Starting with the trivial model structure on the category of simplicial sets (that is the weak equivalences are exactly the isomorphisms and the cofibrations and fibrations are arbitrary maps), is it possible to get the usual Quillen model structure on simplicial sets by performing a number of explicit left and right Bousfield localizations (e.g. by localizing along the inclusions of horns into simplices)?

$\begingroup$ You may run into issues proving that you can do this sequence of left/right Bousfield localizations. Of course, both left and right Bousfield Localization will work on the trivial model structure because it's left and right proper (as all objects are bifibrant) and combinatorial. However, if you do a right Bousfield localization the result will be right proper but may no longer be left proper. By the way, you should also look into the notion of LeftDetermined Model Structure. If you can do a right Bousfield localization and end up with cofibrations=monomorphisms then you should be all set. $\endgroup$ – David White Dec 18 '12 at 15:43

$\begingroup$ Mike Shulman's answer to the following MO question might be useful. It talks about the Left Determined structure on sSet: mathoverflow.net/questions/14266/… $\endgroup$ – David White Dec 18 '12 at 15:43

$\begingroup$ Here is the beginning of an idea: by rightBousfield localizing by all trivial fibrations, you will reduce the class of cofibrations. So the trivial fibrations must be interpreted as colocal equivalences. So I suggest first a right Bousfield localization by the set of simplices, and then if the new model category has exactly the monomorphisms as cofibrations (?), it should be "between" the minimal model structure and the usual model structure by Cisinski's result (so it should be left proper), then a left Bousfield localization by the accessible class of weak equivalences could work. $\endgroup$ – Philippe Gaucher Dec 18 '12 at 15:49
Such sequence of localizations/colocalizations does not exist, since homotopy is not concrete.
Suppose, for contradiction, that such sequence is constructed, then we obtain a corresponding sequence of reflections/coreflections on the level of homotopy categories, producing a fully faithful embedding of $\mathrm{Ho}(sSet_{\mathrm{standard}})\rightarrow \mathrm{Ho}(sSet_{\mathrm{trivial}})\cong sSet$. Next, it is possible to construct a faithful functor $sSet\rightarrow Set$, say, by sending every simplicial set into the product of the sets of its simplices, obtaining a contradiction with Freyd's theorem.

2$\begingroup$ Thanks for this clear answer and the reference to the paper of Freyd, which (unfortunately) I didn't know of. $\endgroup$ – Marc NieperWißkirchen Feb 14 '13 at 11:45

$\begingroup$ You are welcome! I enjoyed thinking about your question. $\endgroup$ – Boris Chorny Feb 14 '13 at 14:32