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The analog of a reflective subcategory-inclusion as adjunctions of functors are replaced by Quillen adjunctions.
Let $\mathcal{C}$ and $\mathcal{D}$ 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 $c$ is just a cofibrant resolution-morphism
but regarded in the model structure $\mathcal{D}_{loc}$. Hence it is sufficient to see that acyclic fibrations in $\mathcal{D}$ 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
$(L \underset{Qu}{\dashv} R)$ is a Quillen reflection precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is a reflective subcategory-inclusion;
$(L \underset{Qu}{\dashv} R)$ is a Quillen co-reflection precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is a co-reflective subcategory-inclusion;
$(L \underset{Qu}{\dashv} R)$ is a Quillen equivalence precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is an equivalence of categories.
By this Prop. the components of the adjunction unit/counit of $(\mathbb{L}L \dashv \mathbb{R}R)$ are precisely the images under localization of the derived adjunction unit/counit of $(L \underset{Qu}{\dashv} R)$. Moreover, by this Prop. the localization functor of a model category inverts precisely the weak equivalences. Hence the adjunction (co-)unit of $(\mathbb{L}L \dashv \mathbb{R}R)$ is an isomorphism if and only if the derived (co-)unit of $(L \underset{Qu}{\dashv} R)$ 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.