on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
Quillen adjunctions are one convenient notion of morphism between model categories. They present adjoint (∞,1)-functors between the (∞,1)-categories presented by the model categories.
For $C$ and $D$ two model categories, a pair $(L,R)$
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 acyclic fibrations;
$L$ preserves cofibrations and $R$ preserves fibrations;
$L$ preserves acyclic cofibrations and $R$ preserves acyclic fibrations.
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.
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.
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.
If
is a Quillen adjunction, $S \subset Mor(D)$ is a set of morphisms such that the left Bousfield localization of $D$ at $S$ exists, and such that the derived image $\mathbb{L}L(S)$ of $S$ lands in the weak equivalences of $C$, then the Quillen adjunction descends to the localization $D_S$
This appears as (Hirschhorn, prop. 3.3.18)
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 $C$ and $D$ be simplicial model categories and let
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
On the decategorified level of the homotopy categories these are the total left and right derived functors, respectively, of $L$ and $R$.
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).
If $C$ and $D$ are simplicial model categories and $D$ is a left proper model category, then an sSet-enriched adjunction
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.
Quillen adjunction
See the references at model category. For instance