related by the Dold-Kan correspondence
For and two model categories, a pair
preserves cofibrations and acyclic cofibrations;
preserves fibrations and acyclic fibrations;
preserves cofibrations and preserves fibrations;
preserves acyclic cofibrations and preserves acyclic fibrations.
The conditions in def. 1 are indeed all equivalent.
We discuss statement (i), statement (ii) is formally dual. So let be an acyclic cofibration in and a fibration in . Then for every commuting diagram as on the left of the following, its -adjunct is a commuting diagram as on the right here:
If preserves acyclic cofibrations, then the diagram on the right has a lift, and so the -adjunct of that lift is a lift of the left diagram. This shows that has the right lifting property against all acylic cofibrations and hence is a fibration. Conversely, if preserves fibrations, the same argument run from right to left gives that preserves acyclic fibrations.
Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.
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.
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)
These present adjoint (∞,1)-functors, as the first proposition below asserts.
Let and be simplicial model categories and let
be an sSet-enriched adjunction whose underlying ordinary adjunction is a Quillen adjunction. Let and be the (∞,1)-categories presented by and (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors
This is proposition 188.8.131.52 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).
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 the references at model category. For instance
and this question is also partially addressed in
The case for simplicial model categories is also in