Left Bousfield localization and Eilenberg-Moore categories (Q2240616)

Left Bousfield localization [\textit{A. K. Bousfield}, Topology 14, 133--150 (1975; Zbl 0309.55013)] has become a fundamental tool in modern abstract homotopy theory. The ability to take a well-behaved model category and a prescribed set of maps, and then produce a new model structure where those maps are weak equivalences, has applications in a variety of settings. \par Several groups of researchers have been applying the machinery of left Bousfield localization to better understand algebras over (colored) operads, especially results regarding when algebraic structure is preserved by localization, e.g., \textit{M. A. Batanin} [Proc. Am. Math. Soc. 145, No. 7, 2785--2798 (2017; Zbl 1375.18048)], \textit{M. A. Hill} and \textit{M. J. Hopkins} [Contemp. Math. 620, 183--199 (2014; Zbl 1342.55007)], \textit{C. Casacuberta} et al. [Proc. Lond. Math. Soc. (3) 101, No. 1, 105--136 (2010; Zbl 1196.18005)], \textit{D. White} and \textit{D. Yau} [Appl. Categ. Struct. 26, No. 1, 153--203 (2018; Zbl 1397.18023)] and others. \par The authors prove that the equivalence of several hypotheses that have appeared recently in the literature for studying left Bousfield localization and algebras over a monad. Conditions so that there is a model structure for local algebras, so that localization preserves algebras, and so that localization lifts to the level of algebras are found. Examples coming from the theory of colored operads, and applications to spaces, spectra, and chain complexes are included.