Bousfield localisations along Quillen bifunctors

TL;DR

This paper develops a general framework for Bousfield localisations along Quillen bifunctors, extending existing localisations to broader model categories and enabling the definition of Postnikov sections in new contexts.

Contribution

It introduces a method to construct localised model structures along Quillen bifunctors, generalising several known localisations and broadening their applicability.

Findings

Constructed localised model structures for Quillen bifunctors.

Extended Bousfield localisations to more general model categories.

Enabled the definition of Postnikov sections in new settings.

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.