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homotopy category of a model category

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Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

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see also algebraic topology

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Contents

Idea

A model category structure (𝒞(W,Fib,Cof))(\mathcal{C}(W,Fib,Cof)) on some category 𝒞\mathcal{C} is a means to guarantee the local smallness and to improve the tractability of the homotopy category of the underlying category with weak equivalences (𝒞,W)(\mathcal{C},W). In particular, if a category with weak equivalences admits a model category structure, then its homotopy category (in the sense of localization 𝒞[W 1]\mathcal{C}[W^{-1}] at the class of weak equivalences) is equivalent to the category whose objects are those objects of 𝒞\mathcal{C} which are both fibrant cofibrant, and whose morphisms are the actual homotopy classes of morphisms between these objects (left homotopy or right homotopy equivalence classes in the sense of homotopy in a model category) .

Definition

Definition

Let 𝒞\mathcal{C} be a model category. Write Ho(𝒞)Ho(\mathcal{C}) for the category whose

This is, up to equivalence of categories, the homotopy category of the model category 𝒞\mathcal{C}.

Universal property

We spell out that def. 1 indeed satisfies the universal property that defines the homotopy category of a category with weak equivalences.

Lemma

(Whitehead theorem in model categories)

Let 𝒞\mathcal{C} be a model category. A weak equivalence between two objects which are both fibrant and cofibrant is a homotopy equivalence.

(e.g. Goerss-Jardine 99, part I, theorem 1.10)

Proof

By the factorization axioms in 𝒞\mathcal{C}, every weak equivalence f:XYf\colon X \longrightarrow Y factors through an object ZZ as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with XX and YY both fibrant and cofibrant, so is ZZ, and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.

So let f:XYf \colon X \longrightarrow Y be an acyclic fibration between fibrant-cofibrant objects, the case of acyclic cofibrations is formally dual. Then in fact it has a genuine right inverse given by a lift f 1f^{-1} in the diagram

X cof f 1 FibW f X = X. \array{ \emptyset &\rightarrow& X \\ {}^{\mathllap{\in cof}}\downarrow &{}^{{f^{-1}}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib \cap W}} \\ X &=& X } \,.

To see that f 1f^{-1} is also a left inverse up to left homotopy, let Cyl(X)Cyl(X) be any cylinder object on XX, hence a factorization of the codiagonal on XX as a cofibration followed by a an acyclic fibration

XXι XCyl(X)σX X \sqcup X \stackrel{\iota_X}{\longrightarrow} Cyl(X) \stackrel{\sigma}{\longrightarrow} X

and consider the square

XX (f 1f,id) X ι X Cof WFib f Cyl(X) fσ y, \array{ X \sqcup X &\stackrel{(f^{-1}\circ f, id)}{\longrightarrow}& X \\ {}^{\mathllap{\iota_X}}{}_{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(X) &\underset{f\circ \sigma}{\longrightarrow}& y } \,,

which commutes due to f 1f^{-1} being a genuine right inverse of ff. By construction, this commuting square now admits a lift η\eta, and that constitutes a left homotopy η:f 1f Lid\eta \colon f^{-1}\circ f \Rightarrow_L id.

Definition

Given a model category 𝒞\mathcal{C}, consider a choice for each object X𝒞X \in \mathcal{C} of

  1. a factorization Cofi XQXWFibp xX\emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_x}{\longrightarrow} X of the initial morphism, such that when XX is already cofibrant then p X=id Xp_X = id_X;

  2. a factorization XWCofj XRXFibq x*X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} R X \underoverset{\in Fib}{q_x}{\longrightarrow} \ast of the terminal morphism, such that when XX is already fibrant then j X=id Xj_X = id_X.

Write then

γ R,Q:𝒞Ho(𝒞) \gamma_{R,Q} \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})

for the functor to the homotopy category, def. 1, which sends an object XX to the object RQXR Q X and sends a morphism f:XYf \colon X \longrightarrow Y to the homotopy class of the result of first lifting in

QY i X Qf p Y QX fp X Y \array{ \emptyset &\longrightarrow& Q Y \\ {}^{\mathllap{i_X}}\downarrow &{}^{Q f}\nearrow& \downarrow^{\mathrlap{p_Y}} \\ Q X &\underset{f\circ p_X}{\longrightarrow}& Y }

and then lifting (here: extending) in

QX j QYQf RQY j QX RQf q QY RQX *. \array{ Q X &\overset{j_{Q Y} \circ Q f}{\longrightarrow}& R Q Y \\ {}^{\mathllap{j_{Q X}}}\downarrow &{}^{R Q f}\nearrow& \downarrow^{\mathrlap{q_{Q Y}}} \\ R Q X &\longrightarrow& \ast } \,.
Lemma

The construction in def. 2 is indeed well defined.

Proof

First of all, the object RQXR Q X is indeed both fibrant and cofibrant (as well as related by a zig-zag of weak equivalences to XX):

Cof Cof QX WCof RQX Fib * W X. \array{ \emptyset \\ {}^{\mathllap{\in Cof}}\downarrow & \searrow^{\mathrlap{\in Cof}} \\ Q X &\underset{\in W \cap Cof}{\longrightarrow}& R Q X &\underset{\in Fib}{\longrightarrow}& \ast \\ {}^{\mathllap{\in W}}\downarrow \\ X } \,.

Now to see that the image on morphisms is well defined. First observe that any two choices (Qf) i(Q f)_{i} of the first lift in the definition are left homotopic to each other, exhibited by lifting in

QXQX ((Qf) 1,(Qf) 2) QY WCof Fib p Y Cyl(QX) fp Xσ QX Y. \array{ Q X \sqcup Q X &\stackrel{((Q f)_1, (Q f)_2 )}{\longrightarrow}& Q Y \\ {}^{\mathllap{\in W \cap Cof}}\downarrow && \downarrow^{\mathrlap{p_{Y}}}_{\mathrlap{\in Fib}} \\ Cyl(Q X) &\underset{f \circ p_{X} \circ \sigma_{Q X}}{\longrightarrow}& Y } \,.

Hence also the composites j QY(Q f) ij_{Q Y}\circ (Q_f)_i are left homotopic to each other, and since their domain is cofibrant, they are also right homotopic (via this lemma) by a right homotopy κ\kappa. This implies finally, by lifting in

QX κ Path(RQY) WCof Fib RQX (R(Qf) 1,R(Qf) 2) RQY×RQY \array{ Q X &\overset{\kappa}{\longrightarrow}& Path(R Q Y) \\ {}^{\mathllap{\in W \cap Cof}}\downarrow && \downarrow^{\mathrlap{\in Fib}} \\ R Q X &\underset{(R (Q f)_1, R (Q f)_2)}{\longrightarrow}& R Q Y \times R Q Y }

that also R(Qf) 1R (Q f)_1 and R(Qf) 2R (Q f)_2 are right homotopic, hence that indeed RQfR Q f represents a well-defined homotopy class.

Finally to see that the assignment is indeed functorial, observe that the commutativity of the lifting diagrams for QfQ f and RQfR Q f imply that also the following diagram commutes

X p X QX j QX RQX f Qf RQf Y p y QY j QY RQY. \array{ X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& R Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}} \\ Y &\underset{p_y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& R Q Y } \,.

Now from the pasting composite

X p X QX j QX RQX f Qf RQf Y p Y QY j QY RQY g Qg RQg Z p Z QZ j QZ RQZ \array{ X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& R Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}} \\ Y &\underset{p_Y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& R Q Y \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{Q g}} && \downarrow^{\mathrlap{R Q g}} \\ Z &\underset{p_Z}{\longleftarrow}& Q Z &\underset{j_{Q Z}}{\longrightarrow}& R Q Z }

one sees that (RQg)(RQf)(R Q g)\circ (R Q f) is a lift of gfg \circ f and hence the same argument as above gives that it is homotopic to the chosen RQ(gf) R Q(g \circ f).

Definition

For 𝒞\mathcal{C} a category with weak equivalences, its homotopy category (or: localization at the weak equivalences) is, if it exists, a category Ho(𝒞)Ho(\mathcal{C}) equipped with a functor

γ:𝒞Ho(C) \gamma \colon \mathcal{C} \longrightarrow Ho(C)

which sends weak equivalences to isomorphisms, and which is universal with this property:

for F:𝒞DF \colon \mathcal{C} \longrightarrow D any functor out of 𝒞\mathcal{C} into any category, such that FF takes weak equivalences to isomorphisms, it factors through γ\gamma up to a natural isomorphism

𝒞 F D γ ρ F˜ Ho(𝒞) \array{ \mathcal{C} && \overset{F}{\longrightarrow} && D \\ & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}} \\ && Ho(\mathcal{C}) }

and this factorization is unique up to unique isomorphism, in that for (F˜ 1,ρ 1)(\tilde F_1, \rho_1) and (F˜ 2,ρ 2)(\tilde F_2, \rho_2) two such factorizations, then there is a unique natural isomorphism κ:F˜ 1F˜ 2\kappa \colon \tilde F_1 \Rightarrow \tilde F_2 making the evident diagram of natural isomorphisms commute.

Theorem

For 𝒞\mathcal{C} a model category, the functor γ R,Q\gamma_{R,Q} in def. 2 (for any choice of RR and QQ) exhibits Ho(𝒞)Ho(\mathcal{C}) as indeed being the homotopy category of the underlying category with weak equivalences, in the sense of def. 3.

Proof

First, to see that that γ\gamma indeed takes weak equivalences to isomorphisms: By two-out-of-three applied to the commuting diagrams shown in the proof of lemma 2 the morphism RQfR Q f is a weak equivalence if ff is:

X p X QX j QX RQX f Qf RQf Y p y QY j QY RQY \array{ X &\underoverset{\simeq}{p_X}{\longleftarrow}& Q X &\underoverset{\simeq}{j_{Q X}}{\longrightarrow}& R Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}} \\ Y &\underoverset{p_y}{\simeq}{\longleftarrow}& Q Y &\underoverset{j_{Q Y}}{\simeq}{\longrightarrow}& R Q Y }

With this the “Whitehead theorem for model categories”, lemma 1, implies that RQfR Q f represents an isomorphism in Ho(𝒞)Ho(\mathcal{C}).

Now let F:𝒞DF \colon \mathcal{C}\longrightarrow D be any functor that sends weak equivalences to isomorphisms. We need to show that it factors as

𝒞 F D γ ρ F˜ Ho(𝒞) \array{ \mathcal{C} && \overset{F}{\longrightarrow} && D \\ & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}} \\ && Ho(\mathcal{C}) }

uniquely up to unique natural isomorphism. Now by construction of RR and QQ in def. 2, γ R,Q\gamma_{R,Q} is the identity on the full subcategory of fibrant-cofibrant objects. It follows that if F˜\tilde F exists at all, it must satisfy for all XfYX \stackrel{f}{\to} Y with XX and YY both fibrant and cofibrant that

F˜([f])F(f). \tilde F([f]) \simeq F(f) \,.

But by def. 1 that already fixes F˜\tilde F on all of Ho(𝒞)Ho(\mathcal{C}), up to unique natural isomorphism. Hence it only remains to check that with this definition of F˜\tilde F there exists any natural isomorphism ρ\rho filling the diagram above.

To that end, apply FF to the above commuting diagram to obtain

F(X) isoF(p X) F(QX) isoF(j QX) F(RQX) F(f) F(Qf) F(RQf) F(Y) F(p y)iso F(QY) F(j QY)iso F(RQY). \array{ F(X) &\underoverset{iso}{F(p_X)}{\longleftarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(R Q X) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{F(Q f)}} && \downarrow^{\mathrlap{F(R Q f)}} \\ F(Y) &\underoverset{F(p_y)}{iso}{\longleftarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(R Q Y) } \,.

Here now all horizontal morphisms are isomorphisms, by assumption on FF. It follows that defining ρ XF(j QX)F(p X) 1\rho_X \coloneqq F(j_{Q X}) \circ F(p_X)^{-1} makes the required natural isomorphism:

ρ X: F(X) isoF(p X) 1 F(QX) isoF(j QX) F(RQX) = F˜(γ R,Q(X)) F(f) F(RQf) F˜(γ R,Q(X)) ρ Y: F(Y) F(p y) 1iso F(QY) F(j QY)iso F(RQY) = F˜(γ R,Q(X)). \array{ \rho_X \colon & F(X) &\underoverset{iso}{F(p_X)^{-1}}{\longrightarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(R Q X) &=& \tilde F(\gamma_{R,Q}(X)) \\ & {}^{\mathllap{F(f)}}\downarrow && && \downarrow^{\mathrlap{F(R Q f)}} && \downarrow^{\tilde F(\gamma_{R,Q}(X))} \\ \rho_Y\colon& F(Y) &\underoverset{F(p_y)^{-1}}{iso}{\longrightarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(R Q Y) &=& \tilde F(\gamma_{R,Q}(X)) } \,.
Remark

Due to theorem 1 we may suppress the choices of cofibrant QQ and fibrant replacement RR in def. 2 and just speak of the localization functor

γ:𝒞Ho(𝒞) \gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})

up to natural isomorphism.

While the construction of the homotopy category in def. 1 combines the restriction to good (fibrant/cofibrant) objects with the passage to homotopy classes of morphisms, it is often useful to consider intermediate stages:

Definition

Given a model category 𝒞\mathcal{C}, write

𝒞 fc 𝒞 c 𝒞 f 𝒞 \array{ && \mathcal{C}_{fc} \\ & \swarrow && \searrow \\ \mathcal{C}_c && && \mathcal{C}_f \\ & \searrow && \swarrow \\ && \mathcal{C} }

for the system of full subcategory inclusions on the cofibrant objects (𝒞 c\mathcal{C}_c), the fibrant objects (𝒞 f\mathcal{C}_f) and the objects which are both fibrant and cofibrant (𝒞 fc\mathcal{C}_{fc}), all regarded a categories with weak equivalences, via the weak equivalences inherited from 𝒞\mathcal{C}.

Remark

Of course the subcategories in def. 4 inherit more structure than just that of categories with weak equivalences from 𝒞\mathcal{C}. 𝒞 f\mathcal{C}_f and 𝒞 c\mathcal{C}_c each inherit “half” of the factorization axioms. One says that 𝒞 f\mathcal{C}_f has the structure of a “fibration category” called a “category of fibrant objects”, while 𝒞 c\mathcal{C}_c has the structure of a “cofibration category”.

The proof of theorem 1 immediately implies the following:

Corollary

For 𝒞\mathcal{C} a model category, the restriction of the localization functor γ:𝒞Ho(𝒞)\gamma\;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C}) from def. 2 (using remark 1) to any of the sub-categories with weak equivalences of def. 4

𝒞 fc 𝒞 c 𝒞 f 𝒞 γ Ho(𝒞) \array{ && \mathcal{C}_{fc} \\ & \swarrow && \searrow \\ \mathcal{C}_c && && \mathcal{C}_f \\ & \searrow && \swarrow \\ && \mathcal{C} \\ && \downarrow^{\mathrlap{\gamma}} \\ && Ho(\mathcal{C}) }

exhibits Ho(𝒞)Ho(\mathcal{C}) equivalently as the homotopy category also of these subcategories. In particular there are equivalences of categories

Ho(𝒞)Ho(𝒞 f)Ho(𝒞 c)Ho(𝒞 fc). Ho(\mathcal{C}) \simeq Ho(\mathcal{C}_{f}) \simeq Ho(\mathcal{C}_{c}) \simeq Ho(\mathcal{C}_{fc}) \,.

Derived functors

Definition

For 𝒞\mathcal{C} and 𝒟\mathcal{D} two categories with weak equivalences, then a functor F:𝒞𝒟F \colon \mathcal{C}\longrightarrow \mathcal{D} is called homotopical functor if it sends weak equivalences to weak equivalences.

Definition

Given a homotopical functor F:𝒞𝒟F \colon \mathcal{C} \longrightarrow \mathcal{D} (def. 5) between categories with weak equivalences whose homotopy categories Ho(𝒞)Ho(\mathcal{C}) and Ho(𝒟)Ho(\mathcal{D}) exist (def. 3), then its derived functor is the functor Ho(F)Ho(F) between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. 3):

𝒞 F 𝒟 γ 𝒞 γ 𝒟 Ho(𝒞) Ho(F) Ho(𝒟). \array{ \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\exists \; Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,.
Remark

While many functors of interest between model categories are not homotopical in the sense of def. 5, many become homotopical after restriction to the full subcategories of fibrant object or of cofibrant objects, def. 4. By corollary 1 this is just as good for the purpose of homotopy theory.

Therefore one considers the following generalization of def. 6:

Definition

Consider a functor F:𝒞𝒟F \colon \mathcal{C} \longrightarrow \mathcal{D} out of a model category 𝒞\mathcal{C} into a category with weak equivalences 𝒟\mathcal{D}.

  1. If the restriction of FF to the full subcategory 𝒞 f\mathcal{C}_f of fibrant object becomes a homotopical functor (def. 5), then the derived functor of that restriction, according to def. 6, is called the right derived functor of FF and denoted by F\mathbb{R}F:

    𝒞 f 𝒞 F 𝒟 γ 𝒞 f γ 𝒞 γ 𝒟 F: Ho(𝒞 f) Ho(𝒞) Ho(F) Ho(𝒟). \array{ & \mathcal{C}_f &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow && \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ \mathbb{R} F \colon & Ho(\mathcal{C}_f) &\simeq& Ho(\mathcal{C}) &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,.

    Here the commuting square on the left is from corollary 1, the square on the right is that of def. 6.

  2. If the restriction of FF to the full subcategory 𝒞 c\mathcal{C}_c of cofibrant object becomes a homotopical functor (def. 5), then the derived functor of that restriction, according to def. 6, is called the left derived functor of FF and denoted by 𝕃F\mathbb{L}F:

    𝒞 c 𝒞 F 𝒟 γ 𝒞 f γ 𝒞 γ 𝒟 𝕃F: Ho(𝒞 c) Ho(𝒞) Ho(F) Ho(𝒟). \array{ & \mathcal{C}_c &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow && \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ \mathbb{L} F \colon & Ho(\mathcal{C}_c) &\simeq& Ho(\mathcal{C}) &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,.

    Here the commuting square on the left is from corollary 1, the square on the right is that of def. 6.

The key fact that makes def. 7 practically useful is the following

Proposition

(Ken Brown's lemma)

Let 𝒞\mathcal{C} be a model category with full subcategories 𝒞 f,𝒞 c\mathcal{C}_f, \mathcal{C}_c of fibrant objects and of cofibrant objects respectively (def. 4). Let 𝒟\mathcal{D} be a category with weak equivalences.

  1. A functor

    F:𝒞 f𝒟 F \;\colon\; \mathcal{C}_f \longrightarrow \mathcal{D}

is a homotopical functor, def. 5, already if it sends acylic fibrations to weak equivalences.

  1. A functor

    F:𝒞 c𝒟 F \;\colon\; \mathcal{C}_c \longrightarrow \mathcal{D}

is a homotopical functor, def. 5, already if it sends acylic cofibrations to weak equivalences.

Corollary

Let 𝒞,𝒟\mathcal{C}, \mathcal{D} be model categories and consider F:𝒞𝒟F \colon \mathcal{C}\longrightarrow \mathcal{D} a functor. Then:

  1. If FF preserves cofibrant objects and acyclic cofibrations between these, then its left derived functor (def. 7) 𝕃F\mathbb{L}F exists, fitting into a diagram

    𝒞 c F 𝒟 c γ 𝒞 γ 𝒟 Ho(𝒞) 𝕃F Ho(𝒟) \array{ \mathcal{C}_{c} &\overset{F}{\longrightarrow}& \mathcal{D}_{c} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\overset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D}) }
  2. If FF preserves fibrant objects and acyclic fibrants between these, then its right derived functor (def. 7) F\mathbb{R}F exists, fitting into a diagram

    𝒞 f F 𝒟 f γ 𝒞 γ 𝒟 Ho(𝒞) F Ho(𝒟). \array{ \mathcal{C}_{f} &\overset{F}{\longrightarrow}& \mathcal{D}_{f} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D}) } \,.

In practice it turns out to be useful to arrange for the assumptions in corollary 2 to be satisfied in the following neat way:

Definition

Let 𝒞,𝒟\mathcal{C}, \mathcal{D} be model categories. A pair of adjoint functors between them

(LR):𝒞RL𝒟 (L \dashv R) \;\colon\; \mathcal{C} \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}} \mathcal{D}

is called a Quillen adjunction (and LL,RR are called left/right Quillen functors, respectively) if the following equivalent conditions are satisfied

  1. LL preserves cofibrations and RR preserves fibrations;

  2. LL preserves acyclic cofibrations and RR preserves acyclic fibrations;

  3. LL preserves cofibrations and acylic cofibrations;

  4. RR preserves fibrations and acyclic fibrations.

Proposition

The conditions in def. 8 are indeed all equivalent.

Proof

Observe that

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

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

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

Definition

For 𝒞,𝒟\mathcal{C}, \mathcal{D} two model categories, a Quillen adjunction (def.8)

(LR):𝒞RL𝒟 (L \dashv R) \;\colon\; \mathcal{C} \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}} \mathcal{D}

is called a Quillen equivalence if the following equivalent conditions hold.

  1. The right derived functor of RR (via prop. 2, corollary 2) is an equivalence of categories

    R:Ho(𝒞)Ho(𝒟). \mathbb{R}R \colon Ho(\mathcal{C}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{D}) \,.
  2. The left derived functor of LL (via prop. 2, corollary 2) is an equivalence of categories

    𝕃L:Ho(𝒟)Ho(𝒞). \mathbb{L}L \colon Ho(\mathcal{D}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{C}) \,.
  3. For every cofibrant object d𝒟d \in \mathcal{D} and every fibrant object c𝒞c \in \mathcal{C}, a morphism dR(c)d \longrightarrow R(c) is a weak equivalence precisely if its adjunct morphism L(c)dL(c) \to d is

    dW 𝒟R(c)L(d)W 𝒞c. \frac{ d \overset{\in W_{\mathcal{D}}}{\longrightarrow} R(c) }{ L(d) \overset{\in W_{\mathcal{C}}}{\longrightarrow} c } \,.
  4. For every cofibrant object d𝒞d\in \mathcal{C}, the composite

    dR(L(d))R(L(d) fib) d \longrightarrow R(L(d)) \longrightarrow R(L(d)^{fib})

    (of the adjunction unit with any fibrant replacement) is a weak equivalence.

  5. For every fibrant object c𝒞c \in \mathcal{C}, the composite

    L(R(c) cof)L(R(c))c L(R(c)^{cof}) \longrightarrow L(R(c)) \longrightarrow c

    (of the adjunction counit with any cofibrant replacement) is a weak equivalence in DD.

References

The original account is due to

  • Daniel Quillen, Axiomatic homotopy theory in Homotopical algebra, Lecture Notes in Mathematics, No. 43 43, Berlin (1967)

Review includes

Revised on April 4, 2016 09:25:29 by Urs Schreiber (82.113.98.109)