Comonadic Coalgebras and Bousfield Localization

TL;DR

This work develops a comprehensive framework for lifting and preserving coalgebraic structures under left Bousfield localization in model categories. It proves a key clockwork result: $\mathsf{Coalg}(K;L_{\mathcal{C}}\mathcal{M})$ has a left-induced model structure via the forgetful functor to $L_{\mathcal{C}}\mathcal{M}$ if and only if a suitably defined $L_{\mathcal{C}'}\mathsf{Coalg}(K;\mathcal{M})$ exists, with these two models coinciding. It then provides general admissibility and preservation criteria for when localization preserves comonadic coalgebras, along with multiple equivalent formulations. The authors apply the theorems across a wide spectrum of settings—chain complexes, (localized) spectra, the stable module category, and simplicial contexts—showing that left Bousfield localization often lifts to coalgebras, comodules, and cooperadic coalgebras, including under smashing localizations. Collectively, the results offer a unifying toolkit for understanding how localization interacts with coalgebraic structures and enable broad applications to operads, Koszul duality, and homotopical algebra in diverse categories.

Abstract

For a model category, we prove that taking the category of coalgebras over a comonad commutes with left Bousfield localization in a suitable sense. Then we prove a general existence result for the left-induced model structure on the category of coalgebras over a comonad in a left Bousfield localization. Next we provide several equivalent characterizations of when a left Bousfield localization preserves coalgebras over a comonad. These results are illustrated with many applications in chain complexes, (localized) spectra, the stable module category, and simplicial settings.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Definition 2.3: bayehgkrhkrs
• Remark 2.4
• Definition 2.5
• Lemma 2.7
• proof