nLab Quillen reflection

Idea

The analog of a reflective subcategory-inclusion as adjunctions of functors are replaced by Quillen adjunctions.

Definition

Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be model categories, and let

$$\mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}$$

be a Quillen adjunction between them. Then this may be called

1. a Quillen reflection if the derived adjunction counit is componentwise a weak equivalence;

2. a Quillen co-reflection if the derived adjunction unit is componentwise a weak equivalence.

Example

(left Bousfield localization is Quillen reflection)

1. A left Bousfield localization is a Quillen reflection.

2. A right Bousfield localization is a Quillen coreflection.

Proof

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)

$$\mathcal{D}_{loc} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {{}_{\phantom{Qu}} \bot_{Qu}} \mathcal{D}$$

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

$$Q(c) \underoverset{ \in W_{\mathcal{D}} \cap Fib_{\mathcal{D}} }{p_c}{\longrightarrow} c$$

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.

Properties

Proposition

Let

$$\mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}$$

be a Quillen adjunction and write

$$Ho(\mathcal{C}) \underoverset {\underset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\bot_{Qu}} Ho(\mathcal{D})$$

for the induced adjoint pair of derived functors on the homotopy categories (this Prop.).

Then

1. $(L \underset{Qu}{\dashv} R)$ is a Quillen reflection precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is a reflective subcategory-inclusion;

2. $(L \underset{Qu}{\dashv} R)$ is a Quillen co-reflection precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is a co-reflective subcategory-inclusion;

3. $(L \underset{Qu}{\dashv} R)$ is a Quillen equivalence precisely if $(\mathbb{L}L \dashv \mathbb{R}R)$ is an equivalence of categories.

Proof

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.).