  We prove the equivalence of several hypotheses that have appeared recently in
the literature for studying left Bousfield localization and algebras over a
monad. We find 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. We include examples coming from the theory of
colored operads, and applications to spaces, spectra, and chain complexes.
