nLab
Quillen adjunction

Idea

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

Definition

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

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

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

• $L$ preserves cofibrations and acyclic cofibrations;

• $R$ preserves fibrations and trivial fibrations;

• $L$ preserves cofibrations and $G$ preserves fibrations;

• $R$ preserves trivial cofibrations and $G$ preserves trivial fibrations.

Properties

General

Proposition

It follows from the definition that

• 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 $R$ and hence by 2-out-of-3 the claim follows.

For $L$ we apply the formally dual argument.

Of $\mathrm{sSet}$-enriched adjunctions

Of particzular interest are SSet-enriched adjunctions between simplicial model categories, as these present adjoint (∞,1)-functors, as the first proposition below asserts.

Proposition

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

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

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

On the decategorified level of the homotopoy 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 undeer 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

The underlying adjunction of an SSet-enriched-adjunction

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

between simplicial model categories $C$ and $D$, where $D$ is a left proper model category is a Quillen adjunction precisely if

• $R$ preserves fibrant objects

• $L$ preserves cofibrations.

Proof

This is proposition A.3.7.2 in HTT.

Remarks

• 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.