We give conditions on a monoidal model category M and on a set of maps C so that the Bousfield localisation of M with respect to C preserves the structure of algebras over various operads. This problem was motivated by an example that demonstrates that, for the model category of equivariant spectra, preservation does not come for free. We discuss this example in detail and provide a general theorem regarding when localisation preserves P-algebra structure for an arbitrary operad P. We characterise the localisations that respect monoidal structure and prove that all such localisations preserve algebras over cofibrant operads. As a special case we recover numerous classical theorems about preservation of algebraic structure under localisation, in the context of spaces, spectra, chain complexes, and equivariant spectra. To demonstrate our preservation result for non-cofibrant operads, we work out when localisation preserves commutative monoids and the commutative monoid axiom. Finally, we provide conditions so that localisation preserves the monoid axiom.

Bousfield localisation, or more specifically, left Bousfield localisation, is an established tool to formally add more weak equivalences to a model category. The most common setting is localisation of spaces or spectra with respect to a homology theory: rather than the weak equivalences being isomorphisms of homotopy groups, one constructs a model structure with the homology isomorphisms as the weak equivalences. As a consequence, the homology isomorphisms become strict isomorphisms in the corresponding homotopy category. Therefore, we can think of Bousfield localisation as a good formal framework for inverting maps in the homotopy category. Typically, information is lost in this process, but some specific aspects may stand out clearer after localisation. We will see an example of this behaviour in the final section when we show that the p-local stable homotopy category has vast computational advantages over working with the stable homotopy category itself. We will also see how Bousfield localisation can help us gain insight into the deeper structure of the stable homotopy category via p-localisation, p-completion, K-theory and chromatic homotopy theory.