The definition of the homotopy limit of a diagram of left Quillen functors of model categories has been useful in a number of applications. In this chapter we review its definition and summarise some of these applications. We conclude with a discussion of why we could work with right Quillen functors instead, but cannot work with a combination of the two.

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.

The definition of the homotopy limit of a diagram of left Quillen functors of model categories has been useful in a number of applications. In this chapter we review its definition and summarise some of these applications. We conclude with a discussion of why we could work with right Quillen functors instead, but cannot work with a combination of the two.