This entry is about the article
Abstract Given a small category , we show that there is a universal way of expanding into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing “relations” into these universal gadgets. The paper develops this formalism and discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes.
The article discusses the projective global and local model structure on simplicial presheaves.
The general discussion is in parts is based on the unfinished but useful notes
In particular this
shows that and how the projective model structure on simplicial presheaves on is something like the free completion of under homotopy colimits. This is the model-category theoretic analog of the fact that the (∞,1)-category of (∞,1)-presheaves on is the
free (∞,1)-cocompletion of ;
announces the statement of Dugger’s theorem which states that every combinatorial model category arises as the left Bousfield localization of the projective global model structure on simplicial presheaves. The full proof of this is in the companion article
discusses cofibrant replacement in the projective model structures (necessarily both for the global as well as for the local structure)
The main theorem of the article is the following. For more details see (∞,1)-category of (∞,1)-presheaves.
two plain functors. Say that a model-category theoretic factorization of through is
Notice that the (∞,1)-category presented by a model category – at least by a combinatorial model category – has all (∞,1)-categorical colimits, and that the Quillen left adjoint functor presents, via its derived functor, a left adjoint (∞,1)-functor that preserves -categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving -functors into -categories that have all colimits.
(model category presentation of free -cocompletion)
This is theorem 1.1 in the article.