nLab
Quillen equivalence

Idea

A model category is a context in which we can do homotopy theory or some generalization thereof; two model categories are ‘the same’ for this purpose if they are Quillen equivalent. For example, the classic version of homotopy theory can be done using either topological spaces or simplicial sets. There is a model category of topological spaces, and a model category of simplicial sets, and they are Quillen equivalent.

In short, Quillen equivalence is the right notion of equivalence for model categories — and most importantly, this notion is weaker than equivalence of categories. The work of Dwyer–Kan, Bergner and others has shown that Quillen equivalent model categories present equivalent (infinity,1)-categories.

Definition

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

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

be a Quillen adjunction with $L$ left adjoint to $R$.

Write $\mathrm{Ho}C$ and $\mathrm{Ho}D$ for the corresponding homotopy categories.

Notice that $\mathrm{Ho}C$ may be regarded as obtained by first passing to the full subcategory on cofibrant objects and then inverting weak equivalences, and $L$ (being a left Quillen adjoint) preserves weak equivalences between cofibrant objects. Thus, $L$ induces a functor $𝕃:\mathrm{Ho}C\to \mathrm{Ho}D$ between the homotopy categories, called its (total) left derived functor. Similarly (but dually), $R$ induces a (total) right derived functor $ℝ:\mathrm{Ho}D\to \mathrm{Ho}C$.

The Quillen adjunction $(L⊣R)$ is a Quillen equivalence if the following equivalent conditions are satisfied.

• The total left derived functor $𝕃:\mathrm{Ho}(C)\to \mathrm{Ho}(D)$ is an equivalence of the homotopy categories;

• The total right derived functor $ℝ:\mathrm{Ho}(D)\to \mathrm{Ho}(C)$ is an equivalence of the homotopy categories;

• For every cofibrant object $c\in C$ and every fibrant object $d\in D$, a morphism $c\to R(d)$ is a weak equivalence in $C$ precisely when the adjunct morphism $L(c)\to d$ is a weak equivalence in $D$.

• For every cofibrant object $c\in C$, the composite $c\to R(L(c))\to R(L(c{)}^{\mathrm{fib}})$ is a weak equivalence in $C$, and for every fibrant object $d\in D$, the composite $L(R(d{)}^{\mathrm{cof}})\to L(R(d))\to d$ is a weak equivalence in $D$, where $(-{)}^{\mathrm{fib}}$ and $(-{)}^{\mathrm{cof}}$ denote fibrant and cofibrant resolutions, respectively.

Properties

2-out-of-3

Since equivalences of categories enjoy the 2-out-of-3-property, so do Quillen equivalences.

Presentation of equivalence of $(\mathrm{\infty },1)$-categories

sSet-enriched Quillen equivalences between combinatorial model categories present equivalences between the corresponding locally presentable (infinity,1)-categories. And every equivalence between these is presented by a Zig-Zag of Quillen equivalences. See there for more details.

Related concepts

• Quillen adjunction

• simplicial Quillen adjunction

• Quillen equivalence

• monoidal Quillen adjunction