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A simplicial Quillen adjunction is an sSet-enriched Quillen adjunction: an enriched adjunction
of sSet-enriched functors between simplicial model categories $C$ and $D$, such that the underlying adjunction of ordinary functors is a Quillen adjunction between the model category structures underlying the simplicial model categories.
Simplicial Quillen adjunctions model pairs of adjoint (∞,1)-functors in a fairly immediate manner: their restriction to fibrant-cofibrant objects is the sSet-enriched functor that presents the $(\infty,1)$-derived functor under the model of (∞,1)-categories by simplicially enriched categories.
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 homotopoy 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 simplicial Quillen adjunction may be detected already from knowing of the right adjoint only that it preserves fibrant objects (instead of all fibrations).
(recognition of simplicial Quillen adjunctions)
If
$\mathcal{C}$ and $\mathcal{D}$ are simplicial model categories
$\mathcal{D}$ is a left proper model category,
then for an sSet-enriched adjunction
to be a Quillen adjunction it is already sufficient that
$L$ preserves cofibrations
$R$ preserves fibrant objects.
This appears as HTT, cor. A.3.7.2.
Prop. is particularly useful for finding simplicial Quillen adjunctions into left Bousfield localizations of left proper model categories: the left Bousfield localization keeps the cofibrations unchanged and preserves left properness, and the fibrant objects in the Bousfield localized structure have a good characterization: they are the fibrant objects in the original model structure that are also local objects with respect to the set of morphisms at which one localizes.
Therefore for $D$ the left Bousfield localization of a simplicial left proper model category $E$ at a class $S$ of morphisms, for checking the Quillen adjunction property of $(L \dashv R)$ it is sufficient to check that $L$ preserves cofibrations, and that $R$ takes fibrant objects $c$ of $C$ to such fibrant objects of $E$ that have the property that for all $f \in S$ the derived hom-space map $\mathbb{R}Hom(f,R(c))$ is a weak equivalence.
simplicial Quillen adjunction
On the enhancement of plain Quillen adjunctions between left proper combinatorial model categories to simplicial Quillen adjunctions:
Last revised on July 4, 2022 at 16:39:18. See the history of this page for a list of all contributions to it.