

In order to find a functorial path-object for fibrant objects in DGcoAlg $_\mathrm {c}$, we explain in this paragraph how to achieve a functorial acyclic cofibration-fibration for a given morphism $f :C \rightarrow D$ in DGcoAlg $_\mathrm {c}$. This is done by using a technique described in [19, Definition 4.17, Lemma 4.19]. Proceeding this way offers a dual advantage, namely Proposition 5.11 and Lemma 5.12.

Lemma 5.5

Let $V$ be an acyclic ($H_*V \cong 0$) connected differential graded vector space and $C$ a connected differential graded coalgebra. Then the projection

$$\begin{aligned} C \sqcap T'_d(V)\longrightarrow C \end{aligned}$$

is an acyclic fibration.

Proof

First note that if $C$ is an object of DGcoAlg $_c$ then its linear dual $C^*=\left\{ \mathbf Vct (C_n, K)\right\}$ becomes a connected differential non-positively graded algebra. We also denote by $T_{C^*}$ the tensor $C^*$-algebra functor defined by

$$\begin{aligned} T_{C^*}(M) = C^*\oplus \bigoplus _{n\ge 1} \underbrace{M\otimes _{C^*} M \otimes _{C^*} \cdots \otimes _{C^*} M}_{n \; \mathrm {times}} \end{aligned}$$

for any differential graded $C^*$-bimodule $M$. With this in hand we obtain

$$\begin{aligned} \left[ C \sqcap T'_d (V)\right] ^*\cong C^*\sqcup \left[ T'_d(V)\right] ^*\cong C^*\sqcup T_d (V^*) \cong T_{C^*}\left( C^*\otimes V^*\otimes C^*\right) . \end{aligned}$$

The cohomological Künneth formula yields

$$\begin{aligned} H^*\left( T_{C^*}\left( C^*\otimes V^*\otimes C^*\right) \right) \cong H^*\left( C^*\right) . \end{aligned}$$

The required homology isomorphism $H_*\left( C \sqcap T'_d(V)\right) \cong H_*C$ results from the application of the cohomological universal coefficient theorem.

We still have to prove that each projection is a fibration. To do so, we consider the diagram

where $i$ is a cofibration in DGcoAlg $_\mathrm {c}$. By adjointness, this amounts to considering the diagram

in DGVct $_\mathrm {c}$. Since $I'_d(i)$ is a cofibration and $V \rightarrow 0$ is an acyclic fibration in the model category of DGVct $_\mathrm {c}$, a lift $I'_d(B) \rightarrow V$ exists. $\square$

We recall from [24, Section 1.5] some useful constructions in DGVct.

Definition 5.6

Let $(V,d)$ be an object in the category DGVct.

• 1. We denote by $\mathrm {cone}(V)$ the object of DGVct with $\left( \mathrm {cone}(V)\right) _n = V_{n-1} \oplus V_n$ and differential given by $d(b, c) = (-d(b), d(c)-b), b \in V_{n-1}, c \in V_n.$

• 2. We denote by $s^{-1}V$ the object of DGVct with $\left( s^{-1} V\right) _n = V_{n+1}$ and differential given by $-dc, c \in V_{n+1}.$

• 3. We denote by $\tau _{\ge 1} V$ the object of DGVct with

  $$\begin{aligned} \left( \tau _{\ge 1} V\right) _n = \left\{ \begin{array}{l@{\quad }l@{\quad }l} 0, &{} \text{ if } n < 1, \\ \ker {(V_1 \rightarrow V_0)}, &{} \text{ if } n =1, \\ V_n, &{} \text{ if } n > 1. \end{array} \right. \end{aligned}$$

Remark 3

If $V \in$ DGVct $_\mathrm {c}$, then $\mathrm {cone}(V) \in$ DGVct $_\mathrm {c}$. This is not the case for the desuspension $s^{-1}V$ since $\left( s^{-1}V\right) _0 =V_1$. We may avoid this problem by considering the object $\tau _{\ge 1} \left( s^{-1} V\right)$ which lies in DGVct $_\mathrm {c}$.

Lemma 5.7

Let $V$ be an object of the category DGVct $_\mathrm {c}$. Then, the canonical map $\tau _{\ge 1} \left( s^{-1} \mathrm {cone}(V)\right) \longrightarrow \tau _{\ge 1} \left( V\right) =V$ is a fibration in the model structure of DGVct $_\mathrm {c}$.

Proof

Recall from Sect. 5.1 that fibrations in DGVct $_\mathrm {c}$ are maps $f$ with $f_n$ surjective for $n \ge 2.$ We may picture our case as follows

Our given map is a fibration since, for $n\ge 2$, we have canonical projections. $\square$

Lemma 5.8

If $C$ is an object of the category DGcoAlg $_\mathrm {c}$, then,

$$\begin{aligned} H_*(C) \cong H_*(I'_d C) \oplus H_*(K[0]). \end{aligned}$$

Proof

Since the object $K[0]$ is both initial and terminal in the category DGcoAlg $_\mathrm {c}$, we deduce that $K[0]$ splits off the object $C$. Hence $C \cong I'_d(C) \oplus K[0]$ and the result follows. $\square$

Furthermore, if $C$ and $D$ are object in DGcoAlg $_\mathrm {c}$ such that $H_*(I'_d C) \cong H_*(I'_d D)$, then, $H_*(C) \cong H_*(D)$.

Smith’s construction Let us factor $f$ as $C \mathop {\longrightarrow }\limits ^{i_0} D \sqcap T'_d(I'_d C) \mathop {\longrightarrow }\limits ^{p_0} D$ where the map $i_0$ is a canonical injection hence a cofibration in DGcoAlg $_\mathrm {c}$ and $p_0$ is a fibration.

Then, construct the maps $\left( i_n\right) _{n\ge 1}$ and $\left( p_n\right) _{n\ge 1}$ as displayed in the diagram

in which

• (1) for $n\ge 0$, the object $G(n+1)$ is a pullback of

  $$\begin{aligned} G(n) \longrightarrow T'_d \Big [\tau _{\ge 1} \left( I'_d H(n)\right) \Big ] \longleftarrow T'_d \Big [\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \Big ] \end{aligned}$$

  where $H(n)$ is a pushout of $G(n) \longleftarrow C \mathop {\longrightarrow }\limits ^{\epsilon _C} K[0]$

• (2) the maps $\left( p_n\right) _{n\ge 1}$ form a tower of fibrations. In fact, as pullbacks, they are built out of DGVct $_\mathrm {c}$ fibrations $\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \longrightarrow \tau _{\ge 1}\left( I'_d H(n)\right)$ as shown in Lemma 5.7.

• (3) the maps $\left( i_n\right) _{n\ge 1}$ are cofibrations since by induction the compositions $p_n \circ i_n$ are injections.

The following result is analogue to [19, Lemma 4.19].

Lemma 5.9

The resulting object $\lim _n G(n) \in$ DGcoAlg $_\mathrm {c}$ described above is weakly equivalent to $C$.

Proof

Following [19, Proof of Lemma 4.19] the idea is to factorize the identity map on $\left( \lim _n G(n) \right) / i_\infty \left( C\right)$ through the acyclic object $T'_d \left[ \tau _{\ge 1} s^{-1} \mathrm {cone} \lim _n I'_d H(n)\right]$. To this end, we observe the following category-theoretic facts. We will denote by $A\times _C B$ a pullback of the diagram $A\rightarrow C \leftarrow B$.

• (1) As a pushout of the diagram $\lim _n G(n) \mathop {\longleftarrow }\limits ^{i_\infty } C \longrightarrow K[0]$, we have that

  $$\begin{aligned} \lim _n H(n) \cong \left( \lim _n G(n) \right) / i_\infty \left( C\right) = \lim _n \left( G(n) / i_n \left( C\right) \right) . \end{aligned}$$

• (2) By construction of $G(n)$ in the preceding paragraph, we have

  $$\begin{aligned} \lim _n G(n) = \lim _n G(n) \times _{T'_d \Big [\tau _{\ge 1} \left( I'_d H(n)\right) \Big ]} T'_d \Big [\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \Big ]. \end{aligned}$$

• (3) Let us consider the canonical projection

  $$\begin{aligned} \lim _n G(n) \times _{T'_d \Big [\tau _{\ge 1} \left( I'_d H(n)\right) \Big ]} T'_d \Big [\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \Big ] \longrightarrow \lim _n G(n). \end{aligned}$$

  Since $\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \longrightarrow \tau _{\ge 1}\left( I'_d H(n)\right)$ is a fibration in DGVct $_\mathrm {c}$, there is a map such that the composite

  $$\begin{aligned} \tau _{\ge 1}\left( I'_d H(n)\right) \longrightarrow \tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \longrightarrow \tau _{\ge 1}\left( I'_d H(n)\right) \end{aligned}$$

  is the identity. Applying the cofree functor $T'_d$ to this map yields a map that splits the canonical projection given above.

Consequently, we obtain a map

$$\begin{aligned} \lim _n G(n) \longrightarrow \lim _n G(n) \times _{T'_d \Big [\tau _{\ge 1} \left( I'_d H(n)\right) \Big ]} T'_d \Big [\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \Big ] \end{aligned}$$

that splits the projection map in (3). Furthermore, we may produce the diagram

that factorizes $\left( \lim _n G(n) \right) / i_\infty \left( C\right) \mathop {\longrightarrow }\limits ^{\mathrm {Id}} \left( \lim _n G(n) \right) / i_\infty \left( C\right)$. Since

$$\begin{aligned}&\lim _n H(n) \times _{ T'_d \Big [\tau _{\ge 1} \left( I'_d H(n)\right) \Big ]} T'_d \Big [\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \Big ] \\&\quad \subseteq \lim _n T'_d\left( \lim _n \Big [\tau _{\ge 1} \left( I'_d H(n)\right) \Big ]\right) \times _{ T'_d \Big [\tau _{\ge 1} \left( I'_d H(n)\right) \Big ]} T'_d \Big [\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \Big ] \\&\quad = T'_d \Big [\tau _{\ge 1}\left( s^{-1}\mathrm {cone}\left( I'_d H(n)\right) \right) \Big ] \end{aligned}$$

we deduce that the identity map $\left( \lim _n G(n) \right) / i_\infty \left( C\right) \longrightarrow \left( \lim _n G(n) \right) / i_\infty \left( C\right)$ may be factorised through $T'_d \left[ \tau _{\ge 1} s^{-1} \mathrm {cone} \lim _n I'_d H(n)\right]$. This latter object is acyclic because $\tau _{\ge 1} s^{-1} \mathrm {cone} \lim _n I'_d H(n)$ is canonically acyclic and homology commutes with the functor $T'_d$ by Künneth theorem. It follows that $H_*\Big [I'_d \left( \left( \lim _n G(n) \right) / i_\infty \!\left( C\right) \right) \Big ]\!=\!0.$ Finally, the long exact sequence of homology resulting from the short exact sequence

$$\begin{aligned} 0 \longrightarrow I'_d C \longrightarrow I'_d\left( \lim _n G(n)\right) \longrightarrow I'_d\left( \left( \lim _n G(n) \right) / i_\infty \left( C\right) \right) \longrightarrow 0 \end{aligned}$$

gives the required weak equivalence by using Lemma 5.8. $\square$

Proposition 5.10

Any morphism $f :C \rightarrow D$ in DGcoAlg $_\mathrm {c}$ can be factored as

$$\begin{aligned} C \mathop {\longrightarrow }\limits ^{i} G \mathop {\longrightarrow }\limits ^{p} D \end{aligned}$$

with $i$ an acyclic cofibration and $p$ a fibration.

Proof

Setting $i= i_\infty$ and taking $p$ to be induced by the $p_i$’s give the required factorization. $\square$

Proposition 5.11

The category of connected differential graded coalgebras has a model category structure. A morphism $f \in \mathbf DGcoAlg _\mathrm {c}$ is a weak equivalence (resp. a fibration) if $U(f) =f$ is a weak equivalence (resp. a fibration) in DGcoAlg.

Proof

By Proposition 5.4, the category DGcoAlg $_\mathrm {c}$ is (co)complete. Moreover, Proposition 5.10 endows DGcoAlg $_\mathrm {c}$ with a fibrant replacement functor and a functorial path-object $X \mathop {\longrightarrow }\limits ^{i} \mathrm {Path}(X) \mathop {\longrightarrow }\limits ^{p} X \sqcap X$ for any fibrant object $X \in$ DGcoAlg $_\mathrm {c}$. Finally, since the category DGcoAlg is cofibrantly generated by Proposition 3.17, the transfer principle applies and DGcoAlg $_\mathrm {c}$ inherits a model category structure. $\square$

Lemma 5.12

Let $C$ be a fibrant connected differential graded coalgebra and

$$\begin{aligned} C \mathop {\longrightarrow }\limits ^{i} G \mathop {\longrightarrow }\limits ^{p} K[0] \end{aligned}$$

be the factorization of $\epsilon _C :C \rightarrow K[0]$ as in Proposition 5.10. Then, $C$ is a retract of $G$. Moreover, $G$ is a cofree connected differential graded coalgebra.

Proof

Since $C$ is fibrant, the counit $\epsilon _C$ is a fibration. It follows that a lift exists in the diagram

and therefore that $C$ is a retract of $G$.

For the second statement, note first that $G(0)$ is cofree in DGcoAlg $_\mathrm {c}$ since

$$\begin{aligned} G(0) = T'_d\left( I'_d C\right) \sqcap K[0] = T'_d\left( I'_d C \right) \sqcap T'_d\left( 0\right) \cong T'_d\left( I'_d C \times 0\right) \cong T'_d\left( I'_d C\right) . \end{aligned}$$

Then, the first step of Proposition 5.10 computes the object $H(0)$ as a pushout

The second step gives $G(1)$ as a pullback in the diagram

Then, the map $C \rightarrow H(0)$ yields the map $I'_d C \rightarrow I'_d H(0) = \tau _{\ge 1} (I'_d H(0))$. By the universal property of the functor $T'_d$, we deduce that the map

$$\begin{aligned} G(0)\cong T'_d\Big [I'_d C\Big ] \longrightarrow T'_d \Big [\tau _{\ge 1} \left( I'_d H(0)\right) \Big ] \end{aligned}$$

is of the form $T'_d \Big [I'_d C \longrightarrow \tau _{\ge 1} \left( I'_d H(0)\right) \Big ].$ Hence, the object $G(1)$ is cofree as a pullback of cofree objects and maps induced by maps in the category DGcoAlg $_\mathrm {c}$. By the same arguments, the objects $G(n)$ are cofree and $G = \lim _n G(n)$ is cofree as required. $\square$

5.3 Connected simplicial coalgebras

This section is the simplicial counterpart of the previous one. An object $V$ in SVct is connected if $V_0 = 0$. A connected simplicial coalgebra $C$ is an object in ScoAlg with $C_0=K$. We denote by SVct $_\mathrm {c}$ and ScoAlg $_\mathrm {c}$ the categories of connected simplicial vector spaces and coalgebras.

Lemma 5.13

Let $C$ be a connected simplicial coalgebra. Then, the constant simplicial object $I(K)$ splits off the object $C$.

Proof

In ScoAlg $_\mathrm {c}$, $I(K)$ is both initial and terminal. Consequently, the canonical map $i :I(K) \rightarrow C$ is an injection and $I(K)$ splits off $C$. $\square$

The following definition is a consequence of the previous Lemma 5.13.

Definition 5.14

Let $C$ be a connected simplicial coalgebra. Then, there is a functor $I'_s :\mathbf ScoAlg _\mathrm {c} \rightarrow \mathbf SVct _\mathrm {c}$ defined by $I'_s(C)= C / I(K).$

Proposition 5.15

The functor $I'_s :\mathbf ScoAlg _\mathrm {c} \rightarrow \mathbf SVct _\mathrm {c}$ has a right adjoint defined by

$$\begin{aligned} T'_s(W) = \bigoplus _{n \ge 0} W^{\widehat{\otimes }n} = I(K) \oplus W \oplus \cdots \oplus W^{\widehat{\otimes }n} \oplus \cdots \end{aligned}$$

for any object $W$ in the category SVct $_{\mathrm c}$.

Proof

The required adjunction is obtained by considering the following bijections with $C \in$ ScoAlg $_\mathrm {c}$ and $W \in$ SVct $_\mathrm {c}$

$$\begin{aligned} \mathbf SVct _\mathrm {c}(I'_s C, V)&\cong \mathbf DGVct _\mathrm {c}(NI'_sC, NW) \\&\cong {\varvec{DGVct}}_\mathrm {c}(I'_dNC, NW)\\&\cong {\varvec{DGcoAlg}}_\mathrm {c}(NC, T'_d(NW))\\&\cong {\varvec{ScoAlg}}_\mathrm {c}(C, RT'_d(NW)) \\&\cong {\varvec{ScoAlg}}_\mathrm {c}(C, T'_s\Gamma (NW)) \\&\cong {\varvec{ScoAlg}}_\mathrm {c}(C, T'_s(W)). \end{aligned}$$

Note that the third bijection comes from considering the diagram

where $\nabla :T'_d(NW) \rightarrow NT'_s(W)$ is the normalized shuffle map and $\pi :T'_s(W) \rightarrow W$ the canonical projection. $\square$

Proposition 5.16

The category of connected simplicial coalgebras is complete and cocomplete.

Proof

For limits in ScoAlg $_\mathrm {c}$ it suffices to extend degreewise the construction in [1, Theorem 1.1] for the category of coalgebras over fields. Hence, a terminal object in ScoAlg $_\mathrm {c}$ is given by $I(K)$, the constant simplicial coalgebra. Notice that, since the field $K$ is a terminal object in coAlg, the product $K \sqcap K$ is isomorphic to $K$. This ensures that the products $C \sqcap D$ of objects in ScoAlg $_\mathrm {c}$ is again connected since $\left( C \sqcap D\right) _0 = C_0 \sqcap D_0 = K \sqcap K = K$.

Colimits in ScoAlg $_\mathrm {c}$ are formed in the same way as for DGcoAlg $_\mathrm {c}$. In this way, an initial object is given by $I(K)$. If $f,g :C \rightarrow D$ are two maps in ScoAlg $_\mathrm {c}$, their coequalizer is given by $D / \mathrm im (f-g)$. Finally, if $C$ and $D$ are two objects in ScoAlg $_\mathrm {c}$, we may form the maps

$$\begin{aligned} I(K) \mathop {\longrightarrow }\limits ^{\varphi _C} C \mathop {\longrightarrow }\limits ^{i_C} C \oplus D \quad \quad \mathrm and \quad \quad I(K) \mathop {\longrightarrow }\limits ^{\varphi _D} D \mathop {\longrightarrow }\limits ^{i_D} C \oplus D. \end{aligned}$$

The coproduct of $C$ and $D$ in ScoAlg $_\mathrm {c}$ is then given by

$$\begin{aligned} C \sqcup D = C \oplus D / \mathrm im \left( i_C \circ \varphi _C - i_D \circ \varphi _D \right) . \end{aligned}$$

Notice that the direct sum is taken degreewise and that the quotient guarantees the connectedness condition. $\square$

With the above-mentioned facts, the category ScoAlg $_\mathrm {c}$ is endowed with a model category structure exactly as in [7, Section 3].

5.4 A Quillen equivalence for connected coalgebras

In this section, we improve the Quillen adjunction $(\widetilde{N}, R)$ to a Quillen equivalence. We were not able to check Hovey’s criterion (see [10, Corollary 1.3.16]) for arbitrary fibrant differential graded coalgebras. However, connectedness is a condition that guarantees such a criterion which yields a Quillen equivalence.

Lemma 5.17

There is an equivalence between the category of connected simplicial vector spaces and the category of connected differential graded vector spaces.

Proof

We notice that the restriction of the normalization functor $N:$ SVct $_{\mathrm {c}} \rightarrow$ DGVct $_{\mathrm {c}}$ is full and faithful since it is induced by the Dold–Kan equivalence. Moreover, if $V \in$ DGVct $_{\mathrm {c}}$, we may find $W \in$ SVct $_{\mathrm {c}}$ so that $NW \cong V$. Since $\Gamma (V)_0 = V_0 = 0$, it follows that $\Gamma (V) \in$ SVct $_{\mathrm {c}}$. Setting $W = \Gamma (V)$ meets the required condition. Therefore, by [15, Section 2.1, Proposition 3], we deduce that the restriction of the normalization functor $N$ induces an equivalence of categories between connected vector spaces with an inverse given by the restriction of $\Gamma$. $\square$

The following result is dual to [17, Part I, Proposition 4.5].

Lemma 5.18

Let $V$ be a differential graded vector space. Then the following maps

$$\begin{aligned} H_*(\widetilde{N} T'_s \Gamma (V)) \longrightarrow H_*(T'_d (V)) \longrightarrow T'_d H_*(V) \end{aligned}$$

of graded coalgebras are isomorphisms.

Proof

These maps are obtained by using the universal properties of the respective tensor coalgebras. Hence we deduce the isomorphisms by applying Künneth and Eilenberg–Zilber theorems. $\square$

Lemma 5.19

If $C$ is a cofree differential graded coalgebra, then the map

$$\begin{aligned} \widetilde{N} R C \longrightarrow C \end{aligned}$$

is a weak equivalence.

Proof

Since $C \cong T'_d(V)$ one has

$$\begin{aligned} H_*(\widetilde{N} R C) \cong H_*(\widetilde{N}T'_s \Gamma (V)) \cong H_*(T'_d (V)) \cong H_*(C) \end{aligned}$$

and hence the required result. $\square$

Theorem 5.20

If $C$ is a fibrant connected differential graded coalgebra, then the map

$$\begin{aligned} \widetilde{N} R C \longrightarrow C \end{aligned}$$

is a weak equivalence. Hence, there is a Quillen equivalence between the category of connected differential graded coalgebras and the category of connected simplicial coalgebras.

Proof

Recall in Lemma 5.12 that $C$ is a retract of a cofree coalgebra $G$, which may be written as $T'_d(V)$ for some $V \in$ DGVct $_\mathrm {c}$. Applying the functors $R^{\mathrm {com}}$, $\widetilde{N}$ and $H_*$ to the retract map $C \rightarrow T'_d(V) \rightarrow C$ we find the diagram

of homology morphisms. Since by Lemma 5.18, $H_*(\widetilde{N} T'_s \Gamma (V)) \cong H_*(T'_d (V))$, we deduce that $H_*(\widetilde{N} R C) \cong H_*C$ with help of [5, Lemma 2.7]. Then, the Quillen equivalence follows from [10, Corollary 1.3.16]. $\square$

6 Appendix on connected differential graded algebras

In this appendix, we consider the category of connected differential graded algebras. Its main interest here is that a particular dual of its limits is used to construct colimits for the category DGcoAlg $_\mathrm {c}$.

Definition 6.1

A connected differential graded algebra $A$ is a differential graded algebra with $A_0 =K$. We denote by DGAlg $_\mathrm {c}$ the category of connected differential graded algebras.

Definition 6.2

Let $A$ be an object in the category DGAlg $_\mathrm {c}$. The isomorphism $\mu _{\mid _{A_0}} :K \rightarrow A_0$ induces a map $\gamma _A :A \rightarrow K[0]$ and we define a functor $I_d :\mathbf DGAlg _\mathrm {c} \rightarrow \mathbf DGVct _\mathrm {c}$ by $I_d (A) = \ker \gamma _A.$

Lemma 6.3

The tensor algebra functor $T_d :\mathbf DGVct _\mathrm {c} \rightarrow \mathbf DGAlg _\mathrm {c}$ is left adjoint to the functor $I_d$.

Proposition 6.4

The category of connected differential graded algebras is complete and cocomplete.

Proof

Constructions of limits are well-known in the category of differential graded algebras. Because of connectedness, some refinements have to be performed. A terminal object in DGAlg $_\mathrm {c}$ is given by $K[0]$. If $f,g :A \rightarrow B$ are two maps in DGAlg $_\mathrm {c}$, their equalizer is given by $\ker \left( f-g\right)$. Now let $A$ and $B$ be objects in DGAlg $_\mathrm {c}$. We may form the maps

$$\begin{aligned} A \times B \mathop {\longrightarrow }\limits ^{\pi _A} A \mathop {\longrightarrow }\limits ^{\gamma _A} K[0] \quad \mathrm and \quad A \times B \mathop {\longrightarrow }\limits ^{\pi _B} B \mathop {\longrightarrow }\limits ^{\gamma _B} K[0]. \end{aligned}$$

Then the product of $A$ and $B$ in DGAlg $_\mathrm {c}$ is given by

$$\begin{aligned} A \sqcap B = \ker \left( \gamma _A \circ \pi _A - \gamma _B \circ \pi _B\right) \!. \end{aligned}$$

For colimits, we first notice that an initial object in DGAlg $_\mathrm {c}$ is given by $K[0]$. Then the coequalizer of the two maps $f, g :A \rightarrow B$ is given by

$$\begin{aligned} B / \left\langle f(a)-g(a), a \in A\right\rangle \end{aligned}$$

where $\left\langle f(a)-g(a), a \in A\right\rangle$ denotes the ideal generated by $f(a)-g(a)$ for $a \in A$. In [11], the coproduct of two differential graded non-commutative algebras $A$ and $B$ is given by factoring out from the tensor algebra $T_d\left( A \otimes B\right)$ the ideal $\mathcal {I}$ which is generated by elements of the form

$$\begin{aligned} (a_1 \otimes b_1)\otimes (1\otimes b_2)- a_1\otimes b_1b_2, \\ (a_1 \otimes 1)\otimes (a_2\otimes b_2)- a_1a_2\otimes b_2. \end{aligned}$$

However, the resulting object need not to be connected even if $A$ and $B$ are connected. To avoid this problem, we define the coproduct of two objects in DGAlg $_\mathrm {c}$ as

$$\begin{aligned} A \sqcup B = T_d\left( \ker \gamma _A \otimes \ker \gamma _B \right) / \mathcal {I}. \end{aligned}$$

Since $\ker \gamma _A$ and $\ker \gamma _A$ are connected differential graded vector spaces, they will not contribute to the degree zero part of the tensor algebra $T_d\left( \ker \gamma _A \otimes \ker \gamma _B \right)$. In this way, we will have $\left( A \sqcup B\right) _0 =K$ and therefore $A \sqcup B \in$ DGAlg $_\mathrm {c}$. $\square$