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

# Contents

## Idea

Quillen adjunctions are one convenient notion of morphisms between model categories. They present adjoint (∞,1)-functors between the (∞,1)-categories presented by the model categories.

## Definition

###### Definition

For $C$ and $D$ two model categories, a pair $(L,R)$

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

of adjoint functors (with $L$ left adjoint and $R$ right adjoint) is a Quillen adjunction if the following equivalent conditions are satisfied:

1. $L$ preserves cofibrations and acyclic cofibrations;

2. $R$ preserves fibrations and acyclic fibrations;

3. $L$ preserves cofibrations and $R$ preserves fibrations;

4. $L$ preserves acyclic cofibrations and $R$ preserves acyclic fibrations.

###### Proposition

The conditions in def. are indeed all equivalent.

###### Proof

Observe that

• (i) A left adjoint $L$ between model categories preserves acyclic cofibrations precisely if its right adjoint $R$ preserves fibrations.

• (ii) A left adjoint $L$ between model categories preserves cofibrations precisely if its right adjoint $R$ preserves acyclic fibrations.

We discuss statement (i), statement (ii) is formally dual. So let $f\colon A \to B$ be an acyclic cofibration in $\mathcal{D}$ and $g \colon X \to Y$ a fibration in $\mathcal{C}$. Then for every commuting diagram as on the left of the following, its $(L\dashv R)$-adjunct is a commuting diagram as on the right here:

$\array{ A &\longrightarrow& R(X) \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}} \\ B &\longrightarrow& R(Y) } \;\;\;\;\;\; \,, \;\;\;\;\;\; \array{ L(A) &\longrightarrow& X \\ {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}} \\ L(B) &\longrightarrow& Y } \,.$

If $L$ preserves acyclic cofibrations, then the diagram on the right has a lift, and so the $(L\dashv R)$-adjunct of that lift is a lift of the left diagram. This shows that $R(g)$ has the right lifting property against all acylic cofibrations and hence is a fibration. Conversely, if $R$ preserves fibrations, the same argument run from right to left gives that $L$ preserves acyclic fibrations.

Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.

###### Remark

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.

## Properties

###### Proposition

(Ken Brown's lemma)

Given a Quillen adjunction $(L \dashv R)$ (def. ), then

• the left adjoint $L$ preserves weak equivalences between cofibrant objects;

• the right adjoint $R$ preserves weak equivalences between fibrant objects.

###### Proof

To show this for instance for $R$, 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 the right Quillen property of $R$ and hence the claim follows by 2-out-of-3.

For $L$ we apply the formally dual argument.

###### Proposition

For $\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D}$ a Quillen adjunction between model categories, also the corresponding left and right derived functors form a pair of adjoint functors

$Ho(\mathcal{C}) \underoverset {\underset{\mathbb{R}}{\longrightarrow}} {\overset{\mathbb{L}}{\longleftarrow}} {\bot} Ho(\mathcal{D})$

between the corresponding homotopy categories.

### Behaviour under Bousfield localization

###### Proposition

$(L \dashv R) \;\;\;\colon\;\;\; \mathcal{C} \underoverset {\underset{\;\;\;\;\;\; R \;\;\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;\;\; L \;\;\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{{}_{Qu}}}} \mathcal{D}$

and $S \subset Mor(\mathcal{D})$ is a set of morphisms such that

1. the left Bousfield localization of $\mathcal{D}$ at $S$ exists,

2. the derived image $\mathbb{L}L(S)$ of $S$ lands in the weak equivalences of $\mathcal{C}$,

then the Quillen adjunction descends to the Bousfield localization $\mathcal{D}_S$

$(L \dashv R) \;\;\;\colon\;\;\; \mathcal{C} \underoverset {\underset{\;\;\;\;\;\; R \;\;\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;\;\; L \;\;\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{{}_{Qu}}}} \mathcal{D}_S \,.$

This appears as (Hirschhorn, prop. 3.3.18)

### Of $sSet$-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.

###### Proposition

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

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

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

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

On the decategorified level of the homotopy 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.

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).

###### Proposition

If $C$ and $D$ are simplicial model categories and $D$ is a left proper model category, then an sSet-enriched adjunction

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

is a Quillen adjunction already if $L$ preserves cofibrations and $R$ just fibrant objects.

This appears as HTT, cor. A.3.7.2.

See simplicial Quillen adjunction for more details.

###### Theorem

Let $F : C \rightleftarrows D : G$ be a Quillen adjunction between model categories (which are not assumed to admit functorial factorizations or infinite (co)limits). Then there is an induced adjunction of (infinity,1)-categories

$F : C[W_C^{-1}] \rightleftarrows D[W_D^{-1}] : G$

where $C[W_C^{-1}]$ and $D[W_D^{-1}]$ denote the respective simplicial localizations at the respective classes of weak equivalences.

See (Mazel-Gee 16, Theorem 2.1). (This is also asserted as (Hinich 14, Proposition 1.5.1), but it is not completely proved there – see (Mazel-Gee 16, Remark 2.3).)

For simplicial model categories with sSet-enriched Quillen adjunctions between them, this is also in (Lurie, prop. 5.2.4.6).

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 this question is also partially addressed in

The case for simplicial model categories is also in