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

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

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

Write $Ho C$ and $Ho D$ for the corresponding homotopy categories.

By the discussion there, $Ho C$ 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

$$\mathbb{L} : Ho C \to Ho D$$

between the homotopy categories, called its (total) left derived functor. Similarly (but dually), $R$ induces a (total) right derived functor $\mathbb{R} : Ho D \to Ho C$. See at homotopy category of a model category – derived functors for more.

Definition

A Quillen adjunction $(L \dashv R)$ is a Quillen equivalence if the following equivalent conditions are satisfied:

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

• The total right derived functor $\mathbb{R} \colon Ho(D) \to Ho(C)$ 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 $D$.

• The following two conditions hold:

  1. The derived adjunction unit is a weak equivalence, in that for every cofibrant object $c\in C$, the composite $c \overset{\eta_c}{\to} R(L(c)) \to R(L(c)^{fib})$ (of the adjunction unit with a fibrant replacement $R(L(c) \stackrel{\simeq}{\to} L(c)^{fib})$) is a weak equivalence in $C$,

  2. The derived adjunction counit is a weak equivalence, in that for every fibrant object $d\in D$, the composite $L(R(d)^{cof}) \to L(R(d)) \overset{\epsilon_d}{\to} d$ (of the adjunction counit with cofibrant replacement $L(R(d)^{cof} \stackrel{\simeq}{\to} R(d))$) is a weak equivalence in $D$.

Remark

Not every equivalence between homotopy categories of model categories lifts to a Quillen equivalence. An interesting counterexample is given for instance by Dugger & Shipley (2009).

Here are further characterizations:

Proposition

If in a Quillen adjunction $\array{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}}$ the right adjoint $R$ “creates weak equivalences” (in that a morphism $f$ in $\mathcal{C}$ is a weak equivalence precisely if $R(f)$ is) then $(L \dashv R)$ is a Quillen equivalence precisely already if for all cofibrant objects $d \in \mathcal{D}$ the plain adjunction unit

$$d \overset{\eta}{\longrightarrow} R (L (d))$$

is a weak equivalence.

(e.g. Erdal-Ilhan 19, Lemma 3.3.)

Proof

Generally, $(L \dashv R)$ is a Quillen equivalence precisely if

1. for every cofibrant object $d\in \mathcal{D}$, the “derived adjunction unit”, hence the composite

   $$d \overset{\eta}{\longrightarrow} R(L(d)) \overset{R(j_{L(d)})}{\longrightarrow} R(P(L(d)))$$

   (of the adjunction unit with image under $R$ of any fibrant replacement $L(d) \underoverset{\in W}{j_{L(d)}}{\longrightarrow} P(L(d))$) is a weak equivalence;

2. for every fibrant object $c \in \mathcal{C}$, the “derived adjunction counit”, hence the composite

   $$L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c$$

   (of the adjunction counit with the image under $L$ of any cofibrant replacement $Q R(c)\underoverset{\in W}{p_{R(c)}}{\longrightarrow} R(c)$ is a weak equivalence in $D$.

Consider the first condition: Since $R$ preserves the weak equivalence $j_{L(d)}$, by two-out-of-three the composite in the first item is a weak equivalence precisely if $\eta$ is.

Hence it is now sufficient to show that in this case the second condition above is automatic.

Since $R$ also reflects weak equivalences, the composite in item two is a weak equivalence precisely if its image

$$R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c)$$

under $R$ is.

Moreover, assuming, by the above, that $\eta_{Q(R(c))}$ on the cofibrant object $Q(R(c))$ is a weak equivalence, then by two-out-of-three this composite is a weak equivalence precisely if the further composite with $\eta$ is

$$Q(R(c)) \overset{\eta_{Q(R(c))}}{\longrightarrow} R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c) \,.$$

But by the formula for adjuncts, this composite is the $(L\dashv R)$-adjunct of the original composite, which is just $p_{R(c)}$

$$\frac{ L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c }{ Q(R(C)) \overset{p_{R(c)}}{\longrightarrow} R(c) } \,.$$

But $p_{R(c)}$ is a weak equivalence by definition of cofibrant replacement.

Properties

2-out-of-3

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

Presentation of equivalence of $(\infty,1)$-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.

Examples

Example

(trivial Quillen equivalence)

Let $\mathcal{C}$ be a model category. Then the identity functor on $\mathcal{C}$ constitutes a Quillen equivalence from $\mathcal{C}$ to itself:

$$\mathcal{C} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C}$$

Proof

From this prop. it is clear that in this case the derived functors $\mathbb{L}id$ and $\mathbb{R}id$ both are themselves the identity functor on the homotopy category of a model category, hence in particular are an equivalence of categories.

Proposition

(left base change Quillen equivalence)

Let $\mathcal{C}$ be a model category, and $\phi \colon S \overset{ \in \mathrm{W} }{\longrightarrow} T$ be a weak equivalence in $\mathcal{C}$.

Then the left base change Quillen adjunction along $\phi$ is a Quillen equivalence

$$\mathcal{C}_{/T} \underoverset {\underset{\phi^*}{\longrightarrow}} {\overset{\phi_!}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C}_{/S}$$

if and only if $\phi$ has this property:

$(\ast)$ The pullback (base change) of $\phi$ along any fibration is still a weak equivalence.

Notice that the property $(\ast)$ of $\phi$ is implied as soon as either:

• $\mathcal{C}$ is right proper, or

• $\phi$ is an acyclic fibration, or

• both $S$ and $T$ are fibrant objects

(for the first this follows by definition; for the second by the fact that $\phi^\ast$ is a right Quillen functor by this Prop.; for the third by this Prop. on recognizing homotopy pullbacks).

Proof

Using the characterization of Quillen equivalences by derived adjuncts (here), the base change adjunction is a Quillen equivalence iff for

• any cofibrant object $X \to S$ in the slice over $S$ (i.e. $X$ is cofibrant in $\mathcal{C}$)

• and a fibrant object $p \colon Y \to T$ in the slice over $T$ (i.e. $p$ is a fibration in $\mathcal{C}$),

we have that

(1) $X \to \phi^*(Y) = S \times_T Y$ is a weak equivalence

iff

(2) $\phi_!(X) \to Y$ is a weak equivalence.

But the latter morphism is the top composite in the following commuting diagram:

$$\array{ X &\longrightarrow& S \times_T Y &\overset{p^\ast \phi}{\longrightarrow}& Y \\ &\searrow& \big\downarrow &{}^{_{(pb)}}& \big\downarrow {}^{\mathrlap{p \in Fib}} \\ && S &\underset{\phi \in \mathrm{W} }{\longrightarrow}& T }$$

Hence the two-out-of-three-property says that (1) is equivalent to (2) if $p^\ast \phi$ is a weak equivalence.

Conversely, taking $X \to \phi^\ast(X)$ to be a weak equivalence (hence a cofibrant resolution of $\phi^\ast(X)$), two-out-of-three implies that if $(\phi_! \dashv \phi^\ast)$ is a Quillen equivalence, then $p^\ast \phi$ is a weak equivalence.

Related concepts

• Quillen adjunction

• Quillen reflection

• simplicial Quillen adjunction

• Quillen equivalence

• monoidal Quillen adjunction

References

For standard references see at model category.

An example of an equivalence of homotopy categories of model categories which does not lift to a Quillen equivalence is in

• Daniel Dugger, Brooke Shipley, A curious example of triangulated-equivalent model categories which are not Quillen equivalent, Algebraic & Geometric Topology 9 (2009) (pdf)

The characterization of Quillen equivalences in the case that one of the adjoints creates equivalences appears for instance in

• Mehmet Akif Erdal, Aslı Güçlükan İlhan, A model structure via orbit spaces for equivariant homotopy, Journal of Homotopy and Related Structures volume 14, pages 1131–1141 (2019) (arXiv:1903.03152, doi:10.1007/s40062-019-00241-4)