which, roughly speaking, does not make a difference when mapping to a C-local object.
is required to be a weak equivalence (of simplicial sets) for any C-local object W. An object W is called C-local if it is fibrant (in M) and is a weak equivalence for all maps
is, for a general model category (not necessarily enriched over simplicial sets) a certain simplicial set whose set of path components agrees with morphisms in the homotopy category of M: If M is a simplicial model category (such as, say, simplicial sets or topological spaces), then "map" above can be taken to be the derived simplicial mapping space of M. This description does not make any claim about the existence of this model structure, for which see below.
Dually, there is a notion of right Bousfield localization, whose definition is obtained by replacing cofibrations by fibrations (and reversing directions of all arrows).
The left Bousfield localization model structure, as described above, is known to exist in various situations, provided that C is a set: Combinatoriality and cellularity of a model category guarantee, in particular, a strong control over the cofibrations of M. Similarly, the right Bousfield localization exists if M is right proper and cellular or combinatorial and C is a set.
of an (ordinary) category C with respect to a class W of morphisms satisfies the following universal property: The Bousfield localization is the appropriate analogous notion for model categories, keeping in mind that isomorphisms in ordinary category theory are replaced by weak equivalences.
The stable model structure is obtained as a left Bousfield localization of the level (or projective) model structure on spectra, whose weak equivalences (fibrations) are those maps which are weak equivalences (fibrations, respectively) in all levels.
[3] Morita model structure on the category of small dg categories is Bousfield localization of the standard model structure (the one for which the weak equivalences are the quasi-equivalences).