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The analog of a reflective subcategory-inclusion as adjunctions of functors are replaced by Quillen adjunctions.
Let and be model categories, and let
be a Quillen adjunction between them. Then this may be called
a Quillen reflection if the derived adjunction counit is componentwise a weak equivalence;
a Quillen co-reflection if the derived adjunction unit is componentwise a weak equivalence.
(left Bousfield localization is Quillen reflection)
A left Bousfield localization is a Quillen reflection.
A right Bousfield localization is a Quillen coreflection.
We consider the case of left Bousfield localizations, the other case is formally dual.
A left Bousfield localization is a Quillen adjunction by identity functors (this Remark)
This means that the ordinary adjunction counit is the identity morphism and hence that the derived adjunction counit on a fibrant object is just a cofibrant resolution-morphism
but regarded in the model structure . Hence it is sufficient to see that acyclic fibrations in remain weak equivalences in the left Bousfield localized model structure. In fact they even remain acyclic fibrations, by this Remark.
Let
be a Quillen adjunction and write
for the induced adjoint pair of derived functors on the homotopy categories (this Prop.).
Then
is a Quillen reflection precisely if is a reflective subcategory-inclusion;
is a Quillen co-reflection precisely if is a co-reflective subcategory-inclusion;
is a Quillen equivalence precisely if is an equivalence of categories.
By this Prop. the components of the adjunction unit/counit of are precisely the images under localization of the derived adjunction unit/counit of . Moreover, by this Prop. the localization functor of a model category inverts precisely the weak equivalences. Hence the adjunction (co-)unit of is an isomorphism if and only if the derived (co-)unit of is a weak equivalence, respectively.
With this the statement reduces to the characterization of (co-)reflections via invertible units/counits, respectively (this Prop.).
Last revised on July 12, 2018 at 08:00:41. See the history of this page for a list of all contributions to it.