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Model structures on Cat

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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

The natural model structure

Natural: endogenous: produced by factors inside the system

Throughout this page, we shall often denote by X 0\mathbf{X}_0 the set of objects of a category X\mathbf{X} and by F 0:X 0Y 0F_0:\mathbf{X}_0\to \mathbf{Y}_0 the map induced by a functor F:XYF:\mathbf{X}\to \mathbf{Y}.

Definition

If X\mathbf{X} and Y\mathbf{Y} are categories, we shall say that a functor F:XYF:\mathbf{X}\to \mathbf{Y} is an isofibration if for every object AXA\in \mathbf{X} and every isomorphism vYv\in \mathbf{Y} with source FAF A, there exists an isomorphism uXu\in \mathbf{X} with source AA such that F(u)=vF(u)=v,

When the isomorphism uu is uniquely determined by a pair (A,v)(A,v), we shall say that the isofibration has the unique lifting property.

The notion of isofibration is self dual: a functor F:XYF:\mathbf{X}\to \mathbf{Y} is an isofibration iff the opposite functor F o:X oY oF^o:\mathbf{X}^o\to \mathbf{Y}^o is an isofibration (exercise 1). Hence a functor F:XYF:\mathbf{X}\to \mathbf{Y} is an isofibration iff for every object AXA\in \mathbf{X} and every isomorphism vYv\in \mathbf{Y} with target FAF A, there exists an isomorphism uXu\in \mathbf{X} with target AA such that F(u)=vF(u)=v,

Let JJ be the groupoid generated by one isomorphism 010\simeq 1. Then a functor F:XYF:\mathbf{X}\to \mathbf{Y} is an isofibration iff it has the right lifting property with respect to the inclusion {0}J\{0\}\subset J (resp. {1}J\{1\}\subset J). (exercise 2).

Definition

We shall say that a functor F:XYF:\mathbf{X}\to\mathbf{Y} is monic (resp. surjective, bijective) on objects if the map X 0Y 0\mathbf{X}_0\to \mathbf{Y}_0 induced by FF is injective (resp. surjective, bijective).

Let us denote by Cat\mathbf{Cat} the category of small categories and by Cat\mathbf{Cat'} the category of locally small categories. The category Cat\mathbf{Cat'} cannot carry a model structure since it is not finitely cocomplete. However,

Theorem

Let us denote by 𝒞\mathcal{C}' the class of functors monic on objects in the category Cat\mathbf{Cat'}, by 𝒲\mathcal{W}' the class of equivalences, and by \mathcal{F}' the class of isofibrations. Then the pairs (𝒞𝒲,)(\mathcal{C}'\,\cap\,\mathcal{W}',\mathcal{F}') and (𝒞,𝒲)(\mathcal{C}',\mathcal{F}'\,\cap\,\mathcal{W}') are weak factorisation systems in Cat\mathbf{Cat'}. If

𝒞=𝒞Cat,𝒲=𝒲Catand=Cat\mathcal{C}=\mathcal{C}'\,\cap \, \mathbf{Cat},\quad \quad \mathcal{W}=\mathcal{W}'\,\cap \, \mathbf{Cat}\quad \mathrm{and} \quad \mathcal{F}=\mathcal{F}'\,\cap \, \mathbf{Cat}

then the triple (𝒞,𝒲,)(\mathcal{C},\mathcal{W},\mathcal{F}) is a model structure on the category Cat\mathbf{Cat}. The model structure is cartesian closed and proper. We shall say that it is the natural model structure on Cat\mathbf{Cat}.

The proof will be given after Proposition 6.

Lemma

The class 𝒲\mathcal{W}' has the three-for-two property and it is closed under retracts.

Proof

Let us write FGF\simeq G to indicate that the functors F,G:XYF,G:\mathbf{X}\to \mathbf{Y} are isomorphic in the category [X,Y][\mathbf{X},\mathbf{Y}]. The relation FGF\simeq G is compatible with composition on both sides: we have

FGLFLGandFRKRF\simeq G \quad \Rightarrow \quad L F\simeq L G \quad \mathrm{and} \quad F R \simeq K R

for every functor L:YYL:\mathbf{Y}\to \mathbf{Y'} and every functor R:XXR:\mathbf{X'}\to \mathbf{X}. We can thus construct a quotient category Ho(Cat)\mathrm{Ho}(\mathbf{Cat'}) by putting

Ho(Cat)(X,Y)=Cat(X,Y)/\mathrm{Ho}(\mathbf{Cat'})(\mathbf{X},\mathbf{Y})=\mathbf{Cat'}(\mathbf{X},\mathbf{Y})/\simeq

for locally small categories X\mathbf{X} and Y\mathbf{Y}. The canonical functor Ho:CatHo(Cat)Ho:\mathbf{Cat'}\to \mathrm{Ho}(\mathbf{Cat'}) takes a functor F:XYF:\mathbf{X}\to \mathbf{Y} to its isomorphism class in the category of functors XY\mathbf{X}\to \mathbf{Y}. The morphism Ho(F):XYHo(F):\mathbf{X}\to \mathbf{Y} is invertible iff the functor FF is an equivalence of categories. It follows that the class 𝒲\mathcal{W}' has the three-for-two property, since this is true of the class of isomorphisms in any category. Similarly, the class 𝒲\mathcal{W}' is closed under retracts, since this is true of the class of isomorphisms in any category.

Lemma

For any functor F:XYF:\mathbf{X}\to \mathbf{Y}, any map G 0:X 0Y 0G_0:\mathbf{X}_0\to \mathbf{Y}_0 and any family of isomorphisms θ=(θ A:F 0AG 0A|AX 0)\theta=(\theta_A:F_0 A\to G_0 A| A\in \mathbf{X}_0) there exists a unique functor G:XYG:\mathbf{X}\to \mathbf{Y} extending G 0G_0 for which the family θ\theta becomes a natural isomorphism FGF\to G.

Proof

The functor GG takes a map f:ABf:A\to B in X\mathbf{X} to the unique map G(f):G 0AG 0BG(f):G_0 A \to G_0 B in Y\mathbf{Y} for which the following diagram commutes,

We shall say that the functor G:XYG:\mathbf{X}\to \mathbf{Y} is obtained by transporting the functor F:XYF:\mathbf{X}\to \mathbf{Y} along the family of isomorphisms (θ A:F 0AG 0A|AX 0)(\theta_A:F_0 A\to G_0 A| A\in \mathbf{X}_0). The possibility of transporting a functor F:XYF:\mathbf{X}\to \mathbf{Y} along a family of isomorphisms θ\theta actually means that the restriction functor [X,Y][X 0,Y][\mathbf{X},\mathbf{Y}]\to [\mathbf{X}_0,\mathbf{Y}] is an isofibration with unique lifting, where X 0\mathbf{X}_0 denote the discrete category whose objects are the elements of X 0\mathbf{X}_0.

A functor is monic iff it is monic on objects and faithful. We shall say that a functor is surjective if it is surjective on objects and full.

Lemma

A surjective equivalence P:XYP:\mathbf{X}\to \mathbf{Y} is a split epimorphism; more precisely, every section S 0:Y 0X 0S_0:\mathbf{Y}_0 \to \mathbf{X}_0 of the map P 0:X 0Y 0P_0:\mathbf{X}_0\to \mathbf{Y}_0 can be extended uniquely as a section S:YXS:\mathbf{Y}\to \mathbf{X} of the functor PP. There then is a unique isomorphism θ:Id XSF\theta:Id_{\mathbf{X}}\to S F such that Pθ=id PP\circ\theta=id_P. Dually, a monic equivalence U:ABU:\mathbf{A}\to \mathbf{B} is a split monomorphism. If R:BAR:\mathbf{B}\to \mathbf{A} is a retraction, then there is a unique isomorphism θ:Id BUR\theta:Id_{\mathbf{B}}\to U R such that θU=id U\theta \circ U=id_U.

Proof

Let us prove the first statement. Let P:XYP:\mathbf{X}\to \mathbf{Y} be an equivalence surjective on objects. The map P 0:X 0Y 0P_0:\mathbf{X}_0\to \mathbf{Y}_0 has a section, since it is surjective. Let S 0:Y 0X 0S_0:\mathbf{Y}_0 \to \mathbf{X}_0 be a section. If f:ABf:A\to B is a morphism in Y\mathbf{Y}, then there is a unique morphism S(f):S 0AS 0BS(f):S_0 A \to S_0 B in Y\mathbf{Y} such that PS(f)=fP S(f)=f, since the map X(S 0A,S 0B)Y(A,B)\mathbf{X}(S_0 A,S_0 B)\to \mathbf{Y}(A,B) induced by PP is bijective. This defines a functor S:YXS:\mathbf{Y}\to \mathbf{X}. It is easy to see that we have PS=Id YP S=Id_{\mathbf{Y}}. The functor SS is an equivalence of categories, since PP is an equivalence. The existence and uniqueness of an isomorphism θ:Id XSP\theta:Id_{\mathbf{X}}\to S P such that Pθ=id PP\circ\theta=id_P follows. Let us prove the second statement. The functor UU is essentially surjective, since it is fully faithful. Thus, for each object BB 0B \in \mathbf{B}_0 we can choose an object R 0BA 0R_0B \in \mathbf{A}_0 together with an isomorphism θ B:BU 0R 0B\theta_B:B\to U_0R_0B, with the proviso that θ B=1 B\theta_B=1_B when BU(A 0)B \in U(\mathbf{A}_0). The proviso implies that R 0U 0A=AR_0U_0A =A for every object AA 0A\in \mathbf{A}_0, since U 0U_0 is monic. There is then a unique functor R:BAR:\mathbf{B}\to \mathbf{A} extending R 0R_0 for which the family (θ B)(\theta_B) becomes a natural isomorphism id BURid_{\mathbf{B}}\to U R. By construction, the functor RR takes a morphism g:BBg:B\to B' in B\mathbf{B} to the unique morphism R(g):R 0BR 0BR(g):R_0 B\to R_0 B' such that the following square commutes,

If g=U(f)g=U(f), where f:AAf:A\to A' is a morphism in A\mathbf{A}, then we have R(g)=fR(g)=f, since the morphisms θ B\theta_{B} are units when BU(A)B\in U( \mathbf{A}) and the following square commutes,

This shows that RU=id AR U=id_{\mathbf{A}}. The functor RR is an equivalence, since UU is an equivalence. The existence and uniqueness of an isomorphism θ:Id BUR\theta:Id_{\mathbf{B}}\to U R such that θU=id U\theta \circ U=id_U follows.

Lemma

The class of isofibrations is closed under composition, retracts and base changes.

Proof

Let JJ be the groupoid generated by one isomorphism 010\simeq 1. Then a functor F:XYF:\mathbf{X}\to \mathbf{Y} is an isofibration iff it has the right lifting property with respect to the inclusion i 0:{0}Ji_0:\{0\}\subset J. Thus, the class of isofibrations is of the form {i 0} \{i_0\}^\pitchfork. The result then follows from the proposition here.

Lemma

An equivalence is an isofibration iff it is surjective on objects. The class of equivalences surjective on objects is closed under composition, retracts and base changes.

Proof

Let us prove the first statement. Let F:XYF:\mathbf{X}\to \mathbf{Y} be an equivalence which is an isofibration. Then for every object BYB\in \mathbf{Y}, there exists an object AXA\in \mathbf{X} together with an isomorphism v:FABv: F A\to B, since an equivalence is essentially surjective. There is then an isomorphism u:AAu:A\to A' such that F(u)=vF(u)=v, since FF is an isofibration. We then have FA=BF A'=B, and this shows that FF is surjective on objects. Conversely, let us show that an equivalence F:XYF:\mathbf{X}\to \mathbf{Y} surjective on objects is an isofibration. If AA is an object of X\mathbf{X} and v:FABv:F A \to B is an isomorphism in YY, then there exists an object AXA'\in X such that FA=BF A'=B, since FF is surjective on object. The map X(A,A)Y(FA,FA)\mathbf{X}(A,A')\to \mathbf{Y}(F A,F A') induced by FF is bijective, since FF is an equivalence. Hence there exists a morphism u:AAu: A\to A' such that F(u)=vF(u)=v. The morphism uu is invertible, since vv is invertible and FF is an equivalence.
This shows that FF is an isofibration. The first statement of the proposition is proved. Let us prove the second statement. Observe that a functor is an equivalence surjective on objects iff it is fully faithful and surjective on objects. Hence the class of equivalences surjective on objects is the intersection of two classes: the class \mathcal{M} of fully faithful functors and the class 𝒩\mathcal{N} of functors surjective on objects. The class \mathcal{M} is the right class of the Gabriel factorisation system by the Example here. It then follows from the propositions here and here that the class \mathcal{M} is closed under composition, retracts and base changes. The class of surjections in the category of sets has the same closure properties, hence also the class 𝒩\mathcal{N}.

Lemma

Suppose that we have a commutative square of categories and functors

in which the functor UU is monic on objects and the functor FF is an isofibration. If UU or FF is an equivalence, then the square has a diagonal filler.

Proof

For this it suffices to show that the left hand square of the following diagram has a diagonal filler,

(1)

The projection pr 1pr_1 is an isofibration by Lemma 4, since it is a base change of the functor PP. Moreover, pr 1pr_1 is an equivalence when PP is an equivalence by Lemma{trivialfibrationclosure}. This shows that problem can be reduced to the case of a square

(2)

Let us first consider the case where the functor UU is an equivalence. It follows from Lemma 3 that the functor UU admits a retraction R:BAR:\mathbf{B}\to \mathbf{A} together with a natural isomorphism θ:Id BUR\theta:Id_{\mathbf{B}}\to U R such that θU=id U\theta\circ U =id_U. This last condition means that we have θ UA=1 UA\theta_{U A}=1_{U A} for every object AA)A\in \mathbf{A}). We have PFRB=URBP F R B=U R B for every object BBB\in \mathbf{B}, since PF=UP F=U. We thus have a diagram,

There exists an isomorphism α B\alpha_B with source FRBF R B such that P(α B)=θ BP(\alpha_B)=\theta_B, since the functor PP is an isofibration by assumption,

This defines a map D 0:B 0X 0D_0:\mathbf{B}_0\to \mathbf{X}_0, where D 0BD_0 B is the target of α B\alpha_B. The isomorphism α B\alpha_B can be taken to be a unit when B=U(A)U(A 0)B=U(A)\in U(\mathbf{A}_0), since θ B\theta_B is a unit in this case. We then have D 0U 0(A)=F 0(A)D_0 U_0(A)=F_0(A) for every AA 0A\in \mathbf{A}_0. If we transport the functor FR:BXF R : \mathbf{B}\to \mathbf{X} by Lemma 2 along the the family of isomorphisms (α B)(\alpha_B) we obtain a functor D:BXD:\mathbf{B}\to \mathbf{X} equipped with an isomorphism α:FRD\alpha:F R\to D. By construction, the functor DD takes a morphism u:BBu:B\to B' in B\mathbf{B} to the unique morphism D(u):DBDBD(u):D B\to D B' in X\mathbf{X} such that the following square commutes,

It is then a routine matter to verify that we have DU=FD U=F and PD=Id BP D =Id_{\mathbf{B}} . We have proved that the square (2) has a diagonal filler in the case where UU is an equivalence. Let us now consider the case where PP is an equivalence. The functor PP is surjective on objects in this case by Lemma 5. Hence the following square in the category of sets

has a diagonal filler D 0:B 0X 0D_0:\mathbf{B}_0\to \mathbf{X}_0, since U 0U_0 is monic on objects and the pair (Inj,Surj)(Inj,Surj) is a weak factorisation system in the category of sets. It then follows from Lemma 3 that the functor PP admits a unique section D:BXD:\mathbf{B}\to \mathbf{X} which extends the map D 0D_0, since PP is a surjective equivalence. It is then a routine matter to verify that we have DU=FD U=F. We have proved that the square (2) has a diagonal filler in the case where PP is an equivalence.

Notation

Let JJ be the groupoid generated by one isomorphism 010\simeq 1. We shall denote the inclusion {0}J\{0\}\subset J as a map d 1:1Jd_1:1\to J and the inclusion {1}J\{1\}\subset J as a map d 0:1Jd_0:1\to J. This is in accordance with the standard notation for the maps d 0,d 1:[0][1]d_0,d_1:[0]\to [1] in the category Δ\Delta, since d 0d_0 takes the value 11 and d 1d_1 takes the value 00.

The path object of a category X\mathbf{X} is defined to be the category X J=[J,X]\mathbf{X}^J=[J,\mathbf{X}]. An object of this category is an isomorphism a:A 0A 1a:A_0\to A_1 in the category X\mathbf{X}, and a morphism u:abu:a\to b between aa and b:B 0B 1b:B_0\to B_1 is a pair (u 0,u 1)(u_0,u_1) of maps in a commutative square

(3)

The functor 1=[d 1,X]:[J,X]X\partial_1=[d_1,\mathbf{X}]:[J,\mathbf{X}]\to \mathbf{X} is the source functor which which takes an isomorphism a:A 0A 1a:A_0\to A_1 to its source A 0A_0, and 0=[d 0,X]\partial_0=[d_0,\mathbf{X}] is the target functor which which takes an isomorphism a:A 0A 1a:A_0\to A_1 to its target A 1A_1. If ss denotes the functor J1J\to 1, then the functor σ=[s,X]:X[J,X]\sigma=[s,\mathbf{X} ]:\mathbf{X}\to [J,\mathbf{X}] is the unit functor which takes an object AXA\in \mathbf{X} to the unit isomorphism 1 A:AA1_A:A\to A. The relation sd 1=id 1=sd 0s d_1 =id_1=s d_0 implies that we have 1σ=id X= 0σ\partial_1 \sigma =id_{\mathbf{X}}=\partial_0 \sigma. The functors 1, 0\partial_1,\partial_0 and σ\sigma are equivalences of categories, since the functors d 1,d 0d_1,d_0 and ss are equivalences.

Proposition

The functor

( 1, 0):X JX×X(\partial_1,\partial_0):\mathbf{X}^J\to \mathbf{X}\,\times\, \mathbf{X}

is an isofibration and the functor σ:XX J\sigma:\mathbf{X}\to \mathbf{X}^J is an equivalence of categories. Moreover, the functors 1\partial_1 and 0\partial_0 are equivalences surjective on objects.

Proof

Let us show that the functor ( 1, 0)(\partial_1,\partial_0) is an isofibration. Let a:A 0A 1a:A_0\to A_1 be an object of X J\mathbf{X}^J and let (u 0,u 1):(A 0,A 1)(B 0,B 1)(u_0,u_1):(A_0,A_1)\to (B_0,B_1) be an isomorphism in X×X\mathbf{X}\times \mathbf{X}. There is then a unique isomorphism b:B 0B 1b:B_0\to B_1 such that the square (3) commutes. The pair u=(u 0,u 1)u=(u_0,u_1) defines an isomorphism aba\to b in the category X J\mathbf{X}^J, and we have ( 1, 0)(u)=(u 0,u 1)(\partial_1,\partial_0)(u)=(u_0,u_1). This proves that ( 1, 0)(\partial_1,\partial_0) is an isofibration. We saw above that the functor 1, 0\partial_1,\partial_0 and σ\sigma are equivalences of categories. The functor 1\partial_1 is surjective on objects, since 1σ=id X\partial_1\sigma=id_{\mathbf{X}}. Similarly, the functor 0\partial_0 is surjective on objects.

The mapping path object of a functor F:XYF:\mathbf{X}\to \mathbf{Y} is the category P(F)\mathbf{P}(F) defined by the following pullback square

(4)

There is a (unique) functor i X:XP(F)i_{\mathbf{X}}:\mathbf{X}\to \mathbf{P}(F) such that Pi X=σFP i_{\mathbf{X}}=\sigma F and P Xi X=id XP_{\mathbf{X}} i_{\mathbf{X}}=id_{\mathbf{X}} since square (4) is cartesian and we have 1σF=id YF=Fid X\partial_1\sigma F=id_{\mathbf{Y}} F =F id_{\mathbf{X}}. Let us put P Y= 0PP_{\mathbf{Y}}=\partial_0 P. Then we have

F=P Yi X:XP(F)YF=P_{\mathbf{Y}} i_{\mathbf{X}}:\mathbf{X}\to \mathbf{P}(F)\to \mathbf{Y}

since P Yi X= 0Pi X= 0σF=id YF=F.P_{\mathbf{Y}} i_{\mathbf{X}}=\partial_0 P i_{\mathbf{X}}=\partial_0 \sigma F=id_{\mathbf{Y}} F =F. This is the mapping path factorisation of the functor FF. Let us describe the category P(F)\mathbf{P}(F) explicitly. Let us first show that we have a pullback square,

(5)

The square commutes, since P Y= 0PP_{\mathbf{Y}}=\partial_0 P. To see that it is cartesian, consider the diagram

The right hand square of this diagram is trivially cartesian. The composite square is cartesian by definition of P(F)\mathbf{P}(F). Hence the left hand square is also cartesian by the lemma here. This shows that the square (5) is cartesian. We now use it for describing the objects and morphisms of the category P(F)\mathbf{P}(F). By construction, an object of P(F)\mathbf{P}(F) is a triple (y,A,B)(y,A,B), where AA is an object of X\mathbf{X}, BB is an object of Y\mathbf{Y} and yy is an isomorphism F(A)BF(A)\to B. The object can be pictured as a leg with the upper part in X\mathbf{X} and with its foot in Y\mathbf{Y}:

We have P(y,A,B)=yP(y,A,B)=y, P X(y,A,B)=AP_{\mathbf{X}}(y,A,B)=A and P Y(y,A,B)=BP_{\mathbf{Y}}(y,A,B)=B. A morphism (y,A,B)(y,A,B)(y,A,B)\to (y',A',B') in the category P(F)\mathbf{P}(F) is a pair of maps u:AAu:A\to A' and v:BBv:B\to B' such that the square foot of the following diagram commutes,

The functor i X:XP(F)i_{\mathbf{X}}:\mathbf{X}\to \mathbf{P}(F) takes an object AXA\in \mathbf{X} to a leg with a very short foot,

Remark

The mapping path category P(F)\mathbf{P}(F) can be constructed as the pseudo-pullback of the functor F:XYF: \mathbf{X}\to \mathbf{Y} with the identity functor YY\mathbf{Y}\to \mathbf{Y}.

Proposition

The functor P YP_{\mathbf{Y}} in the mapping path factorisation

F=P Yi X:XP(F)YF=P_{\mathbf{Y}} i_{\mathbf{X}}:\mathbf{X}\to \mathbf{P}(F)\to \mathbf{Y}

is an isofibration and the functor i X i_{\mathbf{X}} is an equivalence monic on objects.

Proof

Let us show that the functor P YP_{\mathbf{Y}} is an isofibration. We first give a formal proof by using the general argumentation of Quillen. The functor (P X,P Y)(P_{\mathbf{X}},P_{\mathbf{Y}}) is a base change of the functor ( 1, 0)(\partial_1,\partial_0), since the square (5) is cartesian. Hence the functor (P X,P Y)(P_{\mathbf{X}},P_{\mathbf{Y}}) is an isofibration by Proposition 4, since the functor ( 1, 0)(\partial_1,\partial_0) is an isofibration by Proposition 1. The projection pr 2:X×YYpr_2:\mathbf{X}\,\times\, \mathbf{Y}\to \mathbf{Y} is a base change of the functor X1\mathbf{X}\to 1. It is thus an isofibration, since the functor X1\mathbf{X}\to 1 is (trivially) an isofibration. It follows that the composite P Y=pr 2(P X,P Y)P_{\mathbf{Y}}=pr_2(P_{\mathbf{X}},P_{\mathbf{Y}}) is an isofibration, since the class of isofibrations is closed under composition by Proposition 4. Let us now give the bare foot proof. For every object (u,A,B)P(F)(u,A,B)\in \mathbf{P}(F) and every isomorphism y:BBy:B\to B' the pair (1 A,y)(1_A,y) is an isomorphism (u,A,B)(yu,A,B)(u,A,B)\to (y u,A, B') and we have P Y(1 A,y)=yP_{\mathbf{Y}}(1_A,y)=y,

This proves that P YP_{\mathbf{Y}} is an isofibration. It remains to prove that the functor i Xi_{\mathbf{X}} is an equivalence monic on objects. It is certainly monic on objects, since we have P Xi X=id XP_{\mathbf{X}}i_{\mathbf{X}}=id_{\mathbf{X}}. Let us show that it is an equivalence. Again, we shall give two proofs, the first by using the general argumentation of Quillen. It suffices to prove that the functor P XP_{\mathbf{X}} is an equivalence by three-for-two, since we have P Xi X=id XP_{\mathbf{X}}i_{\mathbf{X}}=id_{\mathbf{X}}. But the functor P XP_{\mathbf{X}} is a base change of the functor 1:Y JY\partial_1:\mathbf{Y}^J\to \mathbf{Y}, since the square (4) is cartesian. The functor 1\partial_1 is an equivalence surjective on objects by Proposition 1. It follows that the functor P XP_{\mathbf{X}} is an equivalence by Proposition 5. Let us now gives the bare foot proof that the functor i Xi_{\mathbf{X}} is an equivalence. For this it suffices to exibit a natural isomorphism α:i XP Xid\alpha: i_{\mathbf{X}} P_{\mathbf{X}} \simeq id since we already have P Xi X=id XP_{\mathbf{X}}i_{\mathbf{X}}=id_{\mathbf{X}}. But i XP X(u,A,B)=(1 FA,A,FA)i_{\mathbf{X}} P_{\mathbf{X}}(u,A,B)=(1_{F A},A,F A) and the pair (1 A,u)(1_A,u) defines a natural isomorphism (1 FA,A,FA)(u,A,B)(1_{F A}, A, F A)\to (u,A,B),

Lemma

The class of functors monic on objects is closed under composition, retracts and cobase changes.

Proof

The class of injective maps in the category of sets is closed under composition, retracts and cobase changes. Hence also the class of functors monic on objects.

Proposition

The category Cat\mathbf{Cat'} admits a weak factorisation system (,)(\mathcal{L}, \mathcal{R}) in which \mathcal{L} is the class of equivalences monic on objects and \mathcal{R} is the class of isofibrations.

Proof

We shall use the characterisation of weak factorisation systems here. The mapping path category matbfP(F)\matbf{P}(F) of a functor F:XYF:\mathbf{X} \to \mathbf{Y} is locally small if the category X\mathbf{X} is locally small, since the functor i X:XP(F)i_{\mathbf{X}}:\mathbf{X} \to \mathbf{P}(F) is an equivalence by Proposition 2. It then follows from this proposition that every functor F:XYF:\mathbf{X}\to \mathbf{Y} in Cat\mathbf{Cat'} admits a (,)(\mathcal{L}, \mathcal{R})-factorisation. By Lemma 6 we have \mathcal{L}\,\pitchfork\, \mathcal{R}. It remains to verify that the classes \mathcal{L} and \mathcal{R} are closed under retracts. This is true for the class \mathcal{R} by Proposition 4. And this is true for the class of functors monic on objects by Lemma 7. Moreover, a retract of an equivalence is an equivalence by Lemma 1.

Corollary

The class of equivalences monic on objects is closed under composition, retracts and cobase changes.

Proof

The class of equivalences monic on objects is the left class of a weak factorisation system by Proposition 3. It is thus closed under composition, retracts and cobase changes.

The cylinder of a category A\mathbf{A} is defined to be the category JA=J×AJ \mathbf{A}= J\times \mathbf{A} . We shall denote the inclusion {0}×AJ×A\{0\}\times \mathbf{A} \subseteq J\times \mathbf{A} by d 1:AJAd_1:\mathbf{A}\to J A, the inclusion {1}×AJ×A\{1\}\times \mathbf{A} \subseteq J\times \mathbf{A} by d 0:AJAd_0:\mathbf{A} \to J \mathbf{A}, and the projection pr 2:J×AApr_2:J\times \mathbf{A}\to \mathbf{A} by s:JAAs:J \mathbf{A}\to \mathbf{A}. Notice that sd 1=id A=sd 0s d_1=id_{\mathbf{A}}=s d_0.

Proposition

The functor (d 1,d 0):AAJA(d_1,d_0): \mathbf{A}\sqcup \mathbf{A}\to J \mathbf{A} is monic on objects and the projection s:JAAs:J \mathbf{A}\to \mathbf{A} is an equivalence. Moreover, the functors d 1d_1 and d 0d_0 are equivalences monic on objects.

Proof

I spare you the proof.

The mapping cylinder of a functor F:AAF:\mathbf{A}\to \mathbf{A} is the category C(F)\mathbf{C}(F) defined by the following pushout square:

(6)

There is a (unique) functor Q B:C(F)BQ_{\mathbf{B}}:\mathbf{C}(F)\to \mathbf{B} such that Q BQ=FsQ_{\mathbf{B}}Q=F s and Q Bi B=id BQ_{\mathbf{B}} i_{\mathbf{B}}=id_{\mathbf{B}}, since square (6) is cocartesian and we have Fsi 1=Fid A=id BFF s i_1=F id_{\mathbf{A}}=id_{\mathbf{B}}F . Let us put i A:=Qd 1i_{\mathbf{A}}:=Q d_1. Then we have

F=Q Bi A:AC(F)B,F=Q_{\mathbf{B}} i_{\mathbf{A}}:\mathbf{A}\to \mathbf{C}(F)\to \mathbf{B},

since Q Bi A=Q BQd 1=Fsd 1=Fid A=FQ_{\mathbf{B}} i_{\mathbf{A}}=Q_{\mathbf{B}} Q d_1=F s d_1=F id_{\mathbf{A}}=F. This is the mapping cylinder factorisation of the functor FF. Let us show that we have a pushout square,

(7)

The square commutes since i A=Qd 1i_{\mathbf{A}}=Q d_1. The square is a pushout, since both the top square and the composite square of the following diagram are cocartesian,

(8)
Proposition

In the mapping cylinder factorisation,

F=Q Bi A:AC(F)B,F=Q_{\mathbf{B}} i_{\mathbf{A}}:\mathbf{A}\to \mathbf{C}(F)\to \mathbf{B},

the functor i Ai_{\mathbf{A}} is monic on objects and the functor Q BQ_{\mathbf{B}} is an equivalence surjective on objects.

Proof

The functor (i A,i B)(i_{\mathbf{A}},i_{\mathbf{B}}) is a cobase change of the functor (d 1,d 0)(d_1,d_0), since the square (7) is cocartesian. Hence the functor (i A,i B)(i_A,i_B) is monic on objects by Proposition 7, since the functor (d 1,d 0)(d_1,d_0) is monic on objects by Proposition 4. It follows that the functor i A=(i A,i B)in 1i_{\mathbf{A}}=(i_{\mathbf{A}},i_{\mathbf{B}})in_1 is monic on objects since the functor in 1:AABin_1:A\to A\sqcup B is monic on objects. Let us now show that the functor Q BQ_{\mathbf{B}} is an equivalence surjective on objects. The functor is obviously surjective on objects, since we have Q Bi B=id BQ_{\mathbf{B}}i_{\mathbf{B}}=id_{\mathbf{B}}. In order to show that it is an equivalence, it suffices to show that the functor i Bi_{\mathbf{B}} is an equivalence by three-for-two. But i Bi_{\mathbf{B}} is a cobase change of the functor 0\partial_0, since the square (6) is cocartesian. It follows that the functor i Bi_{\mathbf{B}} is an equivalence by the Proposition 1, since 0\partial_0 is an equivalence monic on objects by Proposition 4.

Proposition

The category Cat\mathbf{Cat'} admits a weak factorisation system (,)(\mathcal{L}, \mathcal{R}) in which \mathcal{L} is the class functors monic on objects and \mathcal{R} is the class of acyclic isofibration.

Proof

We shall use the characterisation of weak factorisation systems here. The mapping cylinder C(F)\mathbf{C}(F) of a functor F:ABF:\mathbf{A} \to \mathbf{B} is locally small if the category B\mathbf{B} is locally small, since the functor Q B:C(F)BQ_{\mathbf{B}}:\mathbf{C}(F)\to\mathbf{B} is an equivalence by Proposition 5. The same proposition shows that every functor F:ABF:\mathbf{A} \to \mathbf{B} admits a (,)(\mathcal{L}, \mathcal{R})-factorisation. It follows from Lemma 6 that we have \mathcal{L}\,\pitchfork\, \mathcal{R}. The class \mathcal{L} is closed under codomain retracts by Corollary 7. Also the class \mathcal{R} by Lemma 5.

Proof of Theorem 1.

The class 𝒲\mathcal{W} of equivalences in Cat\mathbf{Cat} has the three-for-two property by Lemma 1. The pair (𝒞𝒲,)(\mathcal{C}\,\cap\, \mathcal{W}, \mathcal{F}) is a weak factorisation system by Proposition 3, and the pair (𝒞,𝒲)(\mathcal{C}, \mathcal{F}\,\cap\, \mathcal{W}) is a weak factorisation system by Proposition 6. This completes the proof that the triple (𝒞𝒲,)(\mathcal{C}\,\cap\, \mathcal{W}, \mathcal{F}) is a model structure on the category Cat\mathbf{Cat}. Every object X\mathbf{X} of this model structure is fibrant and cofibrant, since the map X1\mathbf{X}\to 1 is an isofibration and the map X\emptyset \to \mathbf{X} is monic on objects. Hence the model structure is proper by a proposition [here]. Let us show that the model structure is cartesian closed. If S:ABS:\mathbf{A}\to \mathbf{B} and T:UVT:\mathbf{U}\to \mathbf{V}, are two functors, consider the functor

S×T:(A×V) A×U(B×U)B×VS\times'T:(\mathbf{A}\times \mathbf{V}) \sqcup_{\mathbf{A}\times \mathbf{U}}(\mathbf{B}\times \mathbf{U})\to \mathbf{B}\times \mathbf{V}

obtained from the commutative square

(9)

Let us show that the functor S×TS\times'T is a cofibration if SS and TT are cofibrations, and that it is a an equivalence if in addition SS or TT is an equivalence. We have (S×T) 0=S 0×T 0(S\times'T)_0=S_0\times'T_0, since the functor Ob:CatSetOb:\mathbf{Cat}\to \mathbf{Set} preserves cartesian products and pushouts (it is actually continuous and cocontinuous). But the map S 0×T 0S_0\times'T_0 is monic, if S 0S_0 and T 0T_0 are monic by a result [here]. This shows that S×TS\times'T is a cofibration if SS and TT are cofibrations. If in addition TT is an equivalence, then so are the vertical maps in the square, since the class of equivalences is closed under products. Consider the following diagram with a pushout square on the left hand side,

The functor in 2in_2 is an acyclic cofibration by cobase change, since the functor A×T\mathbf{A}\times T is. Hence the functor S×TS\times' T is acyclic by three-for-two, since the functors in 2in_2 and B×T\mathbf{B}\times T are acyclic.

If U:ABU:\mathbf{A}\to \mathbf{B} and F:XYF:\mathbf{X}\to \mathbf{Y}, are two functors, consider the functor

[U,F]:[B,X][B,Y]× [A,Y][A,X][ U,F]':[\mathbf{B},\mathbf{X}] \to [\mathbf{B},\mathbf{Y}] \times_{[\mathbf{A},\mathbf{Y}]}[\mathbf{A},\mathbf{X}]

obtained from the commutative square

Recall that the category [A,X][\mathbf{A},\mathbf{X}] is locally small (resp. small) if A\mathbf{A} is small and X\mathbf{X} is locally small (resp. small).

Corollary

The functor [U,F]\mathbf{[} U,F\mathbf{]}' is an isofibration, if the functor FF is an isofibration and the functor UU is monic on objects. Moreover, the functor [U,F]\mathbf{[} U,F\mathbf{]}' is an equivalence, if in addition one of the fonctors UU or FF is an equivalence.

The Karoubian model structure

Definition

Recall that a functor F:ABF:\mathbf{A}\to \mathbf{B} between small categories is said to be a Morita equivalence if the inverse image functor

F *:[B o,Set][A o,Set]F^*:[\mathbf{B}^o,\mathbf{Set}]\to [\mathbf{A}^o,\mathbf{Set}]

is an equivalence of categories. A functor F:ABF:\mathbf{A}\to \mathbf{B} is a Morita equivalence iff it is fully faithful and every object BBB\in \mathbf{B} is a retract of an object in the image of FF.

Definition

Recall an idempotent e:BBe:B\to B in a category C\mathbf{C} is said to split if there exists a pair of morphisms s:ABs:A\to B and r:BAr:B\to A such that e=sre=s r and rs=1 A rs=1_A. We shall say that C\mathbf{C} is Karoubi complete if every idempotent in C\mathbf{C} splits.

Let SplSpl be the category freely generated by two arrows s:01s:0\to 1 and r:10r:1\to 0 such that rs=1 0r s=1_0. And let IdemIdem be the category freely generated by one idempotent e:11e:1\to 1. Then the functor idemSplitidem \to Split which takes ee to srs r is fully faithful.

Definition

We shall say that a functor is an idfibration if it has the right liffting property with respect to the inclusion IdemSplIdem \to Spl. A category C\mathbf{C} is Karoubi complete iff the functor C1\mathbf{C}\to \mathbf{1} is an idfibration.

An idfibration is an isofibration. It is easy to see that an isofibration F:XYF:\mathbf{X}\to \mathbf{Y} is an idfibration iff the functor FF reflects split idempotents, that is, if the implication

F(e)splitsesplitsF(e)\quad \mathrm{splits} \quad \Rightarrow \quad e \quad \mathrm{splits}

is true for every idempotent eXe\in \mathbf{X}.

Theorem

The category of small categories Cat\mathbf{Cat} admits a model structure (𝒞,𝒲,)(\mathcal{C},\mathcal{W},\mathcal{F}) in which 𝒞\mathcal{C} is the class of functors monic on objects, 𝒲\mathcal{W} is the class of Morita equivalences and \mathcal{F} is the class of idfibrations. The model structure is cartesian closed and left proper. We shall say that it is the Karoubian model structure on the category Cat\mathbf{Cat}.

The model structures on Cat/I

The Thomason model structure

Exercises

Exercise

Show that the notion of isofibration is self-dual: a functor F:XYF:\mathbf{X}\to \mathbf{Y} is an isofibration iff the opposite functor F o:X oY oF^o:\mathbf{X}^o\to \mathbf{Y}^o is an isofibration.

Exercise

Show that a functor F:XYF:\mathbf{X}\to \mathbf{Y} is an isofibration iff it has the right lifting property with respect to the inclusion {0}J\{0\}\subset J.

References

Revised on June 20, 2012 at 20:57:13 by revert-spam