The geometric realization--singular set adjunction $|\cdot| \dashv \operatorname{Sing}$ is a Quillen adjunction. The left adjoint $|\cdot|$ preserves cofibrations (monomorphisms become relative CW inclusions) and trivial cofibrations (anodyne extensions become trivial cofibrations in $\mathbf{Top}$).

The Dold--Kan correspondence gives a Quillen adjunction

$N: \mathbf{sAb} \rightleftarrows \mathbf{Ch}_{\geq 0}(\mathbb{Z}) : \Gamma$

where $N$ is the normalized chain complex functor and $\Gamma$ is its right adjoint. This is in fact a Quillen equivalence: simplicial abelian groups and non-negatively graded chain complexes have equivalent homotopy theories.

The free-forgetful adjunction $F: \mathbf{Set} \rightleftarrows \mathbf{sSet} : \operatorname{ev}_0$ (where $F(S)$ is the constant simplicial set and $\operatorname{ev}_0$ evaluates at level $0$) is a Quillen adjunction from the trivial model structure on $\mathbf{Set}$ to the Kan--Quillen model structure on $\mathbf{sSet}$.

For a ring homomorphism $\varphi: R \to S$, the extension-restriction of scalars adjunction

$S \otimes_R -: \mathbf{Ch}(R) \rightleftarrows \mathbf{Ch}(S) : \operatorname{Res}$

is a Quillen adjunction (with appropriate model structures). The derived functors give $\mathbf{L}(S \otimes_R -) = S \otimes_R^{\mathbf{L}} -$ (derived tensor product along $\varphi$).

The tensor-hom adjunction $(-\otimes M) \dashv \operatorname{Hom}(M, -)$ for $R$-modules gives a derived adjunction:

$(-\otimes_R^{\mathbf{L}} M) \dashv \mathbf{R}\operatorname{Hom}_R(M, -)$

on derived categories. This is the fundamental adjunction of homological algebra: $\operatorname{Tor}$ and $\operatorname{Ext}$ are the derived functors of this adjunction.

For a morphism $f: X \to Y$ of schemes, the direct-inverse image adjunction $f^* \dashv f_*$ on sheaf categories gives a derived adjunction:

$\mathbf{L}f^* \dashv \mathbf{R}f_*$

on derived categories of sheaves. Here $\mathbf{R}f_*$ computes higher direct images $R^i f_*\mathcal{F}$, and $\mathbf{L}f^*$ is the derived pullback.

The adjunction $|\cdot| \dashv \operatorname{Sing}: \mathbf{sSet} \rightleftarrows \mathbf{Top}$ is a Quillen equivalence. The unit $X \to \operatorname{Sing}(|X|)$ is a weak equivalence for Kan complexes, and the counit $|\operatorname{Sing}(Y)| \to Y$ is a weak equivalence for all spaces. Thus $\operatorname{Ho}(\mathbf{sSet}) \simeq \operatorname{Ho}(\mathbf{Top})$.

The Dold--Kan adjunction $N \dashv \Gamma: \mathbf{sAb} \rightleftarrows \mathbf{Ch}_{\geq 0}(\mathbb{Z})$ is a Quillen equivalence. The homotopy category of simplicial abelian groups is equivalent to the derived category of non-negatively graded abelian groups.

The identity functor $\mathrm{id}: \mathbf{sSet}_{\mathrm{Joyal}} \to \mathbf{sSet}_{\mathrm{Kan}}$ is NOT a Quillen equivalence (and not even a Quillen adjunction in general). The Joyal model structure has more weak equivalences (categorical equivalences) than the Kan--Quillen model structure, so they capture different homotopy theories. The Joyal model structure models $(\infty, 1)$-categories, while Kan--Quillen models $\infty$-groupoids (spaces).

There is a Quillen equivalence between the Joyal model structure on $\mathbf{sSet}$ (modeling quasi-categories) and the model structure on Segal categories (bisimplicial sets satisfying the homotopy Segal condition). This shows that quasi-categories and Segal categories provide equivalent models for $(\infty, 1)$-categories.

If $F_1 \dashv G_1: \mathcal{M} \rightleftarrows \mathcal{N}$ and $F_2 \dashv G_2: \mathcal{N} \rightleftarrows \mathcal{P}$ are Quillen equivalences, then $F_2 F_1 \dashv G_1 G_2: \mathcal{M} \rightleftarrows \mathcal{P}$ is a Quillen equivalence. This allows "chaining" Quillen equivalences to compare distant model categories.

For instance, the chain $\mathbf{sAb} \simeq_Q \mathbf{Ch}_{\geq 0}(\mathbb{Z})$ and $\mathbf{sAb} \hookrightarrow \mathbf{sSet}$ (forgetful) allow comparison of chain complexes with simplicial sets.

Two model categories $\mathcal{M}$ and $\mathcal{N}$ are Quillen equivalent if they are connected by a zigzag of Quillen equivalences:

$\mathcal{M} \leftarrow \mathcal{M}_1 \to \mathcal{M}_2 \leftarrow \cdots \to \mathcal{N}$

This happens when there is no single Quillen equivalence between them but they still have equivalent homotopy theories. The notion of Quillen equivalent model categories is an equivalence relation on model categories.

A Quillen adjunction $F \dashv G: \mathcal{M} \rightleftarrows \mathcal{N}$ is a Quillen equivalence if and only if:

1. $G$ reflects weak equivalences between fibrant objects: if $Y$ is fibrant in $\mathcal{N}$ and $GY \to GY'$ is a weak equivalence in $\mathcal{M}$, then $Y \to Y'$ is a weak equivalence in $\mathcal{N}$.

2. For every cofibrant $X \in \mathcal{M}$, the map $X \to G(R(FX))$ (unit followed by fibrant replacement) is a weak equivalence.

Alternatively: $F \dashv G$ is a Quillen equivalence iff $\mathbf{L}F$ is essentially surjective on $\operatorname{Ho}(\mathcal{N})$ and for cofibrant $X$, the derived unit $X \to \mathbf{R}G(\mathbf{L}F(X))$ is a weak equivalence.