Yoneda embeddings of stable model categories; composition with Bousfield localizations

Question

For a stable model category $C$ and a set $M$ of object of it I would like to construct a natural functor from $C$ to some stable 'category of functors' on $M$. I suspect that the 'natural' question to do so is to consider the composition of the following functors (yet is it "optimal"?).

1. Using the methods of section 4.2 of Toen-Vezzosi's http://www.math.univ-montp2.fr/~toen/essen.pdf, embed $C$ into a certain category $C^\wedge$ of functors of $C\to SSets$. This yield a full embedding of homotopy categories by Theorem 4.2.1. My questions are: where could one find a (more) complete proof of this statement; could one avoid higher universes here; are there any difficulties with obtaining a full embedding of stable homotopy categories using this method?

2. Next I would like to compose this with the Bousfield localization of $C^\wedge$ by those morphisms that yield weak equivalence when restricted to (elements of) $M$. Are there any references for this functor, or for its composition with the first one? Which restrictions on $C$ (and $M$) are required for its existence? Is the corresponding homotopy functor a full embedding when restricted to elements of $M$?

I would be deeply grateful for any references or hints! I would prefer to avoid higher categories and quasi-categories here (if possible).

Answer 1

Mikhail, I think that what you're looking for is what is known as a presentation of a model category. There's a beautiful paper by Dugger proving that any combinatorial model category has a presentation:

Dugger, Daniel(1-PURD) Combinatorial model categories have presentations. (English summary) Adv. Math. 164 (2001), no. 1, 177–201.

This paper is not particularly concerned with the stable case. You may regard this as an advantage or disadvantage, depending on what you exactly want to do. If you want to consider presheaves of simplicial sets, I'd say it's an advantage. The particular features of stable categories are probably better caught by considering functors to spectra, and more generally modules over ring spectra or spectral categories. If you want to take this direction, then you would find interesting the following paper:

Schwede, Stefan(D-MUNS-GFM); Shipley, Brooke(1-PURD) Stable model categories are categories of modules. (English summary) Topology 42 (2003), no. 1, 103–153.

The hypothesis of being combinatorial is not regarded as a very strong one, since most model categories have a combinatorial model. Actually, under some set theoretical hypotheses (which often appear in homotopy theory) any cofibrantly generated model category has a combinatorial model:

Raptis, George(4-OX) On the cofibrant generation of model categories. (English summary) J. Homotopy Relat. Struct. 4 (2009), no. 1, 245–253.