nLab
Quillen equivalence

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

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general

specific

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Contents

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 CC and DD be model categories and let

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

be a Quillen adjunction with LL left adjoint to RR.

Write HoCHo C and HoDHo D for the corresponding homotopy categories.

By the discussion there, HoCHo C may be regarded as obtained by first passing to the full subcategory on cofibrant objects and then inverting weak equivalences, and LL (being a left Quillen adjoint) preserves weak equivalences between cofibrant objects. Thus, LL induces a functor

𝕃:HoCHoD \mathbb{L} : Ho C \to Ho D

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

Definition

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

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

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

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

  • The following two conditions hold:

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

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

Remark

Not every equivalence between homotopy categories of model categories lifts to a Quillen equivalence. An interesting counterexample is given for instance in (Dugger-Shipley 09).

Here are further characterizations:

Proposition

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

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

is a weak equivalence.

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

Proof

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

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

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

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

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

    L(Q(R(c)))L(p R(c))L(R(c))ϵc 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 LL of any cofibrant replacement QR(c)Wp R(c)R(c)Q R(c)\underoverset{\in W}{p_{R(c)}}{\longrightarrow} R(c) is a weak equivalence in DD.

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

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

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

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

under RR is.

Moreover, assuming, by the above, that η Q(R(c))\eta_{Q(R(c))} on the cofibrant object Q(R(c))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 η\eta is

Q(R(c))η Q(R(c))R(L(Q(R(c))))R(L(p R(c)))R(L(R(c)))R(ϵ)R(c). 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 (LR)(L\dashv R)-adjunct of the original composite, which is just p R(c)p_{R(c)}

L(Q(R(c)))L(p R(c))L(R(c))ϵcQ(R(C))p R(c)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)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 (,1)(\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 𝒞\mathcal{C} be a model category. Then the identity functor on 𝒞\mathcal{C} constitutes a Quillen equivalence from 𝒞\mathcal{C} to itself:

𝒞 Qu Quidid𝒞 \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 𝕃id\mathbb{L}id and id\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 𝒞\mathcal{C} be a model category, and ϕ:SWT\phi \colon S \overset{ \in \mathrm{W} }{\longrightarrow} T be a weak equivalence in 𝒞\mathcal{C}.

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

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

if and only if ϕ\phi has this property:

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

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

(for the first this follows by definition; for the second by the fact that ϕ *\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 XSX \to S in the slice over SS (i.e. XX is cofibrant in 𝒞\mathcal{C})

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

we have that

(1) Xϕ *(Y)=S× TYX \to \phi^*(Y) = S \times_T Y is a weak equivalence

iff

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

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

X S× TY p *ϕ Y (pb) pFib S ϕW T \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 *ϕp^\ast \phi is a weak equivalence.

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

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)

Last revised on October 6, 2021 at 10:22:35. See the history of this page for a list of all contributions to it.