Coalgebras in the Dwyer-Kan localization of a model category

TL;DR

This paper explores how weak monoidal Quillen equivalences lead to equivalences of symmetric monoidal ∞-categories via Dwyer-Kan localization, establishing a Dold-Kan correspondence for coalgebras and analyzing their rigidity properties.

Contribution

It demonstrates that weak monoidal Quillen equivalences induce symmetric monoidal ∞-category equivalences and provides explicit lifts of the stable Dold-Kan correspondence, along with examples of non-rigidifiable homotopy coalgebras.

Findings

Weak monoidal Quillen equivalences induce ∞-category equivalences.

The stable Dold-Kan correspondence lifts to symmetric monoidal ∞-categories.

Certain homotopy coalgebras cannot be rigidified, showing limitations in strictification.

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 lifts to an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal monoidal category. 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.