nLab Quillen adjunction



Model category theory

model category, model \infty -category



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Presentation of (,1)(\infty,1)-categories

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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.



For CC and DD two model categories, a pair (L,R)(L,R)

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

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

  1. LL preserves cofibrations and acyclic cofibrations;

  2. RR preserves fibrations and acyclic fibrations;

  3. LL preserves cofibrations and RR preserves fibrations;

  4. LL preserves acyclic cofibrations and RR preserves acyclic fibrations.


The conditions in def. are indeed all equivalent.


Observe that

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

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

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


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.


Derived adjunction


(Ken Brown's lemma)

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

  • the left adjoint LL preserves weak equivalences between cofibrant objects;

  • the right adjoint RR preserves weak equivalences between fibrant objects.


To show this for instance for RR, 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 RR and hence the claim follows by 2-out-of-3.

For LL we apply the formally dual argument.


(derived adjunction)

For 𝒞 Qu QuRL𝒟\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(𝒞)𝕃Ho(𝒟) 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


Given a Quillen adjunction

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

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

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

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

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

(LR):𝒞 QuRL𝒟 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 sSetsSet-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.


Let CC and DD be simplicial model categories and let

(LR):CLRD (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 C^\circ and D D^\circ be the (∞,1)-categories presented by CC and DD (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors

(𝕃):C D . (\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 LL and RR.


This is proposition 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).


(recognition of simplicial Quillen adjunctions)

If CC and DD are simplicial model categories and DD is a left proper model category, then an sSet-enriched adjunction

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

is a Quillen adjunction already if LL preserves cofibrations and RR just fibrant objects.

This appears as HTT, cor. A.3.7.2.

See simplicial Quillen adjunction for more details.

Associated (infinity,1)-adjunction


Let F:CD:GF : 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]D[W D 1]:G F : C[W_C^{-1}] \rightleftarrows D[W_D^{-1}] : G

where C[W C 1]C[W_C^{-1}] and D[W D 1]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.

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


See the references at model category. For instance

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

and this question is also partially addressed in

The case for simplicial model categories is also in

Last revised on May 13, 2023 at 13:38:57. See the history of this page for a list of all contributions to it.