Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
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 and two model categories, a pair
of adjoint functors (with left adjoint) is a Quillen adjunction if the following equivalent conditions are satisfied:
preserves cofibrations and acyclic cofibrations;
preserves fibrations and acyclic fibrations;
preserves cofibrations and preserves fibrations;
preserves acyclic cofibrations and 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
To show this for instance for , 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 and hence by 2-out-of-3 the claim follows.
For we apply the formally dual argument.
Behaviour under localization
is a Quillen adjunction, is a set of morphisms such that the left Bousfield localization of at exists, and such that the derived image of lands in the weak equivalences of , then the Quillen adjunction descends to the localization
This appears as (Hirschhorn, prop. 3.3.18)
Of -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.
This is proposition 18.104.22.168 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 and are simplicial model categories and is a left proper model category, then an sSet-enriched adjunction
is a Quillen adjunction already if preserves cofibrations and just fibrant objects.
This appears as HTT, cor. A.3.7.2.
See simplicial Quillen adjunction for more details.
See (Hinich 14, Proposition 1.5.1) or (Mazel-Gee 15, Theorem 2.1).
For simplicial model categories with sSet-enriched Quillen adjunctions between them, this is also in (Lurie, prop. 22.214.171.124).
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 also in
The case for simplicial model categories is also in