1 Erratum to: J. Homotopy Relat. Struct. (2016) 11:67–96 DOI 10.1007/s40062-014-0096-1

In Theorem 5.20 of original article, we have claimed to obtain a Quillen equivalence between the category of connected differential graded coalgebras and the category of connected simplicial coalgebras.

But a mistake has been recently discovered, prejudicing the proof of the announced result. Indeed, in the proof of Proposition 5.15 of original article, the identification \(RT'_d = T'_s \Gamma \) that appears in the bijection

$$\begin{aligned} \mathbf{ScoAlg }_{\mathrm {c}}(C, RT'_d(NW)) \cong \mathbf{ScoAlg }_{\mathrm {c}}(C, T'_s\Gamma (NW)) \end{aligned}$$

is wrong. This identification should be considered though in the context of categories of monoids. It leads to a wrong description of the functor

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

as right adjoint to the functor \(I'_s :\mathbf{ScoAlg }_{\mathrm {c}} \rightarrow \mathbf{SVct }_{\mathrm {c}}\).

We should instead consider Sweedler’s cofree coalgebra functor as constructed in [1, Section 6.4] and extend it degreewise for a correct description of the right adjoint functor \(T'_s\).

Consequently, we are not able to establish the isomorphism \(H_*(N T'_s \Gamma (V)) \cong H_*(T'_d (V))\) in Lemma 5.18 of original article and this is a crucial point for proving Theorem 5.20 of original article.