Coalgebras in the Dwyer-Kan localization of a model category

Péroux, Maximilien

doi:10.1090/proc/15949

Coalgebras in the Dwyer-Kan localization of a model category
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Proc. Amer. Math. Soc. 150 (2022), 4173-4190 Request permission

Abstract:

We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover it shows that Shipley’s zig-zag of Quillen equivalences provides an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal model category and we show examples when these coalgebras cannot be rigidified. That is, their $\infty$-categories are not equivalent to the Dwyer-Kan localizations of strict coalgebras in the usual monoidal model categories of spectra and of connective discrete $R$-modules.

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