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Model categories

**Homotopical algebra** ##Contents * Contributors * References * Introduction * Model categories * Homotopy Factorisation Systems * Cisinski's theory

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**Category theory** ##Contents * Contributors * References * Introduction * Basic category theory * Weak factorisation systems * Factorisation systems * Distributors and barrels * Model structures on Cat * Homotopy factorisation systems in Cat * Accessible categories * Locally presentable categories * Algebraic theories and varieties of algebras

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Contents

Model structures

Definitions

Definition

We say that a class 𝒲\mathcal{W} of maps in a category E\mathbf{E} has the three-for-two property if for any commutative triangle

in which two of the three maps belong to 𝒲\mathcal{W}, then so is the third.

Remark

We call this property three-for-two rather than two-for-three because this is like getting three apples for the price of two in a food store.

Definition

We shall say that a triple (π’ž,𝒲,β„±)(\mathcal{C},\mathcal{W},\mathcal{F}) of classes of maps in finitely bicomplete category E\mathbf{E} is a Quillen model structure, or just a model structure, if the following conditions are satisfied: * the class 𝒲\mathcal{W} has the three-for-two property; * The pairs (π’žβˆ©π’²,β„±)( \mathcal{C}\,\cap \,\mathcal{W},\mathcal{F}) and (π’ž,β„±βˆ©π’²)(\mathcal{C},\mathcal{F}\,\cap\, \mathcal{W}) are weak factorisation systems.

A Quillen model category, or just a model category, is a category E\mathbf{E} equipped with a model structure.

Remark

The definition above is equivalent to the notion of closed model structure introduced by Quillen. The proof of the equivalence depends on Tierney’s lemma 1 below.

A map in β„±\mathcal{F} is called a fibration, a map in π’ž\mathcal{C} a cofibration and a map in 𝒲\mathcal{W} a weak equivalence. A map in 𝒲\mathcal{W} is also said to be acyclic. It follows from the axioms that every map f:Aβ†’Bf:A\to B admits a factorisation f=pu:Aβ†’Eβ†’Bf=p u:A\to E\to B with uu an acyclic cofibration and pp a fibration, and also a factorisation f=qv:Aβ†’Fβ†’Bf=q v:A\to F\to B with vv a cofibration and qq an acyclic fibration.

An object X∈EX\in \mathbf{E} is said to be fibrant if the map Xβ†’βŠ€X\to \top is a fibration, where ⊀\top is the terminal object of E\mathbf{E}. Dually, an object A∈EA\in \mathbf{E} is said to be cofibrant if the map βŠ₯β†’A\bot \to A is a cofibration, where βŠ₯\bot is the initial object. We shall say that an object is fibrant-cofibrant if it is both fibrant and cofibrant.

A model structure is said to be right proper if the base change of a weak equivalence along a fibration is a weak equivalence. Dually, a model structure is said to be left proper if the cobase change of a weak equivalence along a cofibration is a weak equivalence. A model structure is said to be proper if it is both left and right proper.

Example

We shall say that a model structure (π’ž,𝒲,β„±)( \mathcal{C},\mathcal{W},\mathcal{F}) on a category E\mathbf{E} is trivial if 𝒲\mathcal{W} is the class of isomorphisms, in which case π’ž=E=β„±\mathcal{C}=\mathbf{E}=\mathcal{F}.

Example

We shall say that a model structure (π’ž,𝒲,β„±)( \mathcal{C},\mathcal{W},\mathcal{F}) on a category E\mathbf{E} is coarse if 𝒲\mathcal{W} is the class of all maps, in which case the pair (π’ž,β„±)(\mathcal{C},\mathcal{F}) is just an arbitrary weak factorisation system in the category E\mathbf{E}.

For less trivial examples, see 1.

Duality

If (π’ž,𝒲,β„±)( \mathcal{C},\mathcal{W},\mathcal{F}) is a model structure on a category E\mathbf{E}, then the triple (β„± o,β„³ o,π’ž o)(\mathcal{F}^o,\mathcal{M}^o,\mathcal{C}^o) is a model structure on the opposite category E o\mathbf{E}^o.

If BB is an object of a category E\mathbf{E} and β„³\mathcal{M} is a class of maps in E\mathbf{E}, we shall denote by β„³/B\mathcal{M}/B the class of maps in E/B\mathbf{E}/B whose underlying map in E\mathbf{E} belongs to β„³\mathcal{M}. Dually, we shall denote by B\β„³B\backslash \mathcal{M} the class of maps in B\EB\backslash \mathbf{E} whose underlying map belongs to β„³\mathcal{M}.

Slice and coslice

If (π’ž,𝒲,β„±)( \mathcal{C},\mathcal{W},\mathcal{F}) is a model structure on a category E\mathbf{E}, then the triple (π’ž/B,𝒲/B,β„±/B)(\mathcal{C}/B,\mathcal{W}/B, \mathcal{F}/B) is a model structure in the slice category E/B\mathbf{E}/B for any object B∈EB\in \mathbf{E}. Dually, the triple (B\π’ž,B\𝒲,B\β„±)(B\backslash \mathcal{C},B\backslash \mathcal{W},B\backslash \mathcal{F}) is a model structure in the coslice category B\EB\backslash \mathbf{E}.

Proof

This follows from the analogous properties of weak factorisation systems here.

Proposition

The classes π’ž\mathcal{C}, π’žβˆ©π’²\mathcal{C}\,\cap \,\mathcal{W}, β„±\mathcal{F} and β„±βˆ©π’²\mathcal{F}\,\cap \,\mathcal{W} are closed under composition and retracts. The classes β„±\mathcal{F} and β„±βˆ©π’²\mathcal{F}\,\cap \,\mathcal{W} are closed under base changes and products. The classes π’ž\mathcal{C} and π’žβˆ©π’²\mathcal{C}\,\cap \,\mathcal{W} are closed under cobase changes and coproducts. The intersection π’žβˆ©π’²βˆ©β„±\mathcal{C}\,\cap \,\mathcal{W}\,\cap \, \mathcal{F} is the class of isomorphisms.

Proof

This follows directly from the general properties of the classes of a weak factorisation system here.

Corollary

A retract of a fibrant object is fibrant. A product of a family of fibrant objects is fibrant. Dually, a retract of a cofibrant object is cofibrant. A coproduct of a family of cofibrant objects is cofibrant.

Proof

If an object XX is a retract of an object YY, then the map Xβ†’βŠ₯X\to \bot is a retract of the map Yβ†’βŠ₯Y\to \bot. If an object XX is the product of a family of object (X i|i∈I)(X_i| i\in I), then the map Xβ†’botX\to bot is the product of the family of maps (X iβ†’βŠ₯|i∈I)(X_i\to \bot| i\in I).

Lemma

(Myles Tierney) The class 𝒲\mathcal{W} is closed under retracts.

Proof

Notice first that the class β„±βˆ©π’²\mathcal{F}\,\cap\,\mathcal{W} is closed under retracts by Proposition 1. Suppose now that a map f:Aβ†’Bf: A \to B is a retract of a map g:Xβ†’Yg: X \to Y in 𝒲\mathcal{W}. We want to show fβˆˆπ’²f \in \mathcal{W}. We have a commutative diagram

with ts=1 At s = 1_A and vu=1 Bv u = 1_B. We suppose first that ff is a fibration. In this case, factor gg as g=qj:X→Z→Yg = q j: X \to Z \to Y with jj an acyclic cofibration and qq a fibration. The map qq is acyclic by three-for-two, since qq and jj are acyclic. The square

has a diagonal filler d:Z→Ad: Z \to A, since ff is a fibration. We get a commutative diagram

The map ff is a retract of qq, since d(js)=ts=1 Ad(j s) = t s =1_A. Thus, ff is acyclic, since qq is an acyclic fibration. In the general case, factor ff as f=pi:A→E→Bf = p i: A \to E \to B with ii an acyclic cofibration and pp a fibration. By taking a pushout we obtain a commutative diagram

where kin 2=gk in_2 =g and rin 1=1 Er in_1 = 1_E. The map in 2in_2 is a cobase change of ii, so in 2 in_2 is an acyclic cofibration, since ii is. Thus, kk is acyclic by three-for-two, since g=kin 2g = k in_2 is acyclic by assumption. So pp is acyclic by the first part, since pp is a fibration. Finally, f=pif = p i is acyclic, since ii is acyclic.

The homotopy category of a model category E\mathbf{E} is defined to be the category of fractions

Ho(E)=𝒲 βˆ’1E.Ho(\mathbf{ E})=\mathcal{W}^{-1}\mathbf{E}.

The canonical functor H:Eβ†’Ho(E)H:\mathbf{ E}\to Ho(\mathbf{ E}) is a localisation with respects to the maps in 𝒲\mathcal{W}. Recall that a functor F:Eβ†’KF:\mathbf{E} \to \mathbf{K} is said to invert a map u:Aβ†’Bu:A\to B if the morphism F(u):FAβ†’FBF(u):F A\to F B is invertible. The functor HH inverts the maps in 𝒲\mathcal{W} and for any functor F:Eβ†’KF:\mathbf{E} \to \mathbf{K} which inverts the maps in 𝒲\mathcal{W}, there is a unique functor Fβ€²:Ho(E)β†’KF': Ho(\mathbf{ E})\to \mathbf{K} such that Fβ€²H=FF'H=F. We shall see in theorem 2 that a map u:Aβ†’Bu:A\to B in the category E\mathbf{E} is acyclic if and only if it is inverted by the functor HH. We shall see in corollary 1 that the category Ho(E)Ho(\mathbf{ E}) is locally small if the category E\mathbf{ E} is locally small.

We denote by E f\mathbf{E}_f (resp. E c\mathbf{E}_c, E fc\mathbf{E}_{f c}) the full subcategory of E\mathbf{ E} spanned by the fibrant (resp. cofibrant, fibrant-cofibrant) objects of E\mathbf{ E}. If β„³\mathcal{M} is a class of maps in E\mathbf{ E}, we shall put

β„³ f=β„³βˆ©E fβ„³ c=β„³βˆ©E candβ„³ fc=β„³βˆ©E fc.\mathcal{M}_f=\mathcal{M} \,\cap\, \mathbf{ E}_f \quad \quad \mathcal{M}_c=\mathcal{M}\,\cap\, \mathbf{ E}_c \quad \mathrm{and} \quad \mathcal{M}_{f c}= \mathcal{M} \,\cap\, \mathbf{ E}_{f c}.

Let us put

Ho(E f)=𝒲 f βˆ’1E f,Ho(E c)=𝒲 c βˆ’1E c,andHo(E fc)=𝒲 fc βˆ’1E fc.Ho(\mathbf{ E}_f)=\mathcal{W}_f^{-1}\mathbf{E}_f, \quad \quad Ho(\mathbf{ E}_c)=\mathcal{W}_c^{-1}\mathbf{E}_c, \quad \mathrm{and}\quad Ho(\mathbf{ E}_{f c})=\mathcal{W}_{f c}^{-1}\mathbf{E}_{f c}.

Then the square of inclusions,

induces a commutative square of categories and canonical functors,

(1)

We shall see in theorem 2 that the four functors in the square are equivalences of categories.

The following result is easy to prove but technically useful:

Proposition

The pair (π’ž fβˆ©π’² f,β„± f)( \mathcal{C}_f\,\cap \,\mathcal{W}_f,\mathcal{F}_f) and the pair (π’ž f,β„± fβˆ©π’² f)(\mathcal{C}_f,\mathcal{F}_f\,\cap\, \mathcal{W}_f) are weak factorisation systems in the category E f\mathbf{ E}_f. The pair (π’ž cβˆ©π’² c,β„± c)( \mathcal{C}_c\,\cap \,\mathcal{W}_c,\mathcal{F}_c) and the pair (π’ž c,β„± cβˆ©π’² c)(\mathcal{C}_c,\mathcal{F}_c\,\cap\, \mathcal{W}_c) are weak factorisation systems in the category E c\mathbf{ E}_c. The pair (π’ž cfβˆ©π’² cf,β„± cf)( \mathcal{C}_{cf}\,\cap \,\mathcal{W}_{cf},\mathcal{F}_{cf}) and the pair (π’ž cf,β„± cfβˆ©π’² cf)(\mathcal{C}_{cf},\mathcal{F}_{cf}\,\cap\, \mathcal{W}_{cf}) are weak factorisation systems in the category E cf\mathbf{ E}_{cf}.

Proof

Let us how that the pair (π’ž fβˆ©π’² f,β„± f)( \mathcal{C}_f\,\cap \,\mathcal{W}_f,\mathcal{F}_f) is a weak factorisation system in E f\mathbf{ E}_f. We shall use the characterisation of a weak factorisation system here. Obviously, we have uβ‹”pu\,\pitchfork\, p for every uβˆˆπ’ž fβˆ©π’² fu\in \mathcal{C}_f\,\cap \,\mathcal{W}_f and pβˆˆβ„± fp\in \mathcal{F}_f. If f:Xβ†’Yf:X\to Y is a map between fibrant objects, let us choose a factoriation f=pu:Xβ†’Eβ†’Yf=pu:X\to E\to Y with uu an acyclic cofibration and pp a fibration. The object EE is fibrant, since pp is a fibration and YY is fibrant. This shows that uβˆˆπ’ž fβˆ©π’² fu\in \mathcal{C}_f\,\cap \,\mathcal{W}_f and pβˆˆβ„± fp\in \mathcal{F}_f. Finally, the class π’ž fβˆ©π’² f\mathcal{C}_f\,\cap \,\mathcal{W}_f is closed under codomain retracts, since a retract of a fibrant object is fibrant. Similarly, the class β„± f\mathcal{F}_f is closed under domain retracts.

Cylinders and left homotopies

Lemma

The inclusion in 1:Aβ†’AβŠ”Bin_1:A\to A\sqcup B is a cofibration if BB is cofibrant, and the inclusion in 2:Bβ†’AβŠ”Bin_2:B\to A\sqcup B is a cofibration if AA is cofibrant.

Proof

The inclusion in 1:Aβ†’AβŠ”Bin_1:A\to A\sqcup B is a cobase change of the map βŠ₯β†’B\bot \to B, since the square

is a pushout. Hence the map in 1in_1 is a cofibration if BB is cofibrant (since the class of cofibrations is closed under cobase changes by Proposition 1)

A cylinder for an object AA is a quadruple (IA,d 1,d 0,s)(I A,d_1,d_0,s) obtained by factoring the codiagonal βˆ‡ A=(1 A,1 A):AβŠ”Aβ†’A\nabla_A=(1_A,1_A):A\sqcup A\to A as a cofibration (d 1,d 0):AβŠ”Aβ†’IA(d_1,d_0):A\sqcup A \to I A followed by a weak equivalence s:IAβ†’As:I A\to A.

Remark

The notation (IA,d 1,d 0,s)(I A,d_1,d_0,s) introduced by Quillen suggests that a cylinder represents the first two terms of a cosimplicial object

I *A:Ξ”β†’AI^* A:\Delta \to \mathbf{A}

with I 0A=AI^0 A =A and I 1A=IAI^1 A =I A. This is the notion of a cosimplicial framing of an object AA. Beware that if d 0d_0 and d 1d_1 are the maps [0]→[1][0]\to [1] in the category Δ\Delta, then d 0(0)=1d_0(0)=1 and d 1(1)=0d_1(1)=0. We may think of a cylinder (IA,d 1,d 0,s)(I A,d_1,d_0,s) has an oriented object with two faces, with the face d 1:A→IAd_1:A\to IA representing the source of the cylinder and the face d 0:A→IAd_0:A\to IA representing the target.

The transpose of a cylinder (IA,d 1,d 0,s)(I A,d_1,d_0,s) is the cylinder (IA,d 0,d 1,s)(I A,d_0,d_1,s).

Lemma

The maps d 1:A→IAd_1:A\to I A and d 0:A→IAd_0:A\to I A are acyclic, and they are acyclic cofibrations when AA is cofibrant.

Proof

The map d 1d_1 and d 0d_0 are acyclic by three-for-two, since we have sd 1=1 A=sd 0s d_1=1_A= s d_0 and ss is acyclic. If AA is cofibrant, then the inclusions in 1:Aβ†’AβŠ”Ain_1:A\to A\sqcup A and in 2:Aβ†’AβŠ”Ain_2:A\to A\sqcup A are cofibrations by Lemma 2. Hence also the composite d 1=(d 1,d 0)in 1d_1=(d_1,d_0)in_1 and d 0=(d 1,d 0)in 2d_0=(d_1,d_0)in_2 (since the class of cofibrations is closed under composition by Proposition 1).

A mapping cylinder of a map f:Aβ†’Bf:A\to B is obtained by factoring the map (f,1 B):AβŠ”Bβ†’B(f,1_B):A\sqcup B\to B as a cofibration (i A,i B):AβŠ”Bβ†’C(f)(i_A,i_B):A\sqcup B\to C(f) followed by a weak equivalence q B:C(f)β†’Bq_B:C(f)\to B. We then have f=q Bi Af=q_B i_A and q Bi B=1 Bq_B i_B=1_B. The factorisation

f=q Bi A:A→C(f)→Bf=q_B i_A:A\to C(f)\to B

is called the mapping cylinder factorisation of the map ff. The map i Bi_B is acyclic by three-for-two, since q Bq_B is acyclic and we have q Bi B=1 Bq_B i_B=1_B.

Lemma

The maps i A:A→C(f)i_A:A\to C(f) is a cofibration when BB is cofibrant, and the map i B:B→C(f)i_B:B\to C(f) is a cofibration when AA is cofibrant.

Proof

The inclusion in 1:Aβ†’AβŠ”Bin_1:A\to A\sqcup B is a cofibration when BB is cofibrant by Lemma 2. Hence also the composite i A=(i A,i B)in 1i_A=(i_A,i_B)in_1 in this case. The inclusion in 2:Bβ†’AβŠ”Bin_2: B\to A\sqcup B is a cofibration when AA is cofibrant by Lemma 2. Hence also the composite i B=(i A,i B)in 2i_B=(i_A,i_B)in_2 in this case.

If AA is cofibrant, then a mapping cylinder for a map f:A→Bf:A\to B can be constructed from a cylinder (IA,d 1,d 0,s)(I A,d_1,d_0,s) by the following diagram with a pushout square

(2)

We have q Bi A=fq_B i_A=f and q Bi B=1 Bq_B i_B=1_B by construction. The map (i A,i B)(i_A,i_B) is a cofibration by cobase change, since the map (d 1,d 0)(d_1,d_0) is a cofibration. Let us show that q Bq_B is acyclic. For this, it suffices to show that i Bi_B is acyclic by three-for-two, since q Bi B=1 Bq_B i_B=1_B. The two squares of the following diagram are pushout,

(3)

hence also their composite,

(4)

by the lemma here. This shows that the map i Bi_B is a cobase change of the map i 1i_1. But i 1i_1 is an acyclic cofibration by Lemma 3, since AA is cofibrant. It follows that i Bi_B is an acyclic cofibration (since the class of acyclic cofibrations is closed under cobase changes by Proposition 1).

Lemma

(Ken Brown 1) Let E\mathbf{E} be a model category and let F:E cβ†’CF:\mathbf{E}_c\to \mathbf{C} be a functor defined on the sub-category of cofibrant objects and taking its values in a category C\mathbf{C} equipped with class 𝒲\mathcal{W} of weak equivalences containing the units and satisfying three-for-two. If the functor FF takes an acyclic cofibration to a weak equivalence, then it takes an acyclic map to a weak equivalence.

Proof

If f:Aβ†’Bf:A\to B is an acyclic map between cofibrant objects, let us choose a mapping cylinder factorisation f=q Bi A:Aβ†’C(f)β†’Bf=q_B i_A:A\to C(f)\to B. The maps i Ai_A and i Bi_B are cofibrations by Lemma 4, since AA and BB are cofibrant. The map i Bi_B is acyclic by three-for-two, since q Bi B=1 Bq_B i_B=1_B and q Bq_B is acyclic. Thus, F(i B)F(i_B) is a weak equivalence. Hence also the map F(q B)F(q_B) by three-for-two since we have F(q B)F(i B)=F(q Bi B)=F(1 B)=1 FBF(q_B)F(i_B)=F(q_B i_B)=F(1_B)=1_{F B} and 𝒲\mathcal{W} contains the units. The map i Ai_A is acyclic by three-for-two, since f=q Bi Af=q_B i_A and ff and q Bq_B are acyclic. Thus, F(i A)F(i_A) is a weak equivalence, and it follows by three-for-two that F(f)F(f) is a weak equivalence since F(f)=F(q Bi A)=F(q B)F(i A)F(f)=F(q_B i_A)=F(q_B)F(i_A).

Recall that a functor is said to invert a morphism in its domain if it takes this morphism to an isomorphism.

Lemma

(Ken Brown 2)

  • If a functor F:E cβ†’CF:\mathbf{E}_c\to \mathbf{C} inverts acyclic cofibrations, then it inverts weak equivalences.

  • If a functor F:E fβ†’CF:\mathbf{E}_f\to \mathbf{C} inverts acyclic fibrations, then it inverts weak equivalences.

Proof

This follows from Lemma 5, if 𝒲\mathcal{W} is the class of isomorphisms in C\mathbf{C}.

Remark

Ken Brown’s lemma implies that the inclusion π’ž cβˆ©π’² cβŠ†π’² c\mathcal{C}_c\, \cap\, \mathcal{W}_c \subseteq \mathcal{W}_c induces an isomorphism of categories,

(π’ž cβˆ©π’² c) βˆ’1E cβ†’Ho(E c).(\mathcal{C}_c\cap \mathcal{W}_c)^{-1}\mathbf{ E}_c\to Ho(\mathbf{ E}_c).

If (IA,d 1,d 0,s)(I A,d_1,d_0,s) is a cyclinder for AA, then a left homotopy h:f→ lgh:f {\rightarrow}_l g between two maps f,g:A→Xf,g:A\to X is a map h:IA→Xh:I A\to X such that and f=hd 1f= h d_1 and g=hd 0g= h d_0. We shall say that hd 1h d_1 is the source of the homotopy hh and that hd 0h d_0 is the target,

The reverse of an homotopy h:f→ lgh:f {\rightarrow}_l g is the homotopy g→ lfg {\rightarrow}_l f defined by the same map h:IA→Xh:I A\to X but on the transpose cylinder (IA,d 0,d 1,s)(I A,d_0,d_1,s). The homotopy unit f= lff=_l f of a map f:A→Xf:A\to X is defined by the map fp:IA→A→Xf p:I A\to A\to X.

Two maps f,g:Aβ†’Xf,g:A\to X are left homotopic, f∼ lgf\sim_l g, if there exists a left homotopy h:fβ†’ lgh:f\to_l g with domain some cylinder object for AA.

Lemma

The left homotopy relation between the maps A→XA\to X can be defined on a fixed cylinder for AA, when XX is fibrant.

Proof

Let us show that if two maps f,g:A→Xf,g:A\to X are homotopic by virtue of a homotopy defined on a cylinder (IA,i 1,i 0,r)(I A,i_1,i_0,r), then then they are homotopic by virtue of a homotopy defined on any another cylinder (JA,j 1,j 0,s)(J A,j_1,j_0,s). By assumption, we have h(i 1,i 0)=(f,g)h(i_1,i_0)=(f,g) for a map h:IA→Xh: I A\to X. Let us choose a factorisation r=r′u:IA→I′A→Ar=r'u:I A\to I' A\to A with uu an acyclic cofibration and r′r' a fibration. The map r′r' is acyclic by three-for-two, since the maps pp and uu are. Hence the square

has a diagonal filler k:JA→I′Ak:J A\to I' A since the map (j 1,j 0)(j_1,j_0) is a cofibration. But the square

has also a diagonal filler d:I′A→Xd:I' A\to X, since uu is an acyclic cofibration and XX is fibrant. The composite h′=dk:JA→Xh'=d k:J A\to X is a left homotopy f→ lgf\to_l g.

Lemma

If a functor F:E→KF:\mathbf{E}\to \mathbf{K} inverts weak equivalences, then the implication

f∼ lgβ‡’F(f)=F(g)f\sim_l g \quad \Rightarrow \quad F(f)=F(g)

is true for any pair of maps f,g:A→Bf,g:A\to B in E\mathbf{E}. The same result is true for a functor defined on E c\mathbf{E}_c or on E fc\mathbf{E}_{f c}.

Proof

If (IA,d 1,d 0,s)(I A,d_1,d_0,s) is a cylinder for AA, then the map F(s)F(s) is invertible by the assumption of FF since ss is acyclic. Hence we have F(d 1)=F(d 0)F(d_1)=F(d_0), since we have

F(s)F(d 1)=F(sd 1)=F(1 A)=F(sd 0)=F(s)F(d 0).F(s) F(d_1)=F(s d_1)=F(1_A)=F(s d_0)=F(s)F(d_0).

If h:IA→Xh:I A\to X is a homotopy between two map f,g:A→Xf,g:A\to X, then

F(f)=F(hd 1)=F(h)F(d 1)=F(h)F(d 0)=F(hd 0)=F(g).F(f)=F(h d_1)=F(h)F(d_1)=F(h)F(d_0)=F(h d_0)=F(g).

Let us now consider the case where the domain of the functor FF is the category E c\mathbf{E}_c. Observe that if AA is cofibrant, then so is the object IAI A in a cylinder (IA,d 1,d 0,s)(I A,d_1,d_0,s), since the map (d 1,d 0):AβŠ”Aβ†’IA(d_1,d_0):A\sqcup A \to IA is a cofibration and the object AβŠ”AA\sqcup A is cofibrant (since a coproduct of cofibrant objects is cofibrant by Corollary 1). Hence the cylinder (IA,d 1,d 0,s)(I A,d_1,d_0,s) belongs to the category E c\mathbf{E}_{c} and the proof above can be repeated in this case. Let us now consider the case where the domain of the functor FF is the category E fc\mathbf{E}_{f c}. In this case the left homotopy relation between the maps Aβ†’BA\to B can defined on a fixed cylinder for AA by Lemma 7, since BB is fibrant. A cylinder for AA can be constructed by factoring the map βˆ‡ A:AβŠ”Aβ†’A\nabla_A:A\sqcup A\to A as an acyclic cofibration (d 1,d 0):AβŠ”Aβ†’IA(d_1,d_0):A\sqcup A \to I A followed by a fibration s:IAβ†’As:I A \to A. The object IAI A is cofibrant, since AA is cofibrant. But IAI A is also fibrant, since ss is a fibration and AA is fibrant. Hence the cylinder (IA,d 1,d 0,p)(I A,d_1,d_0,p) belongs to the category E fc\mathbf{E}_{f c} and the proof above can be repeated.

The left homotopy relation on the set of maps A→XA\to X is reflexive and symmetric. We shall denote by π l(A,X)\pi^l(A,X) the quotient of the set Hom(A,X)Hom(A,X) by the equivalence relation generated by the left homotopy relation. The relation is compatible with composition on the left: the implication

f∼ lgβ‡’pf∼ lpgf\sim_l g \quad \Rightarrow \quad p f\sim_l p g

is true for every maps f,g:A→Xf,g:A\to X and p:X→X′p:X\to X'. This defines a functor

Ο€ l(A,βˆ’):Eβ†’Set.\pi^l(A,-):\mathbf{E}\to \mathbf{Set}.
Lemma

The left homotopy relation between the maps A→XA\to X is an equivalence when AA is cofibrant.

Proof

Cylinders for AA can be composed as cospan. More precisely, the composite of a cylinder (IA,i 1,i 0,r)withacylinder(I A, i_1,i_0,r) with a cylinder (J A,j_1,j_0,s)isthecylinder is the cylinder (K A,k_1,k_0,t)$ defined by the following diagram with a pushout square,

The map t:KA→At:K A\to A is defined by the condition tin 1=rt in_1=r and tin 2=st in_2 =s. Let us show that tt is acyclic. The map j 1:A→JAj_1:A\to J A is an acyclic cofibration by Lemma 3, since AA is cofibrant. It follows that in 1in_1 is an acyclic cofibration, since it is a cobase change of j 1j_1. Hence the map tt is acyclic by three-for-two, since we have tin 1=st in_1=s and the maps in 1in_1 and ss are acyclic. It remains to show that the map (k 0,k 1)(k_0,k_1) is a cofibration. For this, we can use the following diagram with a pushout square,

The map kk in this diagram is a cobase change of the map (i 1,i 0)βŠ”(j 1,j 0)(i_1,i_0)\sqcup (j_1,j_0). But the map (i 1,i 0)βŠ”(j 1,j 0)(i_1,i_0)\sqcup (j_1,j_0) is a cofibration, since the maps (i 1,i 0)(i_1,i_0) and (j 1,j 0)(j_1,j_0) are cofibrations. This proves that the map kk is a cofibration. It follows that the composite (k 1,k 0)=k(in 1βŠ”in 3)(k_1,k_0)=k(in_1\sqcup in_3) is a cofibration, since the map in 1βŠ”in 3in_1\sqcup in_3 is a cofibration by Lemma 2. We have proved that (KA,k 1,k 0,r)(K A,k_1,k_0,r) is a cylinder for AA. We can now prove that the left homotopy relation on the set of maps Aβ†’XA\to X is transitive. Let f 1,f 2f_1,f_2 and f 3f_3 be three maps Aβ†’XA\to X and suppose that h 1:IAβ†’Xh_1:I A\to X is a left homotopy f 1β†’ lf 2f_1\to_l f_2, and h 2:JAβ†’Xh_2:J A\to X is a left homotopy f 2β†’ lf 3f_2\to_l f_3. There is then a unique map h 3:KAβ†’Xh_3: K A\to X such that h 3in 1=h 1h_3 in_1 =h_1 and h 3in 2=h 2h_3 in_2 =h_2, since h 1i 0=f 2=h 2j 1h_1 i_0 =f_2= h_2 j_1. This defines a homotopy h 3:f 1β†’ lf 3h_3:f_1 \to_l f_3, since h 3k 1=h 3in 1i 1=h 1i 1=f 1h_3 k_1= h_3 in_1 i_1 =h_1 i_1=f_1 and h 3k 0=h 3in 2j 0=h 2j 0=f 3h_3 k_0 =h_3 in_2 j_0= h_2 j_0 =f_3.

Lemma

(Covering homotopy theorem) Let AA be cofibrant, let f:X→Yf:X\to Y be a fibration, let a:A→Xa:A\to X, and let h:IA→Yh: I A\to Y be a left homotopy with source fa:A→Yf a:A\to Y. Then there exists a left homotopy H:IA→XH: I A\to X with source aa such that fH=h f H=h.

Proof

The square

has a diagonal filler H:IA→XH: I A\to X, since d 1d_1 is an acyclic cofibration by Lemma 3 and ff is a fibration.

Lemma

(Homotopy lifting lemma) Let f:X→Yf:X\to Y be an acyclic fibration, let a:A→Xa:A\to X and b:A→Xb:A\to X, and let h:IA→Yh: I A\to Y be a left homotopy fa→ lfbf a \to_l f b. Then there exists a map H:IA→XH: I A\to X defining a left homotopy a→ lba \to_l b such that fH=h f H=h.

Proof

The square

has a diagonal filler H:IA→XH: I A\to X, since (d 1,d 0)(d_1,d_0) is a cofibration and ff is an acylic fibration.

Lemma

If AA is cofibrant, then the functor Ο€ l(A,βˆ’):Eβ†’Set\pi^l(A,-):\mathbf{E}\to \mathbf{Set} inverts acyclic maps between fibrant objects.

Proof

Let us first show that the functor Ο€ l(A,βˆ’)\pi^l(A,-) inverts acyclic fibrations. If f:Xβ†’Yf:X\to Y is an acyclic fibration and y:Aβ†’Yy:A\to Y, then the square

has a diagonal filler, since AA is cofibrant and ff is an acyclic fibration. Hence there exists a map x:Aβ†’Xx:A\to X such that fx=yf x=y. This shows that the map Ο€(A,f)\pi(A,f) is surjective. Let us show that it is injective. If a,b:Aβ†’Xa,b:A\to X and fa∼ lfbf a\sim_l f b, then a∼ lba\sim_l b by the homotopy lifting lemma 11. We have proved that the map Ο€ l(A,f)\pi^l(A,f) is bijective. It then follows from Ken Brown’s lemma 6 that that the functor Ο€ l(A,βˆ’)\pi^l(A,-) inverts acyclic maps between fibrant objects.

Lemma

A map which is left homotopic to an acyclic map is acyclic.

Proof

Let h:IA→Bh: I A \to B be a left homotopy between two maps hd 1=uh d_1=u and hd 0=vh d_0=v. If vv is acyclic, then so is hh by three-for-two, since d 0d_0 is acyclic. Hence the composite u=hd 1u=h d_1 is acyclic by three-for-two, since d 1d_1 is acyclic.

Path objects and right homotopies

Lemma*

(Dual to Lemma 2) The projection pr 1:X×Y→Xpr_1:X\times Y \to X is a fibration if YY is fibrant, and the projection pr 2:X×Y→Ypr_2:X\times Y \to Y is a fibration if XX is fibrant

A path object for an object XX is a quadruple (PX,βˆ‚ 1,βˆ‚ 0,Οƒ)(P X,\partial_1,\partial_0,\sigma) obtained by factoring the diagonal Ξ” X=(1 X,1 X):Xβ†’XΓ—X\Delta_X=(1_X,1_X):X\to X\times X as a weak equivalence Οƒ:Xβ†’PX\sigma:X\to P X followed by a fibration (βˆ‚ 1,βˆ‚ 0):PXβ†’XΓ—X(\partial_1,\partial_0):P X\to X\times X.

Remark

The notation (PX,βˆ‚ 1,βˆ‚ 0,Οƒ)(P X,\partial_1,\partial_0,\sigma) introduced by Quillen suggests that a path object represents the first two terms of a simplicial object

P *X:Ξ”β†’AP_* X:\Delta \to \mathbf{A}

with P 0X=XP_0 X = X and P 1X=PXP_1 X= P X. This is the notion of simplicial framing of an object XX. Beware that the source of a 1-simplex ff in a simplicial set SS is the vertex βˆ‚ 1(f)∈S 0\partial_1(f)\in S_0, and that its target is the vertex βˆ‚ 0(f)∈S 0\partial_0(f)\in S_0.

The transpose of a path object (PX,βˆ‚ 1,βˆ‚ 0,Οƒ)(P X,\partial_1,\partial_0,\sigma) is the path object (PX,βˆ‚ 0,βˆ‚ 1,Οƒ)(P X,\partial_0,\partial_1,\sigma).

Lemma*

(Dual to Lemma 3) The maps βˆ‚ 1:PXβ†’X\partial_1:P X \to X and βˆ‚ 0:PXβ†’X\partial_0:P X\to X are acyclic, and they are are acyclic fibrations when XX is fibrant.

A mapping path object of a map f:X→Yf:X\to Y is obtained by factoring the map
(1 X,f):X→X×Y(1_X,f):X\to X\times Y as a weak equivalence i X:X→P(f)i_X: X\to P(f) followed by a fibration (p X,p Y):P(f)→X×Y(p_X,p_Y):P(f)\to X\times Y. By construction, we have f=p Yi Xf=p_Y i_X and p Xi X=1 Xp_X i_X=1_X. The factorisation

f=p Yi X:X→P(f)→Yf=p_Y i_X:X\to P(f)\to Y

is called the mapping path factorisation of the map ff. The map p Xp_X is acyclic by three-for-two, since i Xi_X is acyclic and p Xi X=1 Xp_X i_X=1_X.

Lemma*

(Dual to Lemma 4) The maps p X:PX→Xp_X:P X\to X and p Y:PY→Yp_Y: P Y\to Y are fibrations when XX and YY are fibrant.

If YY is fibrant, then a mapping path object for a map f:Xβ†’Yf:X\to Y can be constructed from a path object (PY,βˆ‚ 1,βˆ‚ 0,Οƒ)(P Y, \partial_1,\partial_0,\sigma) for YY by the following diagram with a pullback square,

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We have p Xi X=1 Xp_X i_X=1_X and p Yi X=fp_Y i_X=f by construction. The map (p X,p Y)(p_X,p_Y) is a fibration by base change, since the map (βˆ‚ 1,βˆ‚ 0)(\partial_1,\partial_0) is. Let us show that the map i Xi_X is acyclic. For this, it suffices to show that p Xp_X is acyclic by three-for-two, since p Xi X=1 Xp_X i_X=1_X. The two squares of the following diagram are cartesian,

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hence also their composite,

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by the lemma here. Hence the map p Xp_X is a base change of the map βˆ‚ 0\partial_0. But βˆ‚ 1\partial_1 is an acyclic fibration by Lemma 15, since YY is fibrant. This shows that the map p Xp_X is acyclic (since the base change of an acyclic fibration is an acyclic fibration by Proposition 1).

If (PX,βˆ‚ 1,βˆ‚ 0,Οƒ)(P X,\partial_1,\partial_0,\sigma) is a path object for XX, then a right homotopy h:fβ†’ rgh:f {\rightarrow}_r g between two maps f,g:Aβ†’Xf,g:A\to X is defined to be a map h:Aβ†’PXh:A\to PX such that f=βˆ‚ 1hf=\partial_1 h and g=βˆ‚ 0hg=\partial_0 h. We shall say that βˆ‚ 1h\partial_1 h is the source of the homotopy hh and that βˆ‚ 0h\partial_0 h is its target .

The reverse of hh is the homotopy th:gβ†’ rh{}^t h:g {\rightarrow}_r h defined by the same map h:Aβ†’PXh: A\to P X but on the transpose path object (PX,βˆ‚ 0,βˆ‚ 1,Οƒ)(P X,\partial_0,\partial_1,\sigma). The unit homotopy f= rff=_r f of a map f:Aβ†’Xf:A\to X is the map Οƒf:Aβ†’Xβ†’PX\sigma f : A\to X \to P X.

Two maps f,g:Aβ†’Xf,g:A\to X are right homotopic, f∼ rgf\sim_r g, if there exists a right homotopy h:fβ†’ rgh:f\rightarrow_r g with codomain a path object for XX.

Lemma*

(Dual to Lemma 7) The right homotopy relation between the maps A→XA\to X can be defined on a fixed path object for XX when AA is cofibrant.

We shall denote by Ο€ r(A,X)\pi^r(A,X) the quotient of Hom(A,X)Hom(A,X) by the equivalence relation generated by the right homotopy relation. The right homotopy relation is compatible with composition on the right:

f∼ rgβ‡’fu∼ rguf\sim_r g \quad \Rightarrow \quad f u\sim_r g u

for every map u:Aβ€²β†’Au:A'\to A. We thus obtain a functor

Ο€ r(βˆ’,X):E oβ†’Set.\pi^r(-,X):\mathbf{E}^o\to \mathbf{Set}.
Lemma*

(Dual to Lemma 9) The right homotopy relation between the maps A→XA\to X is an equivalence when XX is fibrant.

Lemma*

(Homotopy extension theorem, dual to Lemma 10). Let XX be fibrant, let u:A→Bu:A\to B be a cofibration, let b:B→Xb:B\to X, and let h:A→PXh: A \to P X be a right homotopy with source bu:A→Xb u:A\to X. Then there exists a right homotopy H:B→PXH: B \to P X with source bb such that Hu=h H u =h.

Lemma*

(Homotopy prolongation lemma, dual to Lemma 11) Let XX be fibrant, let u:A→Bu:A\to B be an acyclic cofibration, let a:B→Xa:B\to X and b:B→Xb:B\to X, and let h:A→PXh: A\to P X be a right homotopy au→ rbua u\to_r b u. Then there exists a map H:B→PXH: B \to P X defining a right homotopy a→ rba \to_r b such that Hu=hH u =h.

Lemma*

(Dual to Lemma 12) If XX is fibrant, then the functor Ο€ r(βˆ’,X):E oβ†’Set\pi^r(-,X):\mathbf{E}^o\to \mathbf{Set} inverts acyclic maps between cofibrant objects.

Lemma*

(Dual to Lemma 13) A map which is right homotopic to an acyclic map is acyclic

Double homotopies

If (IA,d 1,d 0,s)(I A,d_1,d_0,s) is a cylinder object for AA and (PX,βˆ‚ 1,βˆ‚ 0,Οƒ)(P X, \partial_1,\partial_0,\sigma) is a path object for XX, then a map H:IAβ†’PXH:I A\to P X is double homotopy between four maps Aβ†’XA\to X,

The four corners of the square are representing maps A→XA\to X, the horizontal sides are representing left homotopies, and the vertical sides are representing right homotopies.

Lemma

If XX is fibrant, then every open box of three homotopies, opened at the top, between four maps f ij:A→Xf_{ij}:A\to X,

can be filled by a double homotopy H:IAβ†’PXH:I A\to P X (ie βˆ‚ 0H=h 0\partial_0 H=h_0, Hd 1=v 1H d_1=v_1 and Hd 0=v 0H d_0=v_0).

Proof

The square

has a diagonal filler H:IAβ†’PXH:I A \to P X, since (d 1,d 0)(d_1,d_0) is a cofibration and βˆ‚ 0\partial_0 is an acyclic fibration by Lemma 15.

Lemma

If XX is fibrant, then the right homotopy relation on the set of maps A→XA\to X implies the left homotopy relation. Dually, if AA is cofibrant, then the left homotopy relation implies the right homotopy relation. Hence the two relations coincide when AA is cofibrant and XX is fibrant.

Proof

Let h:f→ rgh:f\to_r g be a right homotopy between two maps A→XA\to X. By Lemma 23, the open box of homotopies

can be filled by a double homotopy H:IAβ†’PXH:I A \to P X, since XX is fibrant. This yields a left homotopy βˆ‚ 1H:fβ†’ lg\partial_1 H :f\to_l g.

When AA is cofibrant and XX is fibrant, then two maps f,g:Aβ†’Xf,g:A\to X are said to be homotopic if they are left (or right) homotopic; we shall denote this relation by f∼gf\sim g. We shall denote by Ο€(A,X)\pi(A,X) the quotient of the set Hom(A,X)Hom(A,X) by the homotopy relation:

Ο€(A,X)=Hom(A,X)/∼.\pi(A,X)=Hom(A,X)/\sim.

By definition, Ο€(A,X)=Ο€ r(A,X)=Ο€ l(A,X).\pi(A,X)=\pi^r(A,X)=\pi^l(A,X). This defines a functor

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π:E c o×E f→Set,\pi:\mathbf{E}_c^o\,\times\, \mathbf{E}_f\to \mathbf{Set},

that is, a distributor π:E c⇒E f\pi:\mathbf{E}_c\Rightarrow \mathbf{E}_f.

The homotopy category

The homotopy relation ∼\sim is compatible with the composition law

Hom(Y,Z)Γ—Hom(X,Y)β†’Hom(X,Z)Hom(Y,Z)\,\times \, Hom(X,Y) \to Hom(X,Z)

if X,YX,Y and ZZ are fibrant-cofibrant objects. It thus induces a composition law

Ο€(Y,Z)Γ—Ο€(X,Y)β†’Ο€(X,Z)\pi(Y,Z)\,\times \, \pi(X,Y)\to \pi(X,Z)

and this defines a category Ο€E fc\pi\mathbf{E}_{f c} if we put

(Ο€E fc)(X,Y)=Ο€(X,Y)(\pi\mathbf{E}_{f c})(X,Y)=\pi(X,Y)

for X,Y∈E fcX,Y\in \mathbf{E}_{f c}.

Definition

We say that a map in E cf\mathbf{E}_{cf} is a homotopy equivalence if it is invertible in the category Ο€E fc\pi\mathbf{E}_{f c}.

It is obvious from the definition that the class of homotopy equivalences has the three-for-two property.

A map f:Xβ†’Yf:X\to Y in E fc\mathbf{E}_{f c} is a homotopy equivalence iff there exists a map g:Yβ†’Xg:Y\to X such that gf∼1 Xg f\sim 1_X and fg∼1 Yf g\sim 1_Y.

Let us denote by H:E→Ho(E)H:\mathbf{E}\to \mathrm{Ho}(\mathbf{E}), H′:E fc→Ho(E fc)H':\mathbf{E}_{f c}\to \mathrm{Ho}(\mathbf{E}_{f c}) and P:E fc→π(E fc)P:\mathbf{E}_{f c}\to \pi(\mathbf{E}_{f c}), the canonical functors. The functor H′H' takes homotopic maps u,v:X→Yu,v:X\to Y to the same morphism by Lemma 8. It follows that there is a unique functor U:πE fc→Ho(E fc)U:\pi\mathbf{E}_{f c}\to \mathrm{Ho}(\mathbf{E}_{f c}) such that the following triangle commutes,

We shall prove in Theorem 1 below that the functor UU is an isomorphism of categories. We will need a lemma:

Lemma

If Ξ£\Sigma is a set of morphisms in a category A\mathbf{A}, then the localisation functor H:Aβ†’Ξ£ βˆ’1AH:\mathbf{A} \to \Sigma^{-1}\mathbf{A} is an epimorphism of category. Moreover, for any category M\mathbf{M}, the functor

H *:[Ξ£ βˆ’1A,M]β†’[A,M]H^*:\mathbf{[}\Sigma^{-1}\mathbf{A},\mathbf{M}\mathbf{]} \to \mathbf{[}\mathbf{A},\mathbf{M}\mathbf{]}

induced by HH is fully faithful and it induces an isomorphism between the category [Ξ£ βˆ’1A,M]\mathbf{[}\Sigma^{-1}\mathbf{A},\mathbf{M}\mathbf{]} and the full subcategory of [A,M]\mathbf{[}\mathbf{A},\mathbf{M}\mathbf{]} spanned by the functors Aβ†’M\mathbf{A}\to \mathbf{M} inverting the elements of Ξ£\Sigma. In particular, two functors Q 0,Q 1:Aβ†’MQ_0,Q_1:\mathbf{A}\to \mathbf{M} are isomorphic iff the functors Q 0HQ_0H and Q 1HQ_1H are isomorphic.

Proof

Left to the reader.

Theorem

The functor U:πE fc→Ho(E fc)U:\pi\mathbf{E}_{f c}\to \mathrm{Ho}(\mathbf{E}_{f c}) defined above is an isomorphism of categories. A map in E fc\mathbf{E}_{fc} is acyclic iff it is a homotopy equivalence.

Proof

(Mark Hovey) Let us first show that if a map f:X→Yf:X\to Y in the category E fc\mathbf{E}_{f c} is acyclic, then it is a homotopy equivalence. The map π(A,f):π(A,X)→π(A,Y)\pi(A,f):\pi(A,X)\to \pi(A,Y) is bijective for every cofibrant object AA by Lemma 12, since ff is an acyclic map between fibrant object. It follows that ff is inverted by the Yoneda functor

πE fc→[πE fc o,Sets].\pi\mathbf{E}_{f c}\to [\pi\mathbf{E}_{f c}^o, \mathbf{Sets}].

But the Yoneda functor is conservative, since it is fully faithful by Yoneda. It follows that ff is invertible in the category πE fc\pi\mathbf{E}_{f c}. This shows that ff is a homotopy equivalence. Let us now prove that the functor UU is an isomorphism of categories. The canonical functor P:E fc→π(E fc)P:\mathbf{E}_{f c}\to \pi(\mathbf{E}_{f c}) inverts acyclic maps by what we just proved. Hence there is a unique functor T:Ho(E fc)→πE fcT:\mathrm{Ho}(\mathbf{E}_{f c})\to \pi\mathbf{E}_{f c} such that the following triangle commutes,

Let us show that the functors UU and TT are mutually inverses. Let us observe that the functor PP is an epimorphism, since it is surjective on objects and full. But we have TUP=THβ€²=PT U P=T H' =P. It follows that we have TU=IdT U =Id, since PP is an epimorphism. On the other hand, the functor Hβ€²H' is an epimorphism by Lemma 25, since it is a localisation. It follows that we have UT=IdU T=Id, since we have UTHβ€²=UP=Hβ€²U T H'=U P =H'. We have proved that the functor UU and TT are mutually inverse. Let us now show that a homotopy equivalence f:Xβ†’Yf:X\to Y is acyclic. We shall first consider the case where ff is a fibration. There exists a map g:Xβ†’Yg:X\to Y such that fg∼1 Yf g\sim 1_Y and gf∼1 Xg f\sim 1_X, since ff is a homotopy equivalence by assumption. There is then a left homotopy h:fgβ†’ l1 Yh:f g\to_l 1_Y defined on a cylinder IYI Y. I then follows from the covering homotopy theorem 10 that there exists a left homotopy H:IYβ†’XH:I Y\to X such that Hi 0=gH i_0=g, since ff is a fibration. Let us put q=Hi 1q=H i_1. Then fq=1 Yf q =1_Y and q∼gq\sim g. Thus, qf∼gf∼1 Xq f\sim g f \sim 1_X, since the homotopy relation is a congruence. Hence the map qfq f is acyclic by Lemma 1, since 1 X1_X is acyclic. But the map f:Xβ†’Yf:X\to Y is retract of the map qf:Xβ†’Xq f:X\to X, since the following diagram commutes and we have fq=1 Yf q =1_Y,

It then follows by Lemma 1 that ff is a weak equivalence. The implication (⇐\Leftarrow) is proved in the case where ff is a fibration. In the general case, let us choose a factorisation f=pu:Xβ†’Eβ†’Yf=p u:X\to E\to Y with uu an acyclic cofibration and pp a fibration. The map uu is a homotopy equivalence by the first part of the proof, since it is acyclic. Thus, pp is a homotopy equivalence by three-for-two for homotopy equivalences, since ff is a homotopy equivalence by assumption. Thus, pp is acyclic since it is a fibration. Hence the composite f=puf= p u is acyclic by three-for-two.

Definition

A fibrant replacement of an object XX is a fibrant object X′X' together with an acyclic cofibration X→X′X\to X'. A cofibrant replacement of an object XX is a cofibrant object X′X' together with an acyclic fibration X′→XX'\to X.

A fibrant replacement of XX is obtained by factoring the map Xβ†’βŠ€X\to \top as an acyclic cofibration i X:Xβ†’RXi_X:X\to R X followed by a fibration RXβ†’βŠ€R X\to \top. If XX is fibrant, we can take RX=XR X =X and i X=1 Xi_X=1_X. Similarly, a cofibrant replacement of XX is obtained by factoring the map βŠ₯β†’X\bot \to X as a cofibration βŠ₯β†’QX\bot \to Q X followed by an acyclic fibration q X:QXβ†’Xq_X:Q X\to X. If XX is cofibrant, we can take QX=XQ X =X and q X=1 Xq_X=1_X.

A fibrant replacement of a cofibrant object is cofibrant; it is thus fibrant-cofibrant. Dually a cofibrant replacement of a fibrant object is fibrant-cofibrant.

The composite q Xi X:QX→RXq_X i_X:Q X\to R X is acyclic. It can thus be factored as an acyclic cofibration j X:QX→WXj_X:Q X\to W X followed by an acyclic fibration p X:WX→RXp_X:W X\to R X,

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The object WXW X is a fibrant-cofibrant replacement of the object XX. If XX is cofibrant, we can take WX=RXW X=R X, and if XX is fibrant, we can take WX=QXW X =Q X (in which case we have WX=XW X =X, when XX is fibrant-cofibrant).

Lemma

For every map u:X→Yu:X\to Y, there exits two maps R(u):RX→RYR(u):R X\to R Y and W(u):WX→WYW(u):W X\to W Y fitting in the following commutative diagram,

(9)

The map R(u)R(u) is unique up to a right homotopy, and the map W(u)W(u) unique up to homotopy. The maps R(u)R(u) and W(u)W(u) are acyclic if uu is acyclic. If u:Xβ†’Yu:X\to Y and v:Yβ†’Zv:Y\to Z, then R(vu)∼ rR(v)R(u)R(v u)\sim_r R(v)R(u) and W(vu)∼W(v)W(u)W(v u)\sim W(v)W(u).

Proof

The map R(u):RX→RYR(u):R X\to R Y exists because RYR Y is fibrant and the map i Xi_X is an acyclic cofibration. After choosing R(u)R(u), we can choose the map W(u):WX→WYW(u):W X\to W Y, since the object WXW X is cofibrant and the map p Yp_Y is an acyclic fibration. The map R(u)R(u) is unique up to a right homotopy by Lemma 20, since RYR Y is fibrant. Hence also the composite R(u)p X:WX→RYR(u)p_X:W X\to R Y, since the right homotopy relation is compatible with composition on the right. But this composite is also unique up to a left homotopy by Lemma 24, since the object RYR Y is fibrant. It follows by Lemma 11 that the map W(u)W(u) is unique up to a left homotopy, since p Yp_Y is an acyclic fibration. The first statement is proved. Let us prove the second statement. If uu is a weak equivalence, then so are the maps R(u)R(u) and W(u)W(u) by three-for-two, since the vertical sides of the diagram (9) are acyclic. Let us prove the third statement. If we compose horizontally the following diagram,

we obtain the following diagram,

which should be compared with the diagram,

It then follows from the homotopy uniqueness that we have R(vu)∼ rR(v)R(u)R(v u)\sim_r R(v)R(u) and W(vu)∼W(v)W(u)W(v u)\sim W(v)W(u).

Theorem

The four canonical functors in the following square are equivalences of categories,

A map u:A→Bu:A\to B is acyclic iff it is inverted by the canonical functor H:E→Ho(E)H:\mathbf{ E}\to \mathrm{Ho}(\mathbf{ E}).

Proof

Let us first show that the diagonal functor F:Ho(E fc)β†’Ho(E)F:\mathrm{Ho}(\mathbf{ E}_{f c}) \to \mathrm{Ho}(\mathbf{ E}) is an equivalence of categories. We shall use the following commutative square of functors,

(10)

where the top functor is the inclusion and the vertical functor are the canonical functors. We have F(Hβ€²(u))=H(u)F(H'(u))=H(u) for every map u∈E fcu\in \mathbf{ E}_{f c}. For every map u:Xβ†’Yu:X\to Y in E\mathbf{ E}, the map W(u):WXβ†’WYW(u):W X\to W Y is well defined up to homotopy by Lemma 26. Hence the morphism Hβ€²W(u):WXβ†’WYH' W(u):W X \to W Y is independant of the choice of W(u)W(u) by Lemma 8. If u:Xβ†’Yu:X\to Y and v:Yβ†’Zv:Y\to Z, then W(vu)∼W(v)W(u)W(v u)\sim W(v)W(u) by Lemma 26. Thus, Hβ€²W(vu)=Hβ€²W(v)Hβ€²W(u)H' W(v u)= H' W(v) H' W(u). This defines a functor Hβ€²W:Eβ†’Ho(E fc)H' W: \mathbf{ E} \to \mathrm{Ho}(\mathbf{ E}_{f c}). The functor Hβ€²WH' W takes a weak equivalence to an isomorphism, since WW takes a weak equivalence to a weak equivalence by Lemma 26. It follows that there is a unique functor G:Ho(E)β†’Ho(E fc)G: \mathrm{Ho}(\mathbf{ E})\to \mathrm{Ho}(\mathbf{ E}_{f c}) such that the following square commutes,

(11)

By construction, have GH(u)=Hβ€²W(u)G H(u)=H' W(u) for every map u:Xβ†’Yu:X\to Y. Let us show that the functors FF and GG are quasi-inverses. Observe first that we have W(u)∼uW(u) \sim u for a map uu in E fc\mathbf{E}_{f c} (we could even suppose that W(u)=uW(u)=u); thus GFHβ€²(u)=GH(u)=Hβ€²W(u)=Hβ€²(u)G F H'(u)=G H(u)=H' W(u)=H'(u) in this case. This shows that GFHβ€²=Hβ€²G F H' =H' and it follows that GF=IdG F =Id, since the functor Hβ€²H' is epic by Lemma 25. Let us now show that the composite FGF G is isomorphic to the identity of the category Ho(E)\mathrm{Ho}(\mathbf{E}). For this, it suffices to show that the functors FGHF G H and HH are isomorphic by Lemma 25, since the functor HH is a localisation. We have FGH(X)=WXF G H(X)= W X for every object XX. Moreover, for every map u:Xβ†’Yu:X\to Y, the map FGH(u):FGH(X)β†’FGH(Y)F G H(u):F G H(X)\to F G H(Y) is equal to the map HW(u):WXβ†’WYH W(u):W X \to W Y, since FGH(u)=FHβ€²W(u)=HW(u)F G H(u)= F H' W (u)= H W (u). The maps i Xβ†’RXi_X\to R X and p X:WXβ†’RXp_X:W X \to R X are invertible in the category Ho(E)\mathrm{Ho}(\mathbf{ E}) since they are acyclic. Let us put ΞΈ X=H(p X) βˆ’1H(i X):Xβ†’WX\theta_X=H(p_X)^{-1}H(i_X):X\to W X in the category Ho(E)\mathrm{Ho}(\mathbf{ E}). This defines a natural isomorphim ΞΈ:Hβ†’FGH\theta: H\to F G H since the image by HH of the commutative diagram (9) is a commutative diagram

(11)

It then follows from Lemma 25 that there is a natural isomorphism Id→FGId \to F G. We have proved that the functor FF is an equivalence of categories. The proof that the canonical functor Ho(E f)→Ho(E)\mathrm{Ho}(\mathbf{ E}_{f}) \to \mathrm{Ho}(\mathbf{ E}) is an equivalence of categories is similar except that it uses the fibrant replacement RR instead of the fibrant-cofibrant replacement WW. It then follows by duality that the canonical functor Ho(E c)→Ho(E)\mathrm{Ho}(\mathbf{ E}_{c}) \to \mathrm{Ho}(\mathbf{ E}) is an equivalence of categories. Let us now prove that a map u:X→Yu:X\to Y is acyclic iff it is inverted by the canonical functor H:E→Ho(E)H:\mathbf{ E}\to \mathrm{Ho}(\mathbf{ E}). The implication (⇒\Rightarrow) is clear by definition of the functor HH. Conversely, if H(u)H(u) is invertible, let us show that uu is acyclic. The square (eq:77) shows that the map HW(u)H W(u) is also invertible since the vertical sides of the square are invertible. But we have HW(u)=FH′W(u)H W(u)=F H' W(u). Hence the map H′W(u)H' W(u) is invertible in the category πE cf\pi\mathbf{ E}_{cf}, since the functor FF is an equivalence. This shows that WuW u is a homotopy equivalence. Thus, WuW u is acyclic by Theorem 1. It then follows by three-for-two that uu is acyclic, since the vertical maps in the diagram (9) are acyclic.

Corollary

If the category E\mathbf{E} is locally small, then so is the category Ho(E)\mathrm{Ho}(\mathbf{ E})

Proof

The category E fc\mathbf{E}_{f c} is locally small since it is a subcategory of E\mathbf{E}. Hence the category Ο€E fc\pi\mathbf{E}_{f c} is locally small, since it is a quotient of the category E fc\mathbf{E}_{f c} by a congruence relation. It follows by Theorem 1 that the category Ho(E fc)\mathrm{Ho}(\mathbf{E}_{f c}) is locally small. It then follows by Theorem 2 that the category Ho(E)\mathrm{Ho}(\mathbf{E}) is locally small.

Corollary

A map between cofibrant objects u:A→Bu:A\to B is acyclic iff the map π(u,X):π(B,X)→π(A,X)\pi(u,X):\pi(B,X)\to \pi(A,X) is bijective for every fibrant-cofibrant object XX.

Proof

The implication (⇒\Rightarrow) was proved in Lemma 21. Conversely, if the map π(u,X):π(B,X)→π(A,X)\pi(u,X):\pi(B,X)\to \pi(A,X) is bijective for every fibrant-cofibrant object XX, let us show that uu is acyclic. For this, let us choose fibrant replacements i A:A→RAi_A:A\to R A and i B:B→RBi_B:B\to R B of the objects AA and BB, together with a map R(u):RA→RBR(u):R A\to R B fitting in a commutative square,

(12)

The horizontal maps of the following square are bijective by Lemma 21, since the maps i Ai_A and i Bi_B are acyclic,

It follows that the map Ο€(R(u),X)\pi(R(u),X) is bijective, since the map Ο€(u,X)\pi(u,X) is bijective. It then follows by using the Yoneda embedding that the map R(u)R(u) is invertible in the category Ο€E fc\pi\mathbf{E}_{f c}. This shows that the map R(u)R(u) is a acyclic by Theorem 1. It then follows by three-for-two applied to the square (12) that the map uu is acyclic.

Determination

Proposition

A model structure (π’ž,𝒲,β„±)(\mathcal{C},\mathcal{W},\mathcal{F}) in a category E\mathbf{E} is determined by any two of its three classes. A map f:Xβ†’Yf:X\to Y is a weak equivalence iff it admits a factorisation f=pu:Xβ†’Eβ†’Yf=p u:X\to E\to Y with uu an acyclic cofibration and pp an acyclic fibration.

Proof

The class π’žβˆ©π’²\mathcal{C}\,\cap\, \mathcal{W} determines the class β„±\mathcal{F}, since the pair (π’žβˆ©π’²,β„±)( \mathcal{C}\,\cap\, \mathcal{W},\mathcal{F}) is a weak factorisation system. Hence the pair (π’ž,𝒲)(\mathcal{C},\mathcal{W}) determines the class β„±\mathcal{F}. Dually, the pair (β„±,𝒲)(\mathcal{F},\mathcal{W}) determines the class π’ž\mathcal{C}. Let us show that the pair (π’ž,β„±)(\mathcal{C},\mathcal{F}) determines the class 𝒲\mathcal{W}. Obviously, the pair (π’ž,β„±)(\mathcal{C},\mathcal{F}) determines the pair (π’žβˆ©π’²,β„±βˆ©π’²)(\mathcal{C}\,\cap\, \mathcal{W},\mathcal{F}\,\cap\, \mathcal{W}) since π’žβˆ©π’²= β‹”β„±\mathcal{C}\,\cap\, \mathcal{W}={}^\pitchfork\mathcal{F} and β„±βˆ©π’²=π’ž β‹”\mathcal{F}\,\cap\, \mathcal{W}=\mathcal{C}^\pitchfork. Hence it suffices to show that a map f:Xβ†’Yf:X\to Y is a weak equivalence iff it admits a factorisation f=pu:Xβ†’Eβ†’Yf=p u:X\to E\to Y with uu an acyclic cofibration and pp an acyclic fibration. The implication (⇐\Leftarrow) is clear, since the class 𝒲\mathcal{W} is closed under composition. Conversely, if f:Xβ†’Yf:X\to Y is a weak equivalence, let us choose a factorisation f=puf=p u with uu an acyclic cofibration and pp a fibration. The map pp is acyclic by three-for-two, since the maps ff and uu are acyclic. Thus, pβˆˆβ„±βˆ©π’²p\in \mathcal{F}\,\cap\,\mathcal{W}.

Let us denote by M fM_f (resp. M cM_c, M fcM_{f c}) the class of fibrant (resp. cofibrant, fibrant-cofibrant) objects of a model structure M=(π’ž,𝒲,β„±)M=(\mathcal{C},\mathcal{W},\mathcal{F}) in a category E\mathbf{E}.

Proposition

A model structure M=(π’ž,𝒲,β„±)M=(\mathcal{C},\mathcal{W},\mathcal{F}) in a category E\mathbf{E} is determined by its class of cofibrations and its class M fM_{f} of fibrant objects (resp. its class M fcM_{f c} of fibrant-cofibrant objects). More precisely, if Mβ€²=(π’ž,𝒲′,β„±β€²)M'=(\mathcal{C},\mathcal{W}',\mathcal{F}') is another model structure with the same cofibrations, then the four conditions

π’²βŠ†π’²β€²,Mβ€² fβŠ†M f,Mβ€² fcβŠ†M fc,𝒲 cβŠ†π’²β€² c,\mathcal{W}\subseteq \mathcal{W}', \quad \quad \quad M'_{f}\subseteq M_{f}, \quad \quad \quad M'_{f c}\subseteq M_{f c}, \quad \quad \quad \mathcal{W}_c\subseteq \mathcal{W}'_c ,

are equivalent.

Proof:

We shall use the equality

β„±βˆ©π’²=π’ž β‹”=β„±β€²βˆ©π’²β€²\mathcal{F}\,\cap\, \mathcal{W}=\mathcal{C}^\pitchfork=\mathcal{F}'\,\cap\, \mathcal{W}'

and the inclusion π’ž β‹”βŠ†π’²βˆ©π’²β€²\mathcal{C}^\pitchfork\subseteq \mathcal{W}\,\cap\, \mathcal{W}'. Let us prove the implication (1)β‡’\Rightarrow(2). If π’²βŠ†π’²β€²\mathcal {W}\subseteq \mathcal {W}', then π’žβˆ©π’²βŠ†π’žβˆ©π’²β€²\mathcal{C}\,\cap\,\mathcal{W}\subseteq \mathcal{C}\,\cap\, \mathcal{W}'. Thus, β„±β€²βŠ†β„±\mathcal {F}'\subseteq \mathcal {F}, since β„±=(π’žβˆ©π’²) β‹”\mathcal{F}=(\mathcal{C}\,\cap\, \mathcal{W})^\pitchfork and β„±β€²=(π’žβˆ©π’²β€²) β‹”\mathcal{F}'=(\mathcal{C}\,\cap\, \mathcal{W}')^\pitchfork . It follows that Mβ€² fβŠ†M fM'_{f}\subseteq M_{f}. The implication (1)β‡’\Rightarrow(2) is proved. The implication (2)β‡’\Rightarrow(3) is obvious, since M c=Mβ€² cM_{c}=M'_{c}. Let us prove the implication (3)β‡’\Rightarrow(4). If AA is cofibrant and XβˆˆβŠ†M fcX\in \subseteq M_{f c} (resp. XβˆˆβŠ†Mβ€² fcX\in \subseteq M'_{f c}) let us denote by Ο€(A,X)\pi(A,X) (resp. Ο€β€²(A,X)\pi'(A,X)) the set of homotopy classes of maps Aβ†’XA\to X with respect to the model structure MM (resp. Mβ€²M'). We claim that if X∈Mβ€² fcX\in M'_{f c}, then the left homotopy relation between the maps Aβ†’XA\to X only depends on the weak factorisation system (π’ž,π’ž β‹”)(\mathcal{C},\mathcal{C}^\pitchfork). To see this, observe that a cylinder object of AA can be constructed by factoring the map βˆ‡ A:AβŠ”Aβ†’A\nabla_A:A\sqcup A \to A as a cofibration (d 1,d 0):AβŠ”Aβ†’IA(d_1,d_0):A\sqcup A\to I A followed by a map s:IAβ†’As:I A\to A in π’ž β‹”\mathcal{C}^\pitchfork. The left homotopy relation between the maps Aβ†’XA\to X can be defined by using this cylinder alone by Lemma 7, since the object XX is fibrant in both model structures under the assumption that Mβ€² fcβŠ†M fcM'_{f c}\subseteq M_{f c}. This shows that the left homotopy relation between the maps Aβ†’XA\to X only depends on the system (π’ž,π’ž β‹”)(\mathcal{C},\mathcal{C}^\pitchfork) if X∈Mβ€² fcX\in M'_{f c}. It follows that we have Ο€β€²(A,X)=Ο€(A,X)\pi'(A,X)=\pi(A,X) if AA is cofibrant and X∈Mβ€² fcX\in M'_{f c}. We can now prove that 𝒲 cβŠ†π’²β€² c\mathcal{W}_c\subseteq \mathcal{W}'_c. If a map u:Aβ†’Bu:A\to B belongs to 𝒲 c\mathcal {W}_c, then the map Ο€(u,X):Ο€(B,X)β†’Ο€(A,X)\pi(u,X):\pi(B,X)\to \pi(A,X) is bijective for every object X∈M fcX\in M_{f c} by Corollary 2, hence also the map Ο€β€²(u,X):Ο€β€²(B,X)β†’Ο€β€²(A,X)\pi'(u,X):\pi'(B,X)\to \pi'(A,X) for every object X∈Mβ€² fcX\in M'_{f c}. This shows that uβˆˆπ’²β€² cu\in \mathcal {W}'_c by the same corollary. The inclusion 𝒲 cβŠ†π’²β€² c\mathcal{W}_c\subseteq \mathcal{W}'_c is proved. Let us now prove the implication (4)β‡’\Rightarrow(1). Let u:Aβ†’Bu:A\to B be a map in 𝒲\mathcal {W}. Let us factor the map βŠ₯β†’A\bot \to A as a cofibration βŠ₯β†’Aβ€²\bot \to A' followed by a map q A:Aβ€²β†’Aq_A:A'\to A in π’ž β‹”\mathcal{C}^\pitchfork, and then factor the composite uq A:Aβ€²β†’Bu q_A:A'\to B as a cofibration uβ€²:Aβ€²β†’Bβ€²u':A'\to B' followed by a map q B:Bβ€²β†’Bq_B:B'\to B in π’ž β‹”\mathcal{C}^\pitchfork. We obtain a commutative square

with q A,q Bβˆˆπ’²βˆ©π’²β€²q_A,q_B\in \mathcal{W}\,\cap\, \mathcal{W}', since π’ž β‹”βŠ†π’²βˆ©π’²β€²\mathcal{C}^\pitchfork\subseteq \mathcal{W}\,\cap\, \mathcal{W}'. We have uβ€²βˆˆπ’² cu'\in \mathcal{W}_c by three-for-two, since uβˆˆπ’²u\in \mathcal{W}. Thus, uβ€²βˆˆπ’²β€² cu'\in \mathcal{W}'_c, since 𝒲 cβŠ†π’²β€² c\mathcal{W}_c\subseteq \mathcal{W}'_c by assumption. It follows by three-for-two that uβˆˆπ’²β€²u\in \mathcal{W}'. The implication (4)β‡’\Rightarrow(1) is proved.

Corollary

A model structure (π’ž,𝒲,β„±)(\mathcal{C},\mathcal{W},\mathcal{F}) in a category E\mathbf{E} is determined by its subcategory of fibrant objects E f\mathbf{E}_{f} together with one of the classes π’ž\mathcal{C} or β„±βˆ©π’².\mathcal{F}\, \cap\, \mathcal{W}. Dually, a model structure (π’ž,𝒲,β„±)(\mathcal{C},\mathcal{W},\mathcal{F}) is determined by its subcategory of cofibrant objects E c\mathbf{E}_{c} together with one of the classes β„±\mathcal{F} or π’žβˆ©π’².\mathcal{C}\, \cap\, \mathcal{W}.

Proof:

The first statement follows from Proposition 4 and by using the fact that π’ž= β‹”(β„±βˆ©π’²)\mathcal{C}={}^\pitchfork(\mathcal{F}\, \cap\, \mathcal{W}). The rest follows by duality.

Corollary

A model structure (π’ž,𝒲,β„±)(\mathcal{C},\mathcal{W},\mathcal{F}) in a category E\mathbf{E} is determined by its subcategory of fibrant-cofibrant objects E fc\mathbf{E}_{f c} together with one of the classes

π’ž,π’žβˆ©π’²,β„±,β„±βˆ©π’².\mathcal{C}, \quad \mathcal{C}\,\cap\, \mathcal{W}, \quad \mathcal{F}, \quad \mathcal{F}\, \cap\, \mathcal{W}.
Proof:

This follows from Proposition 4 in the case of the class π’ž\mathcal{C}, and also in the case of the class β„±βˆ©π’²\mathcal{F}\, \cap\, \mathcal{W}, since we have π’ž= β‹”(β„±βˆ©π’²)\mathcal{C}={}^\pitchfork(\mathcal{F}\, \cap\, \mathcal{W}). The rest follows by duality.

Derived functors

Definitions

Let Ξ£\Sigma be a set of arrows in a category A\mathbf{A}, and let P:Aβ†’Ξ£ βˆ’1AP:\mathbf{A} \to \Sigma^{-1}\mathbf{A} be the localisation functor. We saw in Lemma 25 that for any category M\mathbf{M}, the functor

P *:[Ξ£ βˆ’1A,M]β†’[A,M]P^*:[\Sigma^{-1}\mathbf{A},\mathbf{M}] \to [\mathbf{A},\mathbf{M}]

induced by PP is fully faithful, and that it induces an isomorphism between the category [Ξ£ βˆ’1A,M][\Sigma^{-1}\mathbf{A},\mathbf{M}] and the full subcategory of [A,M][\mathbf{A},\mathbf{M}] spanned by the functors Aβ†’M\mathbf{A}\to \mathbf{M} which inverts the elements of Ξ£\Sigma. We can thus identify these categories, by using the same notation for a functor G:Ξ£ βˆ’1Aβ†’MG:\Sigma^{-1}\mathbf{A}\to \mathbf{M} and the functor GP:Aβ†’BG P:\mathbf{A}\to \mathbf{B}. In which case the category [Ξ£ βˆ’1A,M][\Sigma^{-1}\mathbf{A},\mathbf{M}] becomes a full subcategory of the category [A,M][\mathbf{A},\mathbf{M}]. The left Kan extension of a functor F:Aβ†’MF:\mathbf{A}\to \mathbf{M} along the functor PP is a functor Fβ€²:Ξ£ βˆ’1Aβ†’MF':\Sigma^{-1}\mathbf{A}\to \mathbf{M} equipped with a natural transformation Ξ·:Fβ†’Fβ€²\eta:F\to F' which reflects FF into the full subcategory [Ξ£ βˆ’1A,M][\Sigma^{-1}\mathbf{A},\mathbf{M}]. More precisely, for any functor G:Ξ£ βˆ’1Aβ†’MG:\Sigma^{-1}\mathbf{A}\to \mathbf{M} and any natural transformation Ξ±:Fβ†’G\alpha: F\to G, there exists a unique natural transformation Ξ±β€²:Fβ€²β†’G\alpha':F'\to G such that Ξ±β€²Ξ·=Ξ±\alpha'\eta=\alpha,

The functor Fβ€²F' can thus be regarded as the best right approximation of the functor FF by a functor inverting the elements in Ξ£\Sigma.

Dually, the right Kan extension of a functor F:Aβ†’MF:\mathbf{A}\to \mathbf{M} along the functor PP is a functor Fβ€²:Ξ£ βˆ’1Aβ†’MF':\Sigma^{-1}\mathbf{A}\to \mathbf{M} equipped with a natural transformation Ο΅:Fβ€²β†’F\epsilon:F'\to F which coreflects FF into the full subcategory [Ξ£ βˆ’1A,M][\Sigma^{-1}\mathbf{A},\mathbf{M}]. More precisely, for any functor G:Ξ£ βˆ’1Aβ†’MG:\Sigma^{-1}\mathbf{A}\to \mathbf{M} and any natural transformation Ξ²:Gβ†’F\beta: G\to F, there exists a unique natural transformation Ξ±β€²:Gβ†’Fβ€²\alpha':G\to F' such that ϡα′=Ξ±\epsilon \alpha'=\alpha,

The functor Fβ€²F' can thus be regarded as the best left approximation of the functor FF by a functor inverting the elements in Ξ£\Sigma.

Definition

Let E\mathbf{E} be a category equipped with a class 𝒲\mathcal{W} of weak equivalences. The right derived functor of a functor F:Eβ†’MF:\mathbf{E}\to \mathbf{M} is defined to be the best dexter approximation Ξ·:Fβ†’F R\eta:F\to F^R of the functor FF by a functor F RF^R inverting weak equivalences. Dually, the left derived functor of FF is defined to be the best sinister approximation Ο΅:F Lβ†’F\epsilon:F^L\to F of the functor FF by a functor F LF^L inverting weak equivalences.

We shall say a right derived functor η:F→F R\eta:F\to F^R is absolute if it stays a right derived functor after postcomposing it with any functor U:M→NU: \mathbf{M}\to \mathbf{N}, that is, if the natural transformation U(η):UF→UF RU(\eta):U F\to U F^R exibits the functor UF RU F^R as the the right derived functor (UF) R(U F)^R of UFU F. Dually, We shall say a left derived functor ϡ:F L→F\epsilon:F^L\to F is absolute if it stays a left derived functor after postcomposing it with any functor U:M→NU: \mathbf{M}\to \mathbf{N}, that is, if the natural transformation U(ϡ):UF L→UFU(\epsilon):U F^L\to U F exibits the functor UF LU F^L as the left derived functor (UF) L(U F)^L of UFU F.

Proposition

Let E\mathbf{E} be a model category, and F:E→MF:\mathbf{E}\to \mathbf{M} be a functor which inverts weak equivalences between fibrant objects. Then the right derived functor η:F→F R\eta:F\to F^R exists and it is absolute. Moreover, the map η X:FX→F RX\eta_X:F X\to F^R X is an isomorphism when XX is fibrant.

Proof:

For each object X∈EX\in \mathbf{E}, let us choose a fibrant replacement i X:Xβ†’RXi_X:X\to R X; we shall take i X=1 X:Xβ†’Xi_X =1_X:X\to X when XX is fibrant. By Lemma 26, for every map u:Xβ†’Yu:X\to Y, there exits a map R(u):RXβ†’RYR(u):R X\to R Y fitting in the commutative square,

(13)

The map R(u)R(u) is unique up to a right homotopy by Lemma 26. Hence the map FR(u):FRX→FRYF R(u):F R X \to F R Y does not depends on the choice of R(u)R(u) by Lemma 8. Moreover, if u:X→Yu:X\to Y and v:Y→Zv:Y\to Z, then we have FR(vu)=FR(v)FR(u)F R (v u)=F R(v) F R(u) by the same lemma. We thus obtain a functor F R=FR:E→MF^R=F R:\mathbf{E} \to \mathbf{M}.

The functor F RF^R inverts weak equivalences, since R(u)R(u) is a weak equivalence between fibrant objects when uu is a weak equivalence by Lemma 26. Thus, F R:Ho(E)β†’MF^R:Ho(\mathbf{E}) \to \mathbf{M}. Notice that F RX=FXF^R X =F X when XX is fibrant. The image by FF of the square (13) is a commutative square,

It shows that we can define a natural transformation η:F→F R\eta:F\to F^R by putting η X=F(i X):FX→FRX\eta_X=F(i_X):F X\to F R X for every object XX. Notice that we have η X=1 FX\eta_X=1_{F X} when XX is fibrant. Let us show that the natural transformation η:F→F R\eta:F\to F^R is reflecting the functor FF into the subcategory [Ho(E),M][Ho(\mathbf{E}),\mathbf{M}] of the category [E,M][\mathbf{E},\mathbf{M}]. For this we have to prove that for any functor G:Ho(E)→MG: Ho(\mathbf{E})\to \mathbf{M} and any natural transformation α:F→G\alpha:F\to G there exists a unique natural transformation α′:F R→G\alpha':F^R\to G such that α′η=α\alpha'\eta =\alpha,

The map G(i X):GX→GRXG(i_X):G X\to G R X in the following square

(14)

is invertible by the assumption on GG, since i Xi_X is a weak equivalence. We can thus defines a map Ξ±β€² X:FRXβ†’GX\alpha'_X:F R X\to G X by putting Ξ± X=G(i X) βˆ’1Ξ± RX\alpha_X=G(i_X)^{-1}\alpha_{R X}. Let us verify that this defines a natural transformation Ξ±β€²:F Rβ†’G\alpha':F^R\to G. The right hand square of the following diagram commutes for any map u:Xβ†’Yu:X\to Y by the functoriality of GG, since the square (13) commutes,

And the left hand square commutes by the naturality of Ξ±\alpha. The naturality of Ξ±β€²\alpha' is proved. Observe that we have G(i X) βˆ’1Ξ± RXF(i X)=Ξ± XG(i_X)^{-1} \alpha_{R X} F(i_X)=\alpha_X for every object XX, since the square (14) commutes. This proves that Ξ±β€²Ξ·=Ξ±\alpha'\eta =\alpha. It remains to prove the uniqueness of Ξ±β€²\alpha'. If Ξ²:FRβ†’G\beta:F R\to G is a natural transformation such that Ξ²Ξ·=Ξ±\beta\eta =\alpha, let us show that Ξ²=Ξ±β€²\beta=\alpha'. Notice that Ξ² X=Ξ± X=Ξ±β€² X\beta_X=\alpha_X=\alpha'_X when XX is fibrant, since we have Ξ· X=1 FX\eta_X=1_{F X} in this case. In general, let us choose a weak equivalence w:Xβ†’Yw:X\to Y with codomain a fibrant object YY (for example, w=i X:Xβ†’RXw=i_X:X\to R X). The bottom maps in the following two squares are equal since YY is fibrant,

Hence also the top maps, since G(w)G(w) is invertible by the assumption on GG. Thus, β X=α′ X\beta_X=\alpha'_X. We have proved that F RF^R is the right derived functor of FF. Moreover, the map η X\eta_X is invertible when XX is fibrant, since we have η X=1 FX\eta_X=1_{F X} in this case. For any functor U:M→NU: \mathbf{M}\to \mathbf{N} we have (UF) R=(UF)R=U(FR)=UF R(U F)^R=(U F)R=U(F R)=U F^R and U(η X)=UF(i X)U(\eta_X)=U F(i_X) . This shows that the right derived functor F RF^R is absolute.

Quillen functors

Definition

We shall say that a functor F:U→VF:\mathbf{U}\to \mathbf{V} between two model categories is a left Quillen functor if it takes a cofibration to a cofibration and an acyclic cofibration to an acyclic cofibration. Dually, we shall say that FF is a right Quillen functor if it takes a fibration to a fibration and an acyclic fibration to an acyclic fibration.

Remark

Most left Quillen functors that we shall consider are cocontinuous and have a right adjoint. Dually, most right Quillen functors have a left adjoint.

Lemma

A left Quillen functor takes an acyclic map between cofibrant objects to an acyclic map. A right Quillen functor takes an acyclic map between fibrant objects to an acyclic map.

Proof:

This follows from Ken Brown’s lemma 5.

Lemma

Let F:U↔V:GF:\mathbf{U}\leftrightarrow \mathbf{V}:G be an adjunction F⊒GF\vdash G between two model categories. Then the left adjoint FF is a left Quillen functor iff the right adjoint GG is a right Quillen functor.

Proof:

Let us denote by (π’ž,𝒲,β„±)(\mathcal{C},\mathcal{W},\mathcal{F}) the model structure on U\mathbf{U} and by (π’žβ€²,𝒲′,β„±β€²)(\mathcal{C}',\mathcal{W}',\mathcal{F}') the model structure on V\mathbf{V}. Then the conditions

F(π’ž)βŠ†π’žβ€²andG(β„±β€²βˆ©π’²β€²)βŠ†β„±βˆ©π’²F(\mathcal{C})\subseteq \mathcal{C}' \quad \mathrm{and} \quad G(\mathcal{F}'\,\cap\, \mathcal{W}')\subseteq \mathcal{F}\cap \mathcal{W}

are equivalent by the proposition here. Similarly, the conditions

F(π’žβˆ©π’²)βŠ†π’žβ€²βˆ©π’²β€²andG(β„±β€²)βŠ†β„±F(\mathcal{C}\,\cap\, \mathcal{W})\subseteq \mathcal{C}'\,\cap\, \mathcal{W}' \quad \mathrm{and} \quad G(\mathcal{F}')\subseteq \mathcal{F}

are equivalent.

Definition

We shall say that an adjunction F⊒GF\vdash G between two model categories is a Quillen adjunction if the left adjoint FF is a left Quillen functor, or equivalently, if the right adjoint GG is a right Quillen functor.

Examples of model categories

Example

The category of simplicial sets SSet\mathbf{SSet} admits a model structure, called the Kan model structure, in which the fibrant objects are the Kan complexes and the cofibrations are the monomorphisms. The weak equivalences are the weak homotopy equivalences and the fibrations are the Kan fibrations. The model structure is cartesian closed and proper.

Example

The category Cat\mathbf{Cat} admits a model structure, called the natural model structure in which the cofibrations are the functors monic on objects, the weak equivalences are the equivalences of categories and the fibrations are the isofibrations. The model structure is cartesian closed and proper.

Example

The category of simplicial sets SSet\mathbf{SSet} admits a model structure, called the model structure for quasi-categories, in which the fibrant objects are the quasi-categories and the cofibrations are the monomorphisms. A weak equivalence is called a categorical equivalence and a fibration is called an isofibration. The model structure is cartesian closed and left proper.

Exercises

Exercise

Gives a direct proof (without using the duality) to the lemmata 14 15 16 17 18 19 20 21 22.

Exercise

Recall (from after Definition 4) that a fibrant-cofibrant replacement of an object XX is obtained by factoring the composite q Xi X:QX→RXq_X i_X:Q X\to R X as an acyclic cofibration j X:QX→WXj_X:Q X\to W X followed by an acyclic fibration p X:WX→RXp_X:W X\to R X. Show that for every map u:X→Yu:X\to Y, there are maps QuQ u, RuR u and WuW u fitting in a commutative cube:

Exercise

Show that the distributor π:E⇒E f\pi:\mathbf{E} \Rightarrow \mathbf{E}_f defined in (7) induces a distributor π′:Ho(E c)⇒Ho(E f)\pi':\mathrm{Ho}(\mathbf{E}_c) \Rightarrow \mathrm{Ho}(\mathbf{E}_f). Show that the distributor π′\pi' is representable by an equivalence of categories Ho(E c)→Ho(E f)\mathrm{Ho}(\mathbf{E}_c) \to \mathrm{Ho}(\mathbf{E}_f).

Exercise

Show that the natural model structure on Cat\mathbf{Cat} is characterised by each of the following four groups of conditions:

  • Group 1:

    • the acyclic maps are the equivalences of categories

    • an acyclic map is a fibration iff it is a split epimorphism

  • Group 2:

    • the acyclic maps are the equivalences of categories

    • an acyclic map is a cofibration iff it is a split monomorphism

  • Group 3:

    • every object is fibrant.

    • an acyclic map is a fibration iff it is a surjective equivalence

  • Group 4:

    • every object is cofibrant.

    • an acyclic map is a cofibration iff it is a monic equivalence

References

Papers:

Lecture Notes and Textbooks:

Revised on October 24, 2012 11:38:47 by Raeder