# nLab simplicial Quillen adjunction

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

#### Enriched category theory

enriched category theory

# Contents

## 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.

## References

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.