Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
Say a class is a localizer on if it is a class of weak equivalences in a Cisinski model structure on .
Every small set of morphisms is contained in a smallest localizer, def. 2, .
One says that is the localiser generated by .
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).
Let be a small category. Write or for the category of presheaves over . We introduce here, culminating in def. 11 below, the ingredients of a homotopical structure on , 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 .
For in , a cylinder on in the following means a cylinder object, denoted , factoring the codiagonal
such that the first morphism is a monomorphism.
A homomorphism of such cyclinders is a pair of morphisms and in , making the evident squares commute. This defines a category of cylinder objects on presheaves on , equipped with a forgetful functor that sends a cylinder to its underlying object .
A functorial cylinder object over is a section of this functor.
This is (Cisinski 06, def. 1.3.1).
In the following, let be a choice of functorial cylinder object. Equivalently, this is a choice of endofunctor
equipped with natural transformations
where , such that the composite is the functorial codiagonal, and where the first transformation is a monomorphism.
For two morphisms in , we say an elementary -homotopy from to is left homotopy from to with respect to the chosen cylinder object , hence a morphism fitting into a diagram
We say -homotopy for the equivalence relation generated by this.
(Cisinski 06, def. 1.3.3).
-homotopy is compatible with composition in .
It is sufficient to show that elementary -homotopies are compatible with composition.
an elementary -homotopy , and for
one , we obtain an elementary homotopy by forming
and then an elementary -homotopy by forming
Together this generates a -homotopy .
Hence the following is well defined.
Write for the category whose objects are those of , and whose morphisms are -homotopy equivalence classes of morphisms in – the -homotopy category. Write
for the projection functor.
A morphism in is called a -homotopy equivalence if it is sent by to an isomorphism.
An object is called -contractible if is a -homotopy equivalence.
(Cisinski 06, def. 1.3.4).
A morphism in is called an acyclic fibration if it has the right lifting property against all monomorphisms.
(Cisinski 06, def. 1.2.18).
Every acyclic fibration in is a -homotopy equivalence.
More is true: every trivial fibration
has a section ;
which is also a -homotopy left inverse;
by an elementary -homotopy which satisfies .
(Cisinski 06, lemma 1.3.5).
The existence of the section follows by right lifting against the monomorphism (out of the initial object)
The -homotopy is obtained by lifting in the diagram
An elementary homotopical datum on is a functorial cylinder , def. 3, such that
the functor commutes with small colimits;
for all monomorphisms the diagrams
for is a pullback square.
(Cisinski 06, def. 1.3.6).
a pullback square in of two monomorphisms and , the universal morphism out of the pushout
is also a monomorphism, usually written as the morphism out of the union
All this follows, for instance, from the corresponding statements in Set, over each object of .
for , be the subfunctor which is the image of .
This way for any the boundary inclusions are identified with
(Cisinski 06, remark 1.3.7).
The condition implies that
is a pullback, for , hence that
is a pullback. So the statement follows with remark 3.
Let be any object equipped with two points (global sections) which are disjoint in that
is a pullback square (here is the initial object in ). This induces a functorial cylinder by the assignment
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).
The disjointness of the two points ensures that 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 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.
Let be the subobject classifier in the presheaf topos . This is the presheaf which to an object of assigns the set of subobjects of the representable functor given by (the sieves on )
and which to a morphism assigns the pullback functor .
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).
That the two points are separated, in that
is a pullback, is the defining property of the subobject classifier.
For an elementary homotopical datum on , a class of anodyne extensions is a class such that
there exists a small set with
for a monomorphism, the pushout product morphisms
(by Joyal-Tierney calculus to be thought of as “”)
are in , for ;
if is in , then so is
(Cisinski 06, def. 1.3.10).
A class of anodyne extensions
(Cisinski 06, remark. 1.3.11).
By prop. 11, is -compact. Therefore the set 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 and using that by def. 8 the interval commutes with colimits, so that .
Finally, to see that with also is anodyne, consider the naturality diagram of the endpoint inclusion
factored through the top pushout square, as indicated. Here is anodyne, being a pushout of an anodyne morphism, and is anodyne by the second clause in def. 10. Therefore also their composite is anodyne.
A homotopical structure on is a choice of elementary homotopical datum , def. 8 and a corresponding choice of a class of anodyne extensions , def. 10.
(Cisinski 06, def. 1.3.14).
The model structure
Let be a small category and a homotopical structure, def. 11, on . Define the following classes of objects and morphisms in :
the cofibrations are the monomorphisms;
the fibrant objects are those for which the terminal morphism has the right lifting property against the anodyne extensions (the morphisms in , def. 10);
the weak equivalences are those morphisms , such that for all fibrant objects the induced morphism in the -homotopy category, def. 5,
is an isomorphism (a bijection of sets).
(Cisinski 06, def. 1.3.21).
(Cisinski 06, theorem 1.3.22).
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.
(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.
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).
Every anodyne extension, def. 10, is a weak equivalence in the model structure of theorem 1.
(Cisinski 06, remark 1.3.46).
For the model structure from theorem 1, the following are equivalent:
every acyclic cofibration is an anodyne extension;
every morphism with right lifting against anodyne extensions is a fibration;
every weak equivalence with right lifting against anony extensions is an acyclic fibration;
every morphism with right lifting against anodyne extensions factors as an anodyne extension followed by a fibration.
(Cisinski 06, prop. 1.3.47).
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.
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.
Every -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).
The first statement holds by definition of .
The second statement also follows directly from the definition. If for and fibrant in the composite
two of three are isomorphisms, then so is the third.
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.
A deformation retract in is a retract with retraction , which is also a section of up to a -homotopy . It is strong if .
A dual deformation restract has a section by a morphism and is also a retract up to a -homotopy . Is is strong if .
(Cisinski 06, prop. 1.3.26).
The first statement is a direct consequence of prop 3.
For the second statement, let be an acyclic fibration, and let be a section. This induces a commuting square
where the lift exists by assumption on ( is necessarily a monomorphism, being a section).
The resulting component triangle
exhibits as a deformation retract, and the other resulting component triangle
says that , hence by naturality of cylinders , hence that the deformation retract is indeed strong.
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 -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 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 as a monomorphism followed by an acyclic fibration . By the previous argument, is a weak equivalence, and so by lemma 1 so is . Therefore, since is a fibration, we have a lift in
This equivalently exhibits as a retract of
So by lemma 4 with also is an acyclic fibration.
(Cisinski 06, remark 1.3.28).
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.
(Cisinski 06, prop. 1.2.9).
Let be a category with all small colimits, let be a cardinal, let be a small category and finally let be a functor with values in -compact objects in . Then the colimit is -compact object, for the maximum of and the cardinality of the set of morphisms of .
(Cisinski 06, prop. 1.2.10).
Let be a -filtered diagram.
Then by prop. 9 we have natural isomorphisms
because the -filtered diagram is at least -filtered and hence, by prop. 9, its colimit commutes with the limit over .
Now since each is assumed to be -compact and hence is also -compact, we conclude with the natural isomorphisms
For , write for the category of elements of . Write for the cardinality of the set of morphisms of the category of elements (throughout assuming to be a small category).
For any object, the hom functor preserves -filtered colimits.
In other words: is a -compact object.
(Cisinski 06, cor. 1.2.11).
By the co-Yoneda lemma is the colimit over its elements
Since the image of the functor is in representables, which are maximally compact, the stament follows with prop. 10.
A cellular model on is a choice of a small set of monomorphisms, such that the class of all monomorphisms is generated from it
(Cisinski 06, def. 1.2.26).
The following lemma will be used to show that cellular structures always exist.
Let be a class of morphisms, and a small set of objects, such that
is closed under pushouts, retracts and transfinite composition;
if are two composable morphisms with and in , then also is in ;
every is the union of those of its sub-objects isomorphic to an object in ;
for every in and every sub-object of in , there is a sub-object from , which contains and such that is in .
Then the small set of morphisms in with codomain in , generates as
(Cisinski 06, lemma. 1.2.24).
There exists a cellular structure, def. 15, on .
The set can be chosen to consist of morphisms into quotient objects of representables.
(Cisinski 06, prop. 1.2.27).
Take in lemma 3 to be the class of monomorphisms and to be the class of quotients of representables.
There exists a functorial factorization of morphisms in into a monomorphism followed by an acyclic fibration.
(Cisinski 06, cor. 1.2.28).
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.
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).
If is fibrant, then for any elementary -homotopy, def. 4, is already an equivalence relation on and coincides with -homotopy.
(Cisinski 06, lemma 1.3.30).
Every anodyne extension is a weak equivalence.
(Cisinski 06, lemma 1.3.31).
For an anodyne extension and a fibrant object, we need to show that
is a bijection.
It is surjective by the defining lifting property, which provides in
To see injectivity, let be two morphisms such that and coincide in . By lemma 5 this is the case precisely if there is an elementary -homotopy relating them. This induces the horizontal morphism in the diagram
where the left morphism is anodyne, by the second clause of def. 10, so that the lift denoted exists. This lift exhibits a -homotopy , hence shows that was already equal to in , hence that is injective.
A morphism between fibrant objects is a weak equivalence precisely if it is a -homotopy equivalence, def. 6.
(Cisinski 06, lemma 1.3.32).
Is is clear that every -homotopy is a weak equivalence. Conversely, let be a weak equivalence between fibrant objects. Write for the full subcategory of , def. 6 on the fibrant objects. The localization is by definition in and for all objects the morphism is an isomorphism. By the Yoneda lemma, therefore, itself is an isomorphism in , hence also in , hence is a weak equivalence.
(Cisinski 06, lemma 1.3.33).
That the former implies the latter was the statement of lemma 2. Conversely, let be a dual strong deformation retract, meaning that there is with , as well as a morphism exhibiting a -homotopy . This being strong means that .
We need to show that this implies for
a commuting diagram with a monomorphism, there is a lift. To this end, observe that the given structures induce a morphism
By the second clause of def. 10 the morphism on the left is an anodyne extension, and so this diagram admits a lift . One see that is a lift of the original square above.
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).
We already know from prop. 6 that acyclic fibrations are weak equivalences.
So let be fibrant and let be a weak equivalence that has rlp against anodyne extensions. We need to show that is an acyclic fibration. By lemma 7 it is sufficient to show that it is a dual strong deformation retract.
By lemma 6 is also a -homotopy equivalence. By lemma 5 this is exhibited by an elementary -homotopy , which in particular gives a commuting diagram
from which we obtain a lift . Set then
One finds then .
A cofibration into a fibrant object is a weak equivalence precisely if it is an anodyne extension.
(Cisinski 06, cor 1.3.35).
By lemma 13 we already know that every anodyne extension is a weak equivalence. So we need to show that a cofibration into a fibrant object which is a weak equivalence is also an anodyne extension. By prop. 4 we may factor this as , with an anodyne extension and having RLP against anodyne extensions. Since is a weak equivalence, by 2-out-of-3 so is . By lemma 8 is an acyclic fibration.
Therefore we have a lift in
and this exhibits as a retract of . Hence with also is an anodyne extension.
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).
(Cisinski 06, cor. 1.3.37).
(Cisinski 06, cor. 1.3.38).
(Cisinski 06, lemma 1.3.39).
Pour tout cardinal assez grand , si on pose , pour toute cofibration triviale , et pour tout sous-objet -accessible de , il existe un sous-objet -accessible de , qui contient , tel que l’inclusion canonique soit une cofibration triviale.
(Cisinski 06, prop. 1.3.40).
Three pages of work…
There exists a set of generating acyclic cofibrations.
(Cisinski 06, prop. 1.3.42).
Use lemma 3 with the class of acyclic cofibrations and the set of -accessible presheaves for a sufficiently large cardinal .
There is a functorial factorization of every morphism into an acyclic cofibration followed by a fibration.
(Cisinski 06, cor. 1.3.43).
We list lemmas to show prop. 8.
Let still be a small category and write for its category of presheaves.
An -localizer is a class of morphisms satisfying the following axioms
every acyclic fibration, def. 7, is in ;
The class of monomorphisms that is in is stable under pushout and transfinite composition.
The elements of we call -equivalences. For a class of morphisms, the smallest -localizer containing is called the -localized generated by . If an -localizer is generated from a small set, we call it accessible. The minimal -localizer is .
(Cisinski 06, def. 1.4.1))
For a homotopical structure, def. 11 on , the class of weak equivalences of the induced model category structure of theorem 1 is an -localizer.
If is a small set generating the anodyne extensions, , then .
(Cisinski 06, prop. 1.4.2))
Let . The following are equivalent.
is an -localizer, def. 16;
There is a set of monomorphisms, such that 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 .
There is some homotopical structure, def. 11, on , such that is the class of weak equivalences of the model structure corresponding to it by theorem 1.
There exists a cofibrantly generated model category on such that is its class of weak equivalences, and such that the cofibrations are the monomorphisms.
In particular, admits a model structure whose cofibrations are the monomorphisms and whose weak equivalences are the minimal localizer, . This is called the minimal model structure on . It is generated from the homotopical datum given by the Lawvere cylinder, example 2 and the empty set.
(Cisinski 06, theorem 1.4.3))
(Cisinski 06, scholie 1.4.6))
Give a localizer on , there is a localizer on
See (Ara, p. 9).
The archetypical and motivating example is the standard model structure on simplicial sets, which is a Cisinski model structure on the presheaf topos on the simplex category (Cisinski 06, section 2).
Accordingly, the injective model structure on simplicial presheaves over a site is a Cisinski model structure, namely on the presheaf topos . Moreover, every left Bousfield localization of such a model structure is still a Cisinski model structure, since left Bousfield localization preserves the class of cofibrations.
Notice that, as discussed there, every presentable (infinity,1)-category has a presentation by such a localization, hence by a Cisinski model structure.
Also the model structure for quasi-categories is a Cisinski model structure on sSet, induced by the localizer given by the spine inclusions.
Moreover, the model structure for complete Segal spaces is the simplicial completion of this model structure. (see Ara).
As a cellular set-variant of this, the model structure on cellular sets is a Cisinski model structure on the category of presheaves over the Theta category restricted to -cells.
The model structure on dendroidal sets is not exactly a Cisinki model structure, but is transferred from one that is.
If is the localizer generated by a set in a Grothendieck topos, then (the Cisinski model structure whose weak equivalences are) is right proper if and only if pullback along fibrations between fibrant objects in this model structure takes morphisms in 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.
The original articles are
Further developments are in