Given a
class C of morphisms in a
model category M the left Bousfield localization is a new model structure on the same category as before. Its equivalences,
cofibrations and
fibrations, respectively, are • the
C-local equivalences • the original cofibrations of
M and (necessarily, since cofibrations and weak equivalences determine the fibrations) • the maps having the
right lifting property with respect to the cofibrations in
M which are also
C-local equivalences. In this definition, a
C-local equivalence is a map f\colon X \to Y which, roughly speaking, does not make a difference when mapping to a
C-local object. More precisely, f^* \colon \operatorname{map} (Y, W) \to \operatorname{map} (X, W) 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 :s^* \colon \operatorname{map} (B, W) \to \operatorname{map} (A, W) is a weak equivalence for
all maps s\colon A \to B in
C. The notation \operatorname{map}(-, -) 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: :\pi_0 (\operatorname{map}(X, Y)) = \operatorname{Hom}_{Ho(M)}(X, Y). 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). ==Existence==