We give sufficient conditions for homotopical localization functors to
preserve algebras over coloured operads in monoidal model categories. Our
approach encompasses a number of previous results about preservation of
structures under localizations, such as loop spaces or infinite loop spaces,
and provides new results of the same kind. For instance, under suitable
assumptions, homotopical localizations preserve ring spectra (in the strict
sense, not only up to homotopy), modules over ring spectra, and algebras over
commutative ring spectra, as well as ring maps, module maps, and algebra maps.
It is principally the treatment of module spectra and their maps that led us to
the use of coloured operads (also called enriched multicategories) in this
context.