model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
A combinatorial model category is a particularly tractable model category structure. (Notice however that there is also the, closely related, technical notion of a tractable model category).
Being combinatorial means that there is very strong control over the cofibrations in these model structures: there is a set (meaning small set, not a proper class) of generating (acyclic) cofibrations, and all objects, in particular the domains and codomains of these cofibrations, are small objects.
So as a slogan we have that
A combinatorial model structure is one that is generated from small data: it is generated from a small set of (acyclic) cofibrations between small objects.
In fact, the combinatoriality condition is a bit stronger than that, as it requires even that every object is small and is the colimit over a small set of generating objects.
There exist large classes of model categories that either are combinatorial or, if not, are Quillen equivalent to ones that are. See the list of examples below.
The relevance of combinatorial model categories is given more abstractly by the result that
Combinatorial simplicial model categories are precisely those model categories that model presentable (∞,1)-categories.
For more see at locally presentable categories - introduction.
There is also a more general class of model structures on locally presentable categories, namely accessible model categories, that share many of the properties of combinatorial ones, and are additionally closed under more constructions such as transferred model structures. In addition, every combinatorial model category can be enhanced to an algebraic model category.
The following definition originates with Jeff Smith:
Recall from the discussion at cofibrantly generated model category that this means that has a set (i.e. a small set, not a proper class) of generating cofibrations and a set of generating trivial cofibrations in that
Here is the collection of fibrations and cofibration, respectively, and is the collection of morphisms satisfying the left or right, respectively, lifting property with respect to a collection of morphisms .
Jeff Smith’s theorem, below, gives an equivalent characterization that is usually easier to handle.
There are two powerful theorems that characterize combinatorial model categories in terms of data that is often easier to handle:
Jeff Smith‘s theorem characterizes combinatorial model categories just in terms of weak equivalences and generating cofibrations, hence using only two third of the input data explicitly required. This greatly facilitates identifying combinatorial model category structures. For instance it helps to show that the left Bousfield localization of certain combinatorial model categories is again a combinatorial model category.
Dugger's theorem shows that combinatorial model categories are, up to Quillen equivalence, precisely those model categories that have a presentation in that they are Bousfield localizations of global model structures on simplicial presheaves. This was used by Jacob Lurie to show that simplicial combinatorial model categories are precisely the models for locally presentable (∞,1)-categories.
A central theorem about combinatorial model categories is Jeff Smith‘s theorem which establishes the existence of combinatorial model category structures from a small amount of input data.
(Jeff Smith’s theorem)
For
an accessibly embedded accessible full subcategory of the arrow category on a class ;
a small set (not a proper class) of morphisms of
such that
satisfies 2-out-of-3;
is closed under pushout and transfinite composition.
we have that is a combinatorial model category with
weak equivalences ;
cofibrations .
Moreover, every combinatorial model category arises in this way.
Here the notation is as described at cofibrantly generated model category, so: and .
This statement was announced by Jeff Smith in 1998 at a conference in Barcelona and appararently first appeared in print as (Beke, theorem 1.7). The above formulation follows (Barwick, prop 1.7).
To prove the first part of the statement, that the given data encodes a combinatorial model category, it is sufficient to find a small set such that
With that the statement follows using the small object argument to show the existence of the required factorizations.
To find this small set, we make use of the assumption that the subcategory of weak equivalences and commuting squares in between them is an accessible subcategory of the arrow category . This means that there is a small set such that every element of is a -directed colimit over element in in , for some large enough cardinal number , such that all elements of are -compact.
Using the small object argument factor every morphism in as . Note that by 2-out-of-3 and , the cofibration is in . Let be the set of acyclic cofibratons so obtained.
By the choice of every morphism
in lifts through one of the components in of (this mechanism is described in detail at small object) as
We can refine this to a factoring through as follows: by construction the morphism factors as . Then lifts to and we obtain the factorization
of the original square from an element in to an element in through an element in . (In Beke, following Jeff Smith, this is called the solution set condition: is “a solution set for at ”). It readily follows that .
By the small object argument every morphism in may be factored as
Here by 2-out-of-3, the morphism is in , and hence in .
This we use to show that every morphism is in :
since we may factor as above and since we obtain a lift in
Rearranging this it becomes a retract diagram in
which shows that is a retract of an element in , hence itself in .
And the converse statement is immediate: by definition and is the saturation of under the operation of forming retracts of transfinite compositions of pushouts of elements of , under which is assumed to be closed.
In total we have indeed which shows that the and given determine a combinatorial model category.
To see the converse, that every combinatorial model structure arises this way, it is sufficient to show that for every combinatorial model category the category is an accessible category.
For applications of this theorem, the following auxiliary statements are useful.
For a combinatorial model category, the full subcategory inclusion
of the arrow category on the weak equivalences is an accessible inclusion of an accessible category.
This is due to Smith. A proof appears as Dugger 01, 7.4. See also Barwick, prop. 1.10.
Let
be an accessible functor between arrow categories. Let be equipped with weak equivalences such that the full subcategory inclusion
on the weak equivalences is an accessible embedding of an accessible category. Then so is the full subcategory of on the pre-images in .
By general properties of accessible categories (see there) the full inverse image along an accessible functor of a full accessible subcategory is again accessible.
In the special case when weak equivalences are closed under filtered colimits and the model structure is left proper, the statement of Smith’s theorem can be simplified.
The following formulation can be found as Proposition A.2.6.15 in Lurie.
Suppose is a locally presentable category equipped with a class of weak equivalences that turn it into a relative category and a set of generating cofibrations. If the following conditions are satisfied, then admits a left proper model structure with as the set of generating cofibrations:
;
every element of is an h-cofibration;
the class is perfect: it satisfies the 2-out-of-3 property, is closed under filtered colimits (in the category of morphisms in ), and is generated under filtered colimits by a set of morphisms.
The following theorem is precisely the model-category theory version of the statement that every locally presentable (∞,1)-category arises as the localization of an (∞,1)-category of (∞,1)-presheaves.
Every combinatorial model category is Quillen equivalent to a left Bousfield localization of the global projective model structure on simplicial presheaves on a small category
This is (Dugger 01, theorem 1.1) building on results in (DuggerUniversalHomotopy).
The proof proceeds (the way Dugger presents it, at least) in roughly three steps:
Use that is in some precise sense the homotopy- free cocompletion of . This means that every functor from a plain category to a model category factors in an essentially unique way through the Yoneda embedding by a Quillen adjunction
The detailed definitions and detailed proof of this are discussed at (∞,1)-category of (∞,1)-presheaves.
For a given combinatorial model category , choose the full subcategory on a small set (guaranteed to exist since is locally presentable) of cofibrant -compact objects, for some regular cardinal , and show that the induced Quillen adjunction
induced by the above statement from the inclusion exhibits as a homotopy-reflective subcategory in that the derived counit ( some cofibrant replacement functor) is a natural weak equivalence on fibrant objects (recall from adjoint functor the characterization of adjoints to full and faithful functors).
Define to be the set of morphisms in that the left derived functor of (here is some cofibrant replacement functor) sends to weak equivalences in . Then form the left Bousfield localization with respect to this set of morphisms and prove that this is Quillen equivalent to .
Carrying this program through requires the following intermediate results.
First recall from the discussion at (∞,1)-category of (∞,1)-presheaves that to produce the Quilen adjunction from , we are to choose a cofibrant resolution functor
of .
The adjunct of this is a functor . For each object write for the slice category induced by this functor.
Lemma (Dugger, prop. 4.2)
For every fibrant object we have that the homotopy colimit is weakly equivalent to .
Corollary (Dugger, cor. 4.4) The induced Quillen adjunction
is a homotopy-reflective embedding precisely if the canonical morphisms
are weak equivalences for every fibrant object .
…
Notice that the theorem just mentions plain combinatorial model categories, not simplicial model categories. But of course by basic facts of enriched category theory is an SSet-enriched category and its projective global model structure on functors is compatibly a simplicial model category, as are all its Bousfield localizations. (See model structure on simplicial presheaves for more details.) Therefore an immediate but very useful corollary of the above statement is
Every combinatorial model category is Quillen equivalent to one which is
A combinatorial model category is a tractable model category precisely if the set of generating cofibrations can be chosen such that all elements have a cofibrant object as domain.
A left proper combinatorial model category precisely if the set of generating trivial cofibrations can be chosen with cofibrant domain.
This are corollaries 2.7 and 2..8 in Bar.
In a combinatorial model category, for every sufficiently large regular cardinal the following holds:
-filtered colimits preserve weak equivalences;
hence -filtered colimits are already homotopy colimits.
See also at filtered homotopy colimit.
This appears as proposition 7.3 in Dug00, reproduced for instance as prop. 2.5 in Bar.
The point is to choose such that all domains and codomains of the generating cofibrations are -compact object. This is possible since by assumption that is a locally presentable category all its objects are small objects, hence each a -compact object for some cardinal . Take to be the maximum of these.
Let be -filtered diagrams in and a natural transformation that is degreewise a weak equivalence. Using the functorial factorization provided by the small object argument this may be factored as where the first transformation is objectwise an acyclic cofibration and the second objectwise an acyclic fibration, and by functoriality of the factorization this sits over a factorization
It remains to show that the second morphism is a weak equivalence. But by our factorization and by 2-out-of-3 applied to our componentwise weak equivalences, we have that all its components are acyclic fibrations.
At small object it is described in detail how -smallness of an object implies that morphisms from into a -filtered colimit lift to some component of the colimit
So given a diagram
we are guaranteed, by the -smallness of and that we established above, a lift
into some component at and hence a lift
Thereby is in .
Combinatorial model categories, like cellular model categories have a good theory of Bousfield localizations, at least if in addition they are left proper. See Bousfield localization of model categories for more on this.
Basic examples are
sSet with its standard model structure on simplicial sets;
sSet with the Joyal-model structure for quasi-categories;
notice that this is not directly a simplicial model category, but is enriched over itself. A Quillen equivalent combinatorial simplicial model category is
the category of dendroidal sets with its model structure on dendroidal sets.
the categories of -Theta spaces.
More generally, every Cisinski model structure is combinatorial.
Further classes of examples are obtained from such basic examples by localizing presheaf categories with values in these:
For a combinatorial model category and a small category the injective and projective model structure on functors and are again combinatorial model categories. See there for details.
If is a left or right proper model category then so is and and hence the standard results of the theory of Bousfield localization of model categories applies, which ensures that all left Bousfield localizations are again combinatorial model categories. Such local local model structures on homotopical presheaves includes notably the local model structure on simplicial presheaves.
Not every cofibrantly generated model category is also a combinatorial model category.
For instance:
Top with the standard model structure on topological spaces is cofibrantly generated, but not combinatorial. But it is Quillen equivalent to a combinatorial model structure, namely to the standard model structure on simplicial sets (see homotopy hypothesis).
One might therefore ask which cofibrantly generated model categories are Quillen equivalent to combinatorial ones. It turns out that if one assumes the large-cardinal hypothesis Vopěnka's principle, then every cofibrantly generated model category is Quillen equivalent to a combinatorial one. In fact, if we slightly generalize the notion of “cofibrantly generated,” this statement is equivalent to Vopěnka’s principle. For a discussion of this see
Although Vopěnka’s principle cannot be proven from ZFC, and in fact is fairly strong as large cardinal hypotheses go, this means that looking for cofibrantly generated model categories that are not Quillen equivalent to combinatorial ones is probably a waste of time. Certainly, all known cofibrantly generated model categories are Quillen equivalent to simplicial ones, usually in a fairly natural way.
Those combinatorial model categories that are at the same time simplicial model categories are precisely those that present presentable (∞,1)-categories. See combinatorial simplicial model category.
not related is the notion of combinatorial category
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
Algebraic model structures: Quillen model structures, mainly on locally presentable categories, and their constituent categories with weak equivalences and weak factorization systems, that can be equipped with further algebraic structure and “freely generated” by small data.
structure | small-set-generated | small-category-generated | algebraicized |
---|---|---|---|
weak factorization system | combinatorial wfs | accessible wfs | algebraic wfs |
model category | combinatorial model category | accessible model category | algebraic model category |
construction method | small object argument | same as | algebraic small object argument |
Much of the theory of combinatorial model categories goes back to Jeff Smith. Apparently Smith will eventually present a book on this subject. To date, however, his ideas and results appear reproduced in articles of other authors.
After Jeff Smith presented his recognition theorem at a conference in Barcelona, its first appearance in a publication is apparently
An early explicit account of the notion of combinatorial model categories is in Section II of:
(which goes on to state and proof Dugger's theorem, based on results in Dugger’s Universal homotopy theories).
The definition of combinatorial model categories is recalled also as:
Clark Barwick, Def. 1.3 in: On left and right model categories and left and right Bousfield localizations, Homology, Homotopy and Applications 12 2 (2010) 245–320
[doi:10.4310/hha.2010.v12.n2.a9, subsuming:arXiv:0708.2067, arXiv:0708.2832, arXiv:0708.3435]
Jacob Lurie, Def. A.2.6.1 in: Higher Topos Theory (2009)
Smith’s theorem appears as Lurie, A.2.6.10 and as Barwick, prop. 1.7. For more see at Bousfield localization of model categories.
Futher details are discussed in:
Review:
On the localization of a 2-categoryHo(CombModCat) of combinatorial model categories at the Quillen equivalences and its equivalence to the homotopy 2-category of (locally) presentable derivators:
See also
Last revised on April 25, 2024 at 09:36:16. See the history of this page for a list of all contributions to it.