nLab
Quillen reflection

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Idea

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

Definition

Definition

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

𝒞 QuAARAAL𝒟 \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)

𝒟 loc Qu QuAAidAAid𝒟 \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 cc is just a cofibrant resolution-morphism

Q(c)W 𝒟Fib 𝒟p cc Q(c) \underoverset{ \in W_{\mathcal{D}} \cap Fib_{\mathcal{D}} }{p_c}{\longrightarrow} c

but regarded in the model structure 𝒟 loc\mathcal{D}_{loc}. Hence it is sufficient to see that acyclic fibrations in 𝒟\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

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

be a Quillen adjunction and write

Ho(𝒞) QuAARAA𝕃LHo(𝒟) 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. (LQuR)(L \underset{Qu}{\dashv} R) is a Quillen reflection precisely if (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) is a reflective subcategory-inclusion;

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

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

Proof

By this Prop. the components of the adjunction unit/counit of (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) are precisely the images under localization of the derived adjunction unit/counit of (LQuR)(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 (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) is an isomorphism if and only if the derived (co-)unit of (LQuR)(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 04:00:41. See the history of this page for a list of all contributions to it.