nLab simplicial Quillen adjunction

Definition

A simplicial Quillen adjunction is an sSet-enriched Quillen adjunction: an enriched adjunction

$$(L \dashv R) : C \stackrel{\leftarrow}{\to} D$$

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.

Properties

Presentation of $\infty$-adjunctions

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.

Proposition

Let $C$ and $D$ be simplicial model categories and let

$$(L \dashv R) \;\colon\; C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$$

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

$$(\mathbb{L} \dashv \mathbb{R}) \;\colon\; C^\circ \stackrel{\leftarrow}{\to} D^{\circ} \,.$$

On the decategorified level of the homotopoy categories these are the total left and right derived functors, respectively, of $L$ and $R$.

Proof

This is proposition 5.2.4.6 in HTT.

Recognition

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).

Proposition

(recognition of simplicial Quillen adjunctions)
If

1. $\mathcal{C}$ and $\mathcal{D}$ are simplicial model categories

2. $\mathcal{D}$ is a left proper model category,

then for an sSet-enriched adjunction

$$(L \dashv R) \;\colon\; \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\bot\;\;\;} \mathcal{D}$$

to be a Quillen adjunction it is already sufficient that

1. $L$ preserves cofibrations

2. $R$ preserves fibrant objects.

This appears as HTT, cor. A.3.7.2.

Remark

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.

Related concepts

• Quillen adjunction

• simplicial Quillen adjunction

• Quillen equivalence

• monoidal Quillen adjunction

References

On the enhancement of plain Quillen adjunctions between left proper combinatorial model categories to simplicial Quillen adjunctions:

• Andrew Blumberg, Emily Riehl, Appendix A of: Homotopical resolutions associated to deformable adjunctions, Algebr. Geom. Topol. 14 (2014) 3021-3048 (arXiv:1208.2844)