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 morphisms 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 and $R$ right 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.
The conditions in def. 1 are indeed all equivalent.
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:
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.
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.
Given a Quillen adjunction $(L \dashv R)$ (def. 1), then
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.
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
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
Quillen adjunction
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