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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
is wrong. This identification should be considered though in the context of categories of monoids. It leads to a wrong description of the functor
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.
Reference
Sweedler, M.: Hopf Algebras. W.A. Benjamin, New York (1969)
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Communicated by Guillermo Cortinas.
The online version of the original article can be found under doi:10.1007/s40062-014-0096-1.
The online version of the original article can be found at http://dx.doi.org/10.1007/s40062-014-0096-1.
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Sore, W.H.B. Erratum to: The Dold–Kan correspondence and coalgebra structures. J. Homotopy Relat. Struct. 12, 511–512 (2017). https://doi.org/10.1007/s40062-016-0133-3
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DOI: https://doi.org/10.1007/s40062-016-0133-3