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We show that weak monoidal Quillen equivalences induce equivalences ofsymmetric monoidal $\infty$-categories with respect to the Dwyer-Kanlocalization of the symmetric monoidal model categories. The result will inducea Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover, itshows that Shipley's zig-zag of Quillen equivalences lifts to an explicitsymmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kancorrespondence. We study homotopy coherent coalgebras associated to a monoidalmonoidal category. We show examples when these coalgebras cannot be rigidified.That is, their $\infty$-categories are not equivalent to the Dwyer-Kanlocalizations of strict coalgebras in the usual monoidal model categories ofspectra and of connective discrete $R$-modules.

Coalgebras in the Dwyer-Kan localization of a model category