Bousfield localisations along Quillen bifunctors

Abstract

Consider a Quillen adjunction of two variables between combinatorial model categories from $\mathcal{C}\times\mathcal{D}$ to $\mathcal{E}$, and a set $\mathcal{S}$ of morphisms in $\mathcal{C}$. We prove that there is a localised model structure $L_{\mathcal{S}}\mathcal{E}$ on $\mathcal{E}$, where the local objects are the $\mathcal{S}$-local objects in $\mathcal{E}$ described via the right adjoint. These localised model structures generalise Bousfield localisations of simplicial model categories, Barnes and Roitzheim's familiar model structures, and Barwick's enriched Bousfield localisations. In particular, we can use these model structures to define Postnikov sections in more general left proper combinatorial model categories.