nLab Quillen adjunction

Idea

Quillen adjunctions are one convenient notion of morphisms between model categories. They present adjoint (∞,1)-functors between the (∞,1)-categories presented by the model categories.

Definition

Definition

For $C$ and $D$ two model categories, a pair $(L,R)$

$$(L \dashv R) : C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}} D$$

of adjoint functors (with $L$ left adjoint and $R$ right adjoint) is a Quillen adjunction if the following equivalent conditions are satisfied:

1. $L$ preserves cofibrations and acyclic cofibrations;

2. $R$ preserves fibrations and acyclic fibrations;

3. $L$ preserves cofibrations and $R$ preserves fibrations;

4. $L$ preserves acyclic cofibrations and $R$ preserves acyclic fibrations.

Proposition

The conditions in def. are indeed all equivalent.

Proof

Observe that

• (i) A left adjoint $L$ between model categories preserves acyclic cofibrations precisely if its right adjoint $R$ preserves fibrations.

• (ii) A left adjoint $L$ between model categories preserves cofibrations precisely if its right adjoint $R$ preserves acyclic fibrations.

We discuss statement (i), statement (ii) is formally dual. So let $f\colon A \to B$ be an acyclic cofibration in $\mathcal{D}$ and $g \colon X \to Y$ a fibration in $\mathcal{C}$. Then for every commuting diagram as on the left of the following, its $(L\dashv R)$-adjunct is a commuting diagram as on the right here:

$$\array{ A &\longrightarrow& R(X) \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}} \\ B &\longrightarrow& R(Y) } \;\;\;\;\;\; \,, \;\;\;\;\;\; \array{ L(A) &\longrightarrow& X \\ {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}} \\ L(B) &\longrightarrow& Y } \,.$$

If $L$ preserves acyclic cofibrations, then the diagram on the right has a lift, and so the $(L\dashv R)$-adjunct of that lift is a lift of the left diagram. This shows that $R(g)$ has the right lifting property against all acylic cofibrations and hence is a fibration. Conversely, if $R$ preserves fibrations, the same argument run from right to left gives that $L$ preserves acyclic fibrations.

Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.

Remark

Quillen adjunctions that are analogous to an equivalence of categories are called Quillen equivalences.

In an enriched model category one speaks of enriched Quillen adjunction.

Properties

Derived adjunction

Proposition

(Ken Brown's lemma)

Given a Quillen adjunction $(L \dashv R)$ (def. ), then

• the left adjoint $L$ preserves weak equivalences between cofibrant objects;

• the right adjoint $R$ preserves weak equivalences between fibrant objects.

Proof

To show this for instance for $R$, we may argue as in a category of fibrant objects and apply the factorization lemma which shows that every weak equivalence between fibrant objects may be factored, up to homotopy, as a span of acyclic fibrations.

These weak equivalences are preserved by the right Quillen property of $R$ and hence the claim follows by 2-out-of-3.

For $L$ we apply the formally dual argument.

Proposition

(derived adjunction)

For $\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D}$ a Quillen adjunction between model categories, also the corresponding left and right derived functors form a pair of adjoint functors

$$Ho(\mathcal{C}) \underoverset {\underset{\mathbb{R}}{\longrightarrow}} {\overset{\mathbb{L}}{\longleftarrow}} {\bot} Ho(\mathcal{D})$$

between the corresponding homotopy categories.

Moreover, the adjunction unit and adjunction counit of this derived adjunction are the images of the derived adjunction unit and derived adjunction counit of the original Quillen adjunction.

(Quillen 67, I.4 theorem 3)

Behaviour under Bousfield localization

Proposition

Given a Quillen adjunction

$$(L \dashv R) \;\;\;\colon\;\;\; \mathcal{C} \underoverset {\underset{\;\;\;\;\;\; R \;\;\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;\;\; L \;\;\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{{}_{Qu}}}} \mathcal{D}$$

and $S \subset Mor(\mathcal{D})$ is a set of morphisms such that

1. the left Bousfield localization of $\mathcal{D}$ at $S$ exists,

2. the derived image $\mathbb{L}L(S)$ of $S$ lands in the weak equivalences of $\mathcal{C}$,

then the Quillen adjunction descends to the Bousfield localization $\mathcal{D}_S$

$$(L \dashv R) \;\;\;\colon\;\;\; \mathcal{C} \underoverset {\underset{\;\;\;\;\;\; R \;\;\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;\;\; L \;\;\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{{}_{Qu}}}} \mathcal{D}_S \,.$$

This appears as (Hirschhorn, prop. 3.3.18)

Of $sSet$-enriched adjunctions

Of particular interest are SSet-enriched adjunctions between simplicial model categories: simplicial Quillen adjunctions.

These present adjoint (∞,1)-functors, as the first proposition below asserts.

Proposition

Let $C$ and $D$ be simplicial model categories and let

$$(L \dashv R) : C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}} D$$

be an sSet-enriched adjunction whose underlying ordinary adjunction is a Quillen adjunction. Let $C^\circ$ and $D^\circ$ be the (∞,1)-categories presented by $C$ and $D$ (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors

$$(\mathbb{L} \dashv \mathbb{R}) : C^\circ \stackrel{\leftarrow}{\to} D^{\circ} \,.$$

On the decategorified level of the homotopy categories these are the total left and right derived functors, respectively, of $L$ and $R$.

Proof

This is proposition 5.2.4.6 in HTT.

The following proposition states conditions under which a Quillen adjunction may be detected already from knowing of the right adjoint only that it preserves fibrant objects (instead of all fibrations).

Proposition

(recognition of simplicial Quillen adjunctions)

If $C$ and $D$ are simplicial model categories and $D$ is a left proper model category, then an sSet-enriched adjunction

$$(L \dashv R) : C \stackrel{\leftarrow}{\to} D$$

is a Quillen adjunction already if $L$ preserves cofibrations and $R$ just fibrant objects.

This appears as HTT, cor. A.3.7.2.

See simplicial Quillen adjunction for more details.

Associated (infinity,1)-adjunction

Theorem

Let $F : C \rightleftarrows D : G$ be a Quillen adjunction between model categories (which are not assumed to admit functorial factorizations or infinite (co)limits). Then there is an induced adjunction of (infinity,1)-categories

$$F : C[W_C^{-1}] \rightleftarrows D[W_D^{-1}] : G$$

where $C[W_C^{-1}]$ and $D[W_D^{-1}]$ denote the respective simplicial localizations at the respective classes of weak equivalences.

See (Mazel-Gee 16, Theorem 2.1). (This is also asserted as (Hinich 14, Proposition 1.5.1), but it is not completely proved there – see (Mazel-Gee 16, Remark 2.3).)

For simplicial model categories with sSet-enriched Quillen adjunctions between them, this is also in (Lurie, prop. 5.2.4.6).

See also at derived functor – As functors on infinity-categories

Related concepts

• Quillen reflection

• Quillen adjoint triple

• simplicial Quillen adjunction

• Quillen equivalence

• monoidal Quillen adjunction

References

See the references at model category. For instance

• Philip Hirschhorn, Model categories and their localization

• Peter May, Kate Ponto, Section 16.2 of: More concise algebraic topology, University of Chicago Press (2012) (ISBN:9780226511795, pdf)

The proof that a Quillen adjunction of model categories induces an adjunction of (infinity,1)-categories is recorded in

• Aaron Mazel-Gee, Quillen adjunctions induce adjunctions of quasicategories, New York Journal of Mathematics Volume 22 (2016) 57-93 (arXiv:1501.03146, publisher)

and this question is also partially addressed in

• Vladimir Hinich, Dwyer-Kan Localization Revisited, Homology, Homotopy and Applications Volume 18 (2016) Number 1 (arXiv:1311.4128, doi:10.4310/HHA.2016.v18.n1.a3)

The case for simplicial model categories is also in

• Jacob Lurie, Higher Topos Theory