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Cisinski model structure

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Contents

General

Definition

For 𝒯\mathcal{T} a topos, a Cisinski model structure on 𝒯\mathcal{T} is a model category structure on 𝒯\mathcal{T} such that

  1. the cofibrations are precisely the monomorphisms;

  2. it is a cofibrantly generated model category.

Remark

Since every topos is a locally presentable category, a Cisinski model structure is in particular a combinatorial model category structure.

Remark

Since every topos is an adhesive category, monomorphisms are automatically preserved by pushout.

Definition

Say a class WMor(𝒯)W \in Mor(\mathcal{T}) is a localizer on 𝒯\mathcal{T} if it is a class of weak equivalences in a Cisinski model structure on 𝒯\mathcal{T}.

Proposition

Every small set of morphisms ΣMor(𝒯)\Sigma \in Mor(\mathcal{T}) is contained in a smallest localizer, def. 2, W(Σ)W(\Sigma).

One says that W(Σ)W(\Sigma) is the localiser generated by Σ\Sigma.

So in particular a Cisinski model structure always exists.

On presheaf toposes

We discuss how on a presheaf topos equipped with a suitable notion of cylinder objects Cisinski model structures can be characterized fairly explicitly. After some preliminaries, the main statement is theorem 1 below.

(This follows sections 1.2 and 1.3 of Cisinski 06).

Preliminary notions

Let AA be a small category. Write PSh(A)PSh(A) or [A op,Set][A^{op}, Set] for the category of presheaves over AA. We introduce here, culminating in def. 11 below, the ingredients of a homotopical structure on PSh(A)PSh(A), which is a choice of functorial cylinder object together with a compatible notion of anodyne extensions. Further below in def. 12 this defines a model category structure on PSh(A)PSh(A).

Definition

For XX in PSh(A)PSh(A), a cylinder on XX in the following means a cylinder object, denoted IXI \otimes X, factoring the codiagonal

XX( X 0, X 1)IXσ XX X \coprod X \stackrel{(\partial_X^0, \partial_X^1)}{\to} I \otimes X \stackrel{\sigma_X}{\to} X

such that the first morphism is a monomorphism.

A homomorphism of such cyclinders is a pair of morphisms XYX \to Y and IXIYI \otimes X \to I \otimes Y in PSh(A)PSh(A), making the evident squares commute. This defines a category Cyl(A)Cyl(A) of cylinder objects on presheaves on AA, equipped with a forgetful functor Cyl(A)PSh(A)Cyl(A) \to PSh(A) that sends a cylinder IXI \otimes X to its underlying object XX.

A functorial cylinder object over AA is a section of this functor.

This is (Cisinski 06, def. 1.3.1).

In the following, let J:PSh(A)Cyl(A)J : PSh(A) \to Cyl(A) be a choice of functorial cylinder object. Equivalently, this is a choice of endofunctor

I():PSh(A)PSh(A) I \otimes (-) : PSh(A) \to PSh(A)

equipped with natural transformations

(I)()( () 0, () 1)I()Id (\partial I)\otimes (-) \stackrel{(\partial^0_{(-)}, \partial^1_{(-)})}{\to} I \otimes (-) \to Id

where (I)X:=XX(\partial I) \otimes X := X \coprod X, such that the composite is the functorial codiagonal, and where the first transformation is a monomorphism.

Definition

For f,g:XYf,g : X \to Y two morphisms in PSh(A)PSh(A), we say an elementary JJ-homotopy fgf \Rightarrow g from ff to gg is left homotopy from ff to gg with respect to the chosen cylinder object JJ, hence a morphism η:IXY\eta : I \otimes X \to Y fitting into a diagram

X X 0 f IX η Y X 1 g X. \array{ X && \\ \downarrow^{\mathrlap{\partial^0_X}} & \searrow^f \\ I \otimes X &\stackrel{\eta}{\to}& Y \\ \uparrow^{\mathrlap{\partial^1_X}} & \nearrow_g \\ X } \,.

We say JJ-homotopy for the equivalence relation generated by this.

(Cisinski 06, def. 1.3.3).

Proposition

JJ-homotopy is compatible with composition in PSh(A)PSh(A).

Proof

It is sufficient to show that elementary JJ-homotopies are compatible with composition.

So for

X f IX η Y g X \array{ X && \\ \downarrow & \searrow^f \\ I \otimes X &\stackrel{\eta}{\to}& Y \\ \uparrow & \nearrow_g \\ X }

an elementary JJ-homotopy fgf \Rightarrow g, and for

Y f IY η Z g Y \array{ Y && \\ \downarrow & \searrow^{f'} \\ I \otimes Y &\stackrel{\eta}{\to}& Z \\ \uparrow & \nearrow_{g'} \\ Y }

one fgf' \Rightarrow g', we obtain an elementary homotopy ffgff' \circ f \Rightarrow g' \circ f by forming

X f Y f IX If IY η Z g X f Y \array{ X & \stackrel{f}{\to} & Y && \\ \downarrow && \downarrow & \searrow^{f'} \\ I \otimes X &\stackrel{I \otimes f}{\to}& I \otimes Y &\stackrel{\eta}{\to}& Z \\ \uparrow && \uparrow & \nearrow_{g'} \\ X &\stackrel{f}{\to}& Y }

and then an elementary JJ-homotopy gfggg' \circ f \Rightarrow g '\circ g by forming

X f IX η Y g Z g X. \array{ X && \\ \downarrow & \searrow^f \\ I \otimes X &\stackrel{\eta}{\to}& Y &\stackrel{g'}{\to}& Z \\ \uparrow & \nearrow_g \\ X } \,.

Together this generates a JJ-homotopy ffggf' \circ f \Rightarrow g' \circ g.

Hence the following is well defined.

Definition

Write Ho J(A)Ho_J(A) for the category whose objects are those of PSh(A)PSh(A), and whose morphisms are JJ-homotopy equivalence classes of morphisms in PSh(A)PSh(A) – the JJ-homotopy category. Write

Q:PSh(A)Ho J(A) Q : PSh(A) \to Ho_J(A)

for the projection functor.

Definition

A morphism f:XYf : X \to Y in PSh(A)PSh(A) is called a JJ-homotopy equivalence if it is sent by QQ to an isomorphism.

An object XPSh(A)X \in PSh(A) is called JJ-contractible if X*X \to * is a JJ-homotopy equivalence.

(Cisinski 06, def. 1.3.4).

Definition

A morphism f:XYf : X \to Y in PSh(A)PSh(A) is called an acyclic fibration if it has the right lifting property against all monomorphisms.

(Cisinski 06, def. 1.2.18).

Proposition

Every acyclic fibration in PSh(A)PSh(A) is a JJ-homotopy equivalence.

More is true: every trivial fibration p:XYp : X \to Y

  1. has a section s:YXs : Y \to X;

  2. which is also a JJ-homotopy left inverse;

  3. by an elementary JJ-homotopy h:idsph : id \Rightarrow s \circ p which satisfies ph=pσ Xp \circ h = p \circ \sigma_X.

(Cisinski 06, lemma 1.3.5).

Proof

The existence of the section s:YXs : Y \to X follows by right lifting against the monomorphism Y\emptyset \to Y (out of the initial object)

X s p Y = Y. \array{ \emptyset &\to& X \\ \downarrow &\nearrow_s& \downarrow^{\mathrlap{p}} \\ Y &\stackrel{=}{\to}& Y } \,.

The JJ-homotopy hh is obtained by lifting in the diagram

XX (id X,sp) X ( X 0, X 1) p IX pσ X Y. \array{ X \coprod X &\stackrel{(id_X, s \circ p)}{\to}& X \\ {}^{\mathllap{(\partial^0_X, \partial^1_X)}}\downarrow && \downarrow^{\mathrlap{p}} \\ I \otimes X &\stackrel{p \circ \sigma_X}{\to}& Y } \,.
Definition

An elementary homotopical datum on PSh(A)PSh(A) is a functorial cylinder JJ, def. 3, such that

  1. the functor I()I \otimes (-) commutes with small colimits;

  2. for all monomorphisms f:KLf : K \to L the diagrams

    K (id,f) L (e,id) (e,id) IK If IL \array{ K &\stackrel{(id, f)}{\to}& L \\ {}^{\mathllap{(e,id)}}\downarrow && \downarrow^{\mathrlap{(e,id)}} \\ I \otimes K &\stackrel{I \otimes f}{\to}& I \otimes L }

    for e{0,1}e \in \{0,1\} is a pullback square.

(Cisinski 06, def. 1.3.6).

Remark

For

R S T U \array{ R &\to & S \\ \downarrow && \downarrow \\ T &\to& U }

a pullback square in PSh(A)PSh(A) of two monomorphisms SUS \hookrightarrow U and TUT \hookrightarrow U, the universal morphism out of the pushout

T RSU T \coprod_R S \to U

is also a monomorphism, usually written as the morphism out of the union

TSU. T \cup S \hookrightarrow U \,.
Proof

All this follows, for instance, from the corresponding statements in Set, over each object of AA.

Definition

Let

ϵ:{ϵ}()I(), \partial^\epsilon : \{\epsilon\}\otimes(-) \hookrightarrow I \otimes(-) \,,

for ϵ=0,1\epsilon = 0,1, be the subfunctor which is the image of ϵ:Id PSh(A)I()\partial^\epsilon : Id_{PSh(A)} \to I \otimes (-).

This way for any XPSh(A)X \in PSh(A) the boundary inclusions X ϵ:XIX\partial^\epsilon_X : X \to I \otimes X are identified with

ϵid X:{ϵ}XIX. \partial^\epsilon \otimes id_X : \{\epsilon\} \otimes X \to I \otimes X \,.
Remark

The second condition on an elementary homotopical datum, def. 8 implies that the canonical morphism

JK(J)LJL J \otimes K \cup (\partial J) \otimes L \to J \otimes L

is a monomorphism.

(Cisinski 06, remark 1.3.7).

Proof

The condition implies that

{ϵ}K id {ϵ}j {ϵ}L ϵid K ϵid L IK (id I)j IL \array{ \{\epsilon\} \otimes K &\stackrel{id_\{\epsilon\} \otimes j}{\to}& \{\epsilon\} \otimes L \\ {}^{\mathllap{\partial^\epsilon \otimes id_K}}\downarrow && \downarrow^{\mathrlap{\partial^\epsilon \otimes id_L}} \\ I \otimes K &\stackrel{(id_I) \otimes j}{\to}& I \otimes L }

is a pullback, for ϵ=0,1\epsilon = 0,1, hence that

IK id Ij IL iid K iid L IK (id I)j IL \array{ \partial I \otimes K &\stackrel{id_{\partial I} \otimes j}{\to}& \partial I \otimes L \\ {}^{\mathllap{i \otimes id_K}}\downarrow && \downarrow^{\mathrlap{i \otimes id_L}} \\ I \otimes K &\stackrel{(id_I) \otimes j}{\to}& I \otimes L }

is a pullback. So the statement follows with remark 3.

Example

Let IPSh(A)I \in PSh(A) be any object equipped with two points (global sections) 0, 1:*I\partial^0, \partial^1 : * \to I which are disjoint in that

* 0 * 1 I \array{ \emptyset &\to& * \\ \downarrow && \downarrow^{\mathrlap{\partial^0}} \\ * &\stackrel{\partial^1}{\to}& I }

is a pullback square (here \emptyset is the initial object in PSh(A)PSh(A)). This induces a functorial cylinder by the assignment

XI×X, X \mapsto I \times X \,,

where on the right we have the cartesian product.

This defines an elementary homotopical datum in the sense of def. 8.

(Cisinski 06, example 1.3.8).

Proof

The disjointness of the two points ensures that **( 0, 1)I* \coprod * \stackrel{(\partial^0, \partial^1)}{\to} I is a monomorphism.

The interval commutes with colimits as these and the product are computed objectwise, and products in Set commute with colimits. More abstractly: by the Giraud theorem valid in the presheaf topos PSh(A)PSh(A) we have “universal colimits”: they are preserved by pullback, and in particular by cartesian product. Therefore the first clause of def. 8 is satisfied.

Similarly, the second axiom of def. 8 holds because limits commute over each other.

Example

(Lawvere cylinder)

Let I:=ΩI := \Omega be the subobject classifier in the presheaf topos PSh(A)PSh(A). This is the presheaf which to an object UU of AA assigns the set of subobjects of the representable functor given by UU (the sieves on UU)

Ω:USub(U) \Omega : U \mapsto Sub(U)

and which to a morphism f:U 1U 2f : U_1 \to U_2 assigns the pullback functor f *:Sub(U 2)Sub(U 1)f^* : Sub(U_2) \to Sub(U_1).

Let then

0, 1:=,:*Ω \partial^0, \partial^1 := \top, \bot: * \to \Omega

be the morphisms that classify top and bottom, respectively, the terminal and the initial subobject of the terminal object.

This is a segment object in the sense of example 1 (the “Lawvere-segment”).

(Cisinski 06, example 1.3.9).

Proof

That the two points are separated, in that

* * Ω \array{ \emptyset &\to& * \\ \downarrow && \downarrow^{\mathrlap{\top}} \\ * &\stackrel{\bot}{\to}& \Omega }

is a pullback, is the defining property of the subobject classifier.

Definition

For JJ an elementary homotopical datum on PSh(A)PSh(A), a class of anodyne extensions is a class AnExtMor(PSh(A))AnExt \subset Mor(PSh(A)) such that

  1. there exists a small set SS with

    AnExt=LLP(RLP(S)) AnExt = LLP(RLP(S))
  2. for KLK \hookrightarrow L a monomorphism, the pushout product morphisms

    (IK)({e}L)IL (I \otimes K) \cup (\{e\} \otimes L) \to I \otimes L

    (by Joyal-Tierney calculus to be thought of as “(KL)¯({e}I)(K \to L) \bar \otimes (\{e\} \to I)”)

    are in AnAxtAnAxt, for e{0,1}e \in \{0,1\};

  3. if KLK \to L is in AnExtAnExt, then so is

    (IK)((I)L)IL. (I \otimes K) \cup ((\partial I) \otimes L) \to I \otimes L \,.

    (hence “(KL)¯(II)(K \to L) \bar \otimes (\partial I \to I)”).

(Cisinski 06, def. 1.3.10).

Proposition

A class of anodyne extensions

(Cisinski 06, remark. 1.3.11).

Proof

By prop. 11, XPSh(A)X \in PSh(A) is |Mor(A/X)||Mor(A/X)|-compact. Therefore the set Λ\Lambda admits a small object argument, which shows the first statement (see there).

Since monomorphisms are closed under these operations, the second statement follows.

The third statement follows by choosing the morphism in the second item of def. 10 to be K\emptyset \to K and using that by def. 8 the interval commutes with colimits, so that II \otimes \emptyset \simeq \emptyset.

Finally, to see that with j:KLj : K \to L also Ij:ILILI \otimes j : I \otimes L \to I \otimes L is anodyne, consider the naturality diagram of the endpoint inclusion

K j L k 0 IK j (IK){0}L Ij k IL, \array{ K &\stackrel{j}{\to}& L \\ \downarrow^{\mathrlap{\partial^0_k}} && \downarrow \\ I \otimes K &\stackrel{j'}{\to}& (I \otimes K) \cup \{0\}\otimes L \\ & {}_{I \otimes j} \searrow & \downarrow^{\mathrlap{k}} \\ && I \otimes L } \,,

factored through the top pushout square, as indicated. Here jj' is anodyne, being a pushout of an anodyne morphism, and kk is anodyne by the second clause in def. 10. Therefore also their composite IjI \otimes j is anodyne.

Definition

A homotopical structure on PSh(A)PSh(A) is a choice of elementary homotopical datum JJ, def. 8 and a corresponding choice of a class of anodyne extensions AnExtAnExt, def. 10.

(Cisinski 06, def. 1.3.14).

The model structure

Definition

Let AA be a small category and (J,AnExt)(J, AnExt) a homotopical structure, def. 11, on PSh(A)PSh(A). Define the following classes of objects and morphisms in PSh(A)PSh(A):

  • the cofibrations are the monomorphisms;

  • the fibrant objects are those XPSh(A)X \in PSh(A) for which the terminal morphism X*X \to * has the right lifting property against the anodyne extensions (the morphisms in AnExtAnExt, def. 10);

  • the weak equivalences are those morphisms f:ABf : A \to B, such that for all fibrant objects XX the induced morphism in the JJ-homotopy category, def. 5,

    Ho J(f,X):Ho J(B,X)Ho J(A,X) Ho_J(f,X) : Ho_J(B,X) \to Ho_J(A, X)

    is an isomorphism (a bijection of sets).

(Cisinski 06, def. 1.3.21).

Theorem

With the classes of morphisms as in def. 12, PSh(A)PSh(A) is a cofibrantly generated model category.

(Cisinski 06, theorem 1.3.22).

Proof

The retract properties are clear, as is the 2-out-of-3 property for weak equivalences, see lemma 1 below.

The lifting properties hold by prop. 6 below, the proof of which is in the section Lifting below.

The factorization properties hold by cor. 1 and cor. 4 below, which are in the section Factorization below.

The existence of a set of generating cofibrations is prop. 12 below, that of generating acyclic cofibrations is prop. 15 below.

Proposition

The homotopy category of (PSh(A),W)(PSh(A), W) is (up to equivalence) the full subcategory of Ho JHo_J, def. 5, on the fibrant objects.

(Cisinski 06, 1.3.23).

Before coming to the proof of these lemmas, the following two statements say that the terminology introduced so far is indeed consistent with the meaning of this theorem.

Proposition

The morphisms called acyclic fibrations in def. 7 are indeed precisely the acyclic fibrations with respect to the model structure of theorem 1.

(Cisinski 06, theorem 1.3.27).

Proposition

Every anodyne extension, def. 10, is a weak equivalence in the model structure of theorem 1.

Remark

What is not true in general is the converse of prop. 13, that every acyclic cofibration is an anodyne extension. (A counterexample derives from chapter XI, remark 2.4 in Goerss, Jardine Simplicial homotopy theory.)

(Cisinski 06, remark 1.3.46).

Proposition

For the model structure from theorem 1, the following are equivalent:

  1. every acyclic cofibration is an anodyne extension;

  2. every morphism with right lifting against anodyne extensions is a fibration;

  3. every weak equivalence with right lifting against anony extensions is an acyclic fibration;

  4. every morphism with right lifting against anodyne extensions factors as an anodyne extension followed by a fibration.

(Cisinski 06, prop. 1.3.47).

Definition

A homotopical structure, def. 11, on a presheaf category is called complete if the model structure from theorem 1 satisfies the equivalent conditions of prop. 8.

(Cisinski 06, def. 1.3.48).

We discuss the proof of prop. 8 below in Completeness.

Proof

We collect lemmas to prove theorem 1 and related statements. A little bit of work is required for demonstrating the lifting axioms, which we do below in Lifting. A little bit more work is required for demonstrating the factorization axioms, which we do below in Factorization. Finally, the proof of the equivalence of the conditions of completeness is in Completeness.

Lemma

Every JJ-homotopy equivalence, def. 6, is a weak equivalence.

The weak equivalences satisfy the two-out-of-three-property and are stable under retracts.

(Cisinski 06, remark 1.3.24).

Proof

The first statement holds by definition of Ho JHo_J.

The second statement also follows directly from the definition. If for AfBgCA \stackrel{f}{\to} B \stackrel{g}{\to} C and fibrant XX in the composite

Ho J(C,X)g *Ho J(B,X)f *Ho J(A,X) Ho_J(C,X) \stackrel{g^*}{\to} Ho_J(B,X) \stackrel{f^*}{\to} Ho_J(A,X)

two of three are isomorphisms, then so is the third.

Lifting

We discuss the lifting properties in the model structure of def. 12.

Since fibrations are defined to be the morphisms satisfying the right lifting property against acyclic cofibration, we only need to show that the fibrations which are also weak equivalences have the right lifting property against the monomorphisms. For this it is sufficient to show prop. 6. This we do now, after a lemma.

Definition

A deformation retract f:XYf : X \to Y in PSh(A)PSh(A) is a retract with retraction g:YXg : Y \to X, which is also a section of ff up to a JJ-homotopy h:id Yfgh : id_Y \Rightarrow f \circ g. It is strong if h(If)=σ Y(If)h \circ (I \otimes f) = \sigma_Y \circ (I \otimes f).

A dual deformation restract f:XYf : X \to Y has a section by a morphism g:YXg : Y \to X and is also a retract up to a JJ-homotopy k:id Xgfk : id_X \Rightarrow g \circ f. Is is strong if fk=fσ Xf \circ k = f \circ \sigma_X.

Lemma

Every acyclic fibration is a dual strong deformation retract, def. 14.

Every section of an acyclic fibration is a strong deformation retract.

(Cisinski 06, prop. 1.3.26).

Proof

The first statement is a direct consequence of prop 3.

For the second statement, let p:XYp : X \to Y be an acyclic fibration, and let s:YXs : Y \to X be a section. This induces a commuting square

(IY)((I)X) (sσ Y,(id X,sp)) X h p IX pσ X Y, \array{ (I \otimes Y) \cup ((\partial I) \otimes X) &\stackrel{(s\circ \sigma_Y,(id_X, s \circ p))}{\to}& X \\ \downarrow &\nearrow_{h}& \downarrow^{\mathrlap{p}} \\ I \otimes X &\stackrel{ p \circ \sigma_X}{\to}& Y } \,,

where the lift hh exists by assumption on pp (ss is necessarily a monomorphism, being a section).

The resulting component triangle

(I)X (id X,sp) X h IX \array{ (\partial I) \otimes X &\stackrel{(id_X, s \circ p)}{\to}& X \\ \downarrow &\nearrow_{h}& \\ I \otimes X }

exhibits ss as a deformation retract, and the other resulting component triangle

IY sσ Y X Is h IX \array{ I \otimes Y &\stackrel{s\circ \sigma_Y}{\to}& X \\ \downarrow^{\mathrlap{I \otimes s}} &\nearrow_{h} \\ I \otimes X }

says that h(Is)=sσ Yh\circ (I \otimes s) = s \circ \sigma_Y, hence by naturality of cylinders =σ X(Is)\cdots = \sigma_X \circ (I \otimes s), hence that the deformation retract is indeed strong.

Proof

of prop. 6

First to see that the acyclic fibrations of def. 7 are indeed fibrations and weak equivalences:

By lemma 2 every acyclic fibration is in particular a JJ-homotopy equivalence, hence by lemma 1 a weak equivalence. Moreover, by def. 7 the acyclic fibrations right-lift against monomorphisms, hence in particular against the acyclic cofibrations, hence are fibrations.

Conversely, let p:XYp : X \to Y be a fibration which is also a weak equivalence. We need to show that it has the right lifting property against all monomorphisms.

By prop. 12, proven below, we may apply the small object argument to factor p=qjp = q \circ j as a monomorphism jj followed by an acyclic fibration qq. By the previous argument, qq is a weak equivalence, and so by lemma 1 so is jj. Therefore, since pp is a fibration, we have a lift σ\sigma in

id j σ p q . \array{ &\stackrel{id}{\to}& \\ \downarrow^{\mathrlap{j}} & \nearrow_\sigma& \downarrow^{p} \\ &\stackrel{q}{\to}& } \,.

This equivalently exhibits pp as a retract of qq

j σ p q p id id . \array{ &\stackrel{j}{\to} & & \stackrel{\sigma}{\to} & \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{q}} && \downarrow^{\mathrlap{p}} \\ & \stackrel{id}{\to} & & \stackrel{id}{\to} & } \,.

So by lemma 4 with qq also pp is an acyclic fibration.

(Cisinski 06, remark 1.3.28).

Factorization

We discuss the two factorization axioms for the model category structure from def. 12 to be established. First for factorizations into cofibrations followed by acyclic fibrations, then for factorizations into acyclic cofibrations followed by fibrations.

Cofibration followed by acyclic fibration

For showing that every morphism factors as a monomorphism followed by an acyclic fibration, it is by prop. 6 sufficient to show that the monomorphisms are generated by a small set that admits the small object argument. This we do now.

This section follows (Cisinski 06, section 1.2).

We start with some entirely general statements about compact objects.

Proposition

For AA a small category, let α=|Mor(A)|\alpha = {\vert Mor(A)\vert} be the smallest regular cardinal \geq the cardinality of the set of morphisms of AA. Then the limit functor

lim :Func(A,Set)Set \lim_\leftarrow : Func(A,Set) \to Set

commutes with α\alpha-filtered colimits / α\alpha-directed colimits.

(Cisinski 06, prop. 1.2.9).

Proposition

Let CC be a category with all small colimits, let α\alpha be a cardinal, let AA be a small category and finally let F:ACF : A \to C be a functor with values in α\alpha-compact objects in CC. Then the colimit lim F\lim_\to F is λ\lambda-compact object, for λ\lambda the maximum of α\alpha and the cardinality |MorA|{\vert Mor A\vert} of the set of morphisms of AA.

(Cisinski 06, prop. 1.2.10).

Proof

Let G:ICG : I \to C be a λ\lambda-filtered diagram.

Then by prop. 9 we have natural isomorphisms

lim IC(lim AF,G) lim Ilim AC(F,G) lim Alim IC(F,G), \begin{aligned} \lim_{\underset{I}{\to}} C(\lim_{\underset{A}{\to}} F, G) & \simeq \lim_{\underset{I}{\to}} \lim_{\underset{A}{\leftarrow}} C(F, G) \\ & \simeq \lim_{\underset{A}{\leftarrow}} \lim_{\underset{I}{\to}} C(F, G) \end{aligned} \,,

because the λ\lambda-filtered diagram is at least |Mor(A)|{\vert Mor(A)\vert}-filtered and hence, by prop. 9, its colimit commutes with the limit over AA.

Now since each F(a)F(a) is assumed to be α\alpha-compact and hence is also λ\lambda-compact, we conclude with the natural isomorphisms

lim AC(F,lim IG) C(lim AF,lim IG). \begin{aligned} \cdots & \simeq \lim_{\underset{A}{\leftarrow}} C(F, \lim_{\underset{I}{\to}} G) \\ & \simeq C(\lim_{\underset{A}{\to}} F, \lim_{\underset{I}{\to}} G) \end{aligned} \,.

For XPSh(A)X \in PSh(A), write A/XA/X for the category of elements of XX. Write |Mor(A/X)|{\vert Mor(A/X)\vert} for the cardinality of the set of morphisms of the category of elements (throughout assuming AA to be a small category).

Proposition

For XPSh(A)X \in PSh(A) any object, the hom functor Hom(X,):PSh(A)SetHom(X, -) : PSh(A) \to Set preserves |Mor(A/X)|{\vert Mor(A/X)\vert}-filtered colimits.

In other words: XX is a |Mor(A/X)|{\vert Mor(A/X)\vert}-compact object.

(Cisinski 06, cor. 1.2.11).

Proof

By the co-Yoneda lemma XX is the colimit over its elements

Xlim (A/XPSh(A)). X \simeq \lim_\to (A/X \to PSh(A)) \,.

Since the image of the functor A/XPSh(A)A/X \to PSh(A) is in representables, which are maximally compact, the stament follows with prop. 10.

Definition

A cellular model on PSh(A)PSh(A) is a choice of a small set IMor(PSh(A))I \subset Mor(PSh(A)) of monomorphisms, such that the class of all monomorphisms is generated from it

Monos=LLP(RLP(I)). Monos = LLP(RLP(I)) \,.

(Cisinski 06, def. 1.2.26).

Remark

By prop. 11 we may apply the small object argument in PSh(A)PSh(A) and so it follows that LLP(RLP(I))LLP(RLP(I)) is the smallest class containing II that is closed under pushout, retract and transfinite composition.

The following lemma will be used to show that cellular structures always exist.

Lemma

Let CMor(PSh(A))C \subset Mor(PSh(A)) be a class of morphisms, and DPSh(A)D \subset PSh(A) a small set of objects, such that

  1. CC is closed under pushouts, retracts and transfinite composition;

  2. if f,gMor(PSh(A))f,g \in Mor(PSh(A)) are two composable morphisms with ff and gfg \circ f in CC, then also gg is in CC;

  3. every XPSh(A)X \in PSh(A) is the union of those of its sub-objects isomorphic to an object in DD;

  4. for every XYX \to Y in CC and every sub-object ZZ of YY in DD, there is a sub-object TYT \hookrightarrow Y from DD, which contains ZZ and such that TXTT \cap X \to T is in CC.

Then the small set IMor(PSh(A))I \subset Mor(PSh(A)) of morphisms in CC with codomain in DD, generates CC as

C=LLP(RLP(I)). C = LLP(RLP(I)) \,.

(Cisinski 06, lemma. 1.2.24).

Proof

A bit of work…

Proposition

There exists a cellular structure, def. 15, on PSh(A)PSh(A).

The set II can be chosen to consist of morphisms into quotient objects of representables.

(Cisinski 06, prop. 1.2.27).

Proof

Take in lemma 3 CC to be the class of monomorphisms and DD to be the class of quotients of representables.

Corollary

There exists a functorial factorization of morphisms in PSh(A)PSh(A) into a monomorphism followed by an acyclic fibration.

(Cisinski 06, cor. 1.2.28).

Proof

By prop. 11 every object is α\alpha-small, for some α\alpha. Therefore by prop. 12 we can apply the small object argument.

Acyclic cofibration followed by fibration

We show now for def. 12 that every morphism factors as an acyclic cofibration followed by a fibration. Since the fibrations are defined by right lifting against acylcic cofibrations, for this it is sufficient to establish a set of generating acyclic cofibrations. This is the statement of prop. 15 below. Establishing this takes a few technical lemmas.

Lemma

Every morphism admits a factorization into an anodyne extension, followed by a morphism having the right lifting property against anodyne extensions.

(Cisinski 06, remark 1.3.29).

Proof

By the small object argument, in view of prop. 4.

Lemma

If TPSh(A)T \in PSh(A) is fibrant, then for any KPSh(A)K \in PSh(A) elementary JJ-homotopy, def. 4, is already an equivalence relation on Hom PSh(A)(K,T)Hom_{PSh(A)}(K,T) and coincides with JJ-homotopy.

(Cisinski 06, lemma 1.3.30).

Proposition

Every anodyne extension is a weak equivalence.

(Cisinski 06, lemma 1.3.31).

Proof

For j:KLj : K \to L an anodyne extension and TT a fibrant object, we need to show that

Ho J(j,T):Ho J(L,T)Ho J(K,T) Ho_J(j,T) : Ho_J(L,T) \to Ho_J(K,T)

is a bijection.

It is surjective by the defining lifting property, which provides σ\sigma in

K T j σ L. \array{ K &\stackrel{}{\to}& T \\ \downarrow^{\mathrlap{j}} & \nearrow_{\mathrlap{\sigma}} \\ L } \,.

To see injectivity, let l 0,l 1:LTl_0, l_1 : L \to T be two morphisms such that l 0jl_0 \circ j and l 1jl_1 \circ j coincide in Ho JHo_J. By lemma 5 this is the case precisely if there is an elementary JJ-homotopy h:IKTh : I \otimes K \to T relating them. This induces the horizontal morphism in the diagram

(IK)((I)L) (h,(l 0,l 1)) T η IL, \array{ (I \otimes K) \cup ((\partial I) \otimes L) &\stackrel{(h,(l_0,l_1))}{\to}& T \\ \downarrow & \nearrow_{\mathrlap{\eta}} \\ I \otimes L } \,,

where the left morphism is anodyne, by the second clause of def. 10, so that the lift denoted η\eta exists. This lift exhibits a JJ-homotopy l 0l 1l_0 \Rightarrow l_1, hence shows that l 0l_0 was already equal to l 1l_1 in Ho JHo_J, hence that j *j^* is injective.

Lemma

A morphism between fibrant objects is a weak equivalence precisely if it is a JJ-homotopy equivalence, def. 6.

(Cisinski 06, lemma 1.3.32).

Proof

Is is clear that every JJ-homotopy is a weak equivalence. Conversely, let f:XYf : X \to Y be a weak equivalence between fibrant objects. Write Ho J fibHo JHo_J^{fib} \hookrightarrow Ho_J for the full subcategory of Ho JHo_J, def. 6 on the fibrant objects. The localization Q(f)Q(f) is by definition in Ho J fibHo_J^{fib} and for all objects THo J fibT \in Ho_J^{fib} the morphism Ho J fib(Q(f),T)Ho_J^{fib}(Q(f), T) is an isomorphism. By the Yoneda lemma, therefore, Q(f)Q(f) itself is an isomorphism in Ho J fibHo_J^{fib}, hence also in Ho JHo_J, hence is a weak equivalence.

Lemma

If a morphism has the right lifting property against the anodyne extensions, then it is an acyclic fibration precisely if it is a dual strong deformation retract.

(Cisinski 06, lemma 1.3.33).

Proof

That the former implies the latter was the statement of lemma 2. Conversely, let pp be a dual strong deformation retract, meaning that there is s:YXs : Y \to X with ps=idp \circ s = id, as well as a morphism k:IXXk : I \otimes X \to X exhibiting a JJ-homotopy idspid \Rightarrow s \circ p. This being strong means that pk=pσ Xp \circ k = p \circ \sigma_X.

We need to show that this implies for

K a X i p L b Y \array{ K &\stackrel{a}{\to}& X \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{p}} \\ L &\stackrel{b}{\to}& Y }

a commuting diagram with ii a monomorphism, there is a lift. To this end, observe that the given structures induce a morphism

(IK)({1}L)X (I \otimes K) \cup (\{1\} \otimes L) \to X

such that

(IK)({1}L) (k(Ia),sb) X j p IL σ L L b Y. \array{ (I \otimes K) \cup (\{1\} \otimes L) &&\stackrel{(k \circ (I \otimes a), s \circ b)}{\to}&& X \\ \downarrow^{\mathrlap{j}} && && \downarrow^{\mathrlap{p}} \\ I \otimes L &\stackrel{\sigma_L}{\to}& L &\stackrel{b}{\to}& Y } \,.

By the second clause of def. 10 the morphism on the left is an anodyne extension, and so this diagram admits a lift h:ILXh : I \otimes L \to X. One see that l:=h L 0l := h \circ \partial^0_L is a lift of the original square above.

Lemma

A morphism into a fibrant object with right lifting property against anodyne extensions is a weak equivalence precisely if it is an acyclic fibration.

(Cisinski 06, lemma 1.3.34).

Proof

We already know from prop. 6 that acyclic fibrations are weak equivalences.

So let YY be fibrant and let p:XYp : X \to Y be a weak equivalence that has rlp against anodyne extensions. We need to show that pp is an acyclic fibration. By lemma 7 it is sufficient to show that it is a dual strong deformation retract.

By lemma 6 pp is also a JJ-homotopy equivalence. By lemma 5 this is exhibited by an elementary JJ-homotopy k:IYYk : I \otimes Y \to Y, which in particular gives a commuting diagram

Y t X Y 1 p IY k Y, \array{ Y &\stackrel{t}{\to}& X \\ \downarrow^{\mathrlap{\partial^1_Y}} && \downarrow^{\mathrlap{p}} \\ I \otimes Y &\stackrel{k}{\to}& Y } \,,

from which we obtain a lift k:IYXk' : I \otimes Y \to X. Set then

s:=k Y 0. s := k' \circ \partial^0_Y \,.

One finds then ps=id Yp \circ s = id_Y.

Next …

Corollary

A cofibration into a fibrant object is a weak equivalence precisely if it is an anodyne extension.

(Cisinski 06, cor 1.3.35).

Proof

By lemma 13 we already know that every anodyne extension is a weak equivalence. So we need to show that a cofibration i:ATi : A \to T into a fibrant object TT which is a weak equivalence is also an anodyne extension. By prop. 4 we may factor this as i=qji = q \circ j, with jj an anodyne extension and QQ having RLP against anodyne extensions. Since jj is a weak equivalence, by 2-out-of-3 so is qq. By lemma 8 qq is an acyclic fibration.

Therefore we have a lift ss in

id q s i j \array{ &\stackrel{id}{\to}& \\ \downarrow^{\mathrlap{q}} &\nearrow_s& \downarrow^{\mathrlap{i}} \\ &\stackrel{j}{\to}& }

and this exhibits ii as a retract of jj. Hence with jj also ii is an anodyne extension.

Proposition

A cofibration is a weak equivalence precisely if it has the left lifting property against morphisms into a fibrant object that have the right lifting property against anodyne extensions.

(Cisinski 06, prop. 1.3.36).

Proof

(…)

Corollary

The acyclic cofibrations are stable under transfinite composition and pushouts

(Cisinski 06, cor. 1.3.37).

Lemma

Every strong deformation retract is an anodyne extension.

(Cisinski 06, cor. 1.3.38).

Lemma

Every anodyne extension between fibrant objects is a strong deformation retract.

(Cisinski 06, lemma 1.3.39).

Proposition

Pour tout cardinal assez grand α\alpha, si on pose β=2 α\beta = 2^\alpha, pour toute cofibration triviale i:CDi : C \to D, et pour tout sous-objet β\beta-accessible JJ de DD, il existe un sous-objet β\beta-accessible KK de DD, qui contient JJ, tel que l’inclusion canonique CKKC \cap K \to K soit une cofibration triviale.

(Cisinski 06, prop. 1.3.40).

Proof

Three pages of work…

Proposition

There exists a set of generating acyclic cofibrations.

(Cisinski 06, prop. 1.3.42).

Proof

Use lemma 3 with CC the class of acyclic cofibrations and D=Acc α(A)D = Acc_\alpha(A) the set of α\alpha-accessible presheaves for a sufficiently large cardinal α\alpha.

Corollary

There is a functorial factorization of every morphism into an acyclic cofibration followed by a fibration.

(Cisinski 06, cor. 1.3.43).

Proof

By prop. 15 and prop. 11 we may apply the small object argument.

Completeness

We list lemmas to show prop. 8.

(…)

Localizers

Let still AA be a small category and write PSh(A)PSh(A) for its category of presheaves.

Definition

An AA-localizer is a class of morphisms WMor(PSh(A))W \subset Mor(PSh(A)) satisfying the following axioms

  1. 2-out-of-3;

  2. every acyclic fibration, def. 7, is in WW;

  3. The class of monomorphisms that is in WW is stable under pushout and transfinite composition.

The elements of WW we call WW-equivalences. For SMor(PSh(A))S \subset Mor(PSh(A)) a class of morphisms, the smallest AA-localizer containing SS is called the AA-localized generated by SS. If an AA-localizer is generated from a small set, we call it accessible. The minimal AA-localizer is W()W(\emptyset).

(Cisinski 06, def. 1.4.1))

Proposition

For (J,AnExt)(J,AnExt) a homotopical structure, def. 11 on PSh(A)PSh(A), the class WW of weak equivalences of the induced model category structure of theorem 1 is an AA-localizer.

If Λ\Lambda is a small set generating the anodyne extensions, AnExt=LLP(RLP(Λ))AnExt = LLP(RLP(\Lambda)), then W=W(Λ)W = W(\Lambda).

(Cisinski 06, prop. 1.4.2))

Theorem

Let WMor(PSh(A))W \subset Mor(PSh(A)). The following are equivalent.

  1. WW is an AA-localizer, def. 16;

  2. There is a set SMor(PSh(A))S \subset Mor(PSh(A)) of monomorphisms, such that WW is the class of weak equivalences of the model structure induced by theorem 1 from the homotopical structure, def. 11, given by the Lawvere cylinder, def. 2, and AnExt:=SAnExt := S.

  3. There is some homotopical structure, def. 11, on PSh(A)PSh(A), such that WW is the class of weak equivalences of the model structure corresponding to it by theorem 1.

  4. There exists a cofibrantly generated model category on PSh(A)PSh(A) such that WW is its class of weak equivalences, and such that the cofibrations are the monomorphisms.

In particular, PSh(A)PSh(A) admits a model structure whose cofibrations are the monomorphisms and whose weak equivalences are the minimal localizer, W()W(\emptyset). This is called the minimal model structure on PSh(A)PSh(A). It is generated from the homotopical datum given by the Lawvere cylinder, example 2 and the empty set.

(Cisinski 06, theorem 1.4.3))

Remark

This may be compared to Jeff Smith's theorem, which constructs a model structure on a locally presentable category.

(Cisinski 06, scholie 1.4.6))

Simplicial completion

Give a localizer WW on PSh(A)PSh(A), there is a localizer W horizW_{horiz} on PSh(A×Δ)PSh(A \times \Delta)

W horiz{X×q *(Δ 1)X}. W_{horiz} \coloneqq \{ X \times q^* (\Delta_1) \to X \} \,.

(…)

See (Ara, p. 9).

Examples

Properties

Properness

Theorem

If W(Σ)W(\Sigma) is the localizer generated by a set Σ\Sigma in a Grothendieck topos, then (the Cisinski model structure whose weak equivalences are) W(Σ)W(\Sigma) is right proper if and only if pullback along fibrations between fibrant objects in this model structure takes morphisms in Σ\Sigma to weak equivalences.

(Cisinski 02, Théorème 4.8)

We can use this to prove that a locally presentable (∞,1)-category is locally cartesian closed (∞,1)-category if and only if it has a presentation by a right proper Cisinski model structure. See locally cartesian closed (∞,1)-category for details.

References

The original articles are

Further developments are in

See also

Revised on July 2, 2014 01:23:15 by Ilan Barnea (78.50.78.74)