Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
Model structures on simplicial presheaves present (∞,1)-categories of (∞,1)-presheaves and localizations of these, such as notably the left exact localizations that are (∞,1)-categories of (∞,1)-sheaves: these model structures are models for ∞-stack (∞,1)-toposes.
This suggests that the (∞,1)-category of (∞,1)-sheaves on some site can be presented by a model category structure on the ordinary functor category
– the category of simplicial presheaves .
Various interrelated flavors of model structures on the category of simplicial presheaves on have been introduced and studied since the 1970s, originally by K. Brown and Andre Joyal and then developed in detail by Rick Jardine.
Notice that when regarded as a presentation of an (∞,1)-sheaf, i.e. of an ∞-stack, a simplicial presheaf – being an ordinary functor instead of a pseudofunctor – corresponds to a rectified ∞-stack. It might therefore seem that a model given by simplicial presheaves is too restrictive to capture the full expected flexibility of a notion of ∞-stack. But this is not so.
a fully general definition of a (∞,1)-category of ∞-stacks is given it is shown – proposition 18.104.22.168 – that, indeed, the Brown–Joyal–Jardine model is a presentation of that.
Originally K. Brown had considered in BrownAHT not a model structure on simplicial presheaves but
and Joyal had originally considered a
Joyal’s local model structure on simplicial sheaves is Quillen equivalent to the injective local model structure on simplicial presheaves.
By repackaging Kan complexes as simplicial groupoids one obtains a model structure on presheaves of simplicial groupoids which is also Quillen equivalent to the above.
If K. Brown’s category of fibrant objects on locally Kan simplicial sheaves is restricted to globally Kan simplicial sheaves on a topos with enough point then it is the full subcategory on the fibrant objects in the projective local model structure on simplicial sheaves.
But since in all cases the weak equivalences are the same (where they apply, for Brown’s model if the topos has enough points), all these local homotopical categories define equivalent homotopy categories.
By Lurie’s result these are in each case in turn equivalent to the homotopy category of the (∞,1)-topos of ∞-stacks. So in particular they serve as a home for general cohomology.
Various old results appear in a new light this way. For instance using the old result of BrownAHT on the way ordinary abelian sheaf cohomology is embedded in the homotopy theory of simplicial sheaves, one sees that the old right derived functor definition of abelian sheaf cohomology really computes the ∞-stackification of a sheaf of chain complexes regarded under the Dold–Kan correspondence as a simplicial sheaf.
The different model structures and their interrelation
It is the very point of model category structures on a given homotopical category that there may be several of them: each presenting the same (∞,1)-category but also each suited for different computational purposes.
So it is good that there are many model structures on simplicial (pre)sheaves, as there are.
Injective/projective - local/global - presheaves/sheaves
The following diagram is a map for part of the territory:
“inj” denotes the injective model structure: cofibrations are objectwise cofibrations
“proj” denotes the projective model structure: fibrations are objectwise fibrations
no “loc” subscript means global model structure: weak equivalences are the objectwise weak equivalences:
“l loc” denotes left Bousfield localization at hypercovers (at stalkwise acyclic fibrations if the topos has enough points)
The identity functor on the category of simplicial presheaves is a Quillen adjunction for the projective and injective global model structure and this is a Quillen equivalence.
The local model structures on simplicial sheaves are just the restrictions of the those on simplicial presheaves. (For the injective structure this is in Jardine, for the projective one in Blander, theorem 2.1, 2.2).
These are related by a Quillen adjunction given by the usual geometric embedding of the category of sheaves as a full subcategory of that of presheaves – with sheafification the left adjoint – and this is also Quillen equivalence.
The characteristic of the left Bousfield localizations is that for them the fibrant objects are those that satisfy descent: see descent for simplicial presheaves.
In either case
The following diagram collection model categories that are presentations for the (∞,1)-category of (∞,1)-sheaves. All indicated morphism pairs are Quillen equivalences.
On the right this lists the model structures on simplicial (pre)sheaves, here displayed as (pre)sheaves with values in simplicial sets, using .
On the left we have the Joyal–Tierney and Luo–Bubenik–Tim model structures on presheaves of simplicial groupoids.
(…have to check here the relation )
To some extent the injective and projective model structures on simplicial presheaves are the two extremes of a larger family of model structures on simplicial presheaves that all have the same weak equivalences but different classes of cofibrations.
Notably if the domain has the special property that it is a Reedy category there is the Reedy model structure on . Its class of cofibrations is intermediate that of the projective and the injective model structure on functors and we have Quillen equivalences
For general , there is still a whole family of model structures on that interpolates between the injective and the projective model structure. See intermediate model structure.
Dependency on the underlying site
Let be sites and let be a functor that induces a morphism of sites in that preserves sheaves and its left adjoint (given by left Kan extension) is left exact functor in that it preserves finite limits.
Then the induced adjunction
is a Quillen adjunction for the local injective model structure on presheaves on both sides.
Let be a site and a full dense sub-site. Then right Kan extension along yields a simplicial Quillen adjunction
between the left Bousfield localizations of the projective model structures at the sieve inclusions for each covering family .
It is immediate that we have a simplicial Quillen adjunction on the global injective model structure: by definition of right Kan extension we have an sSet-adjunction and the left adjoint restriction functor trivially preserves injective cofibrations and acyclic cofibrations.
Since we have left proper model categories it is sufficient (by the discussion at recognition of simplicial Quillen adjunctions) for deducing that the Quillen adjunction descends to the local strucuture to check that preserves locally fibrant objects, which in turn by properties of left Bousfield localization is equivalent to checking that sends covering sieve inclusions to weak equivalences in .
By the result on generalized covers, for this it is sufficient to check that for every covering sieve and every representable and morphism , there is a covering in and local lifts
This follows directly from the single defining condition on a coverage on .
Presentation of -toposes
Let be a site. Let be the projective model structure on simplicial presheaves over .
Let be the set of Cech nerve projections in for each covering in .
for the left Bousfield localization at .
Write for the full sub-simplicially enriched category on the fibrant-cofibrant objects, similarly for .
We have an equivalence of (∞,1)-categories
where at the bottom we have the left and right derived functors of the identity functors, as discussed at simplicial Quillen adjunction.
Fibrant and cofibrant objects
The fibrant objects in the local model structure on simplicial presheaves are those that
This descent condition is the analog in this model of the sheaf-condition and the stack-condition. In fact, it reduces to these for truncated simplicial presheaves.
Since the fibrancy condition in the global projective model structure is simple – it just requires that the simplicial presheaf is in fact a presheaf of Kan complexes – the local projective model structure has slightly more immediate characterization of fibrant objects than the local injective model structures. (In fact, for suitable choices of sites it may become very simple, as the above discussion of site dependence of the model structure shows).
On the other hand the cofibrancy condition on objects is entirely trivial in the global and local injective model structure: since a cofibration there is just an objectwise cofibration, and since every simplicial set is cofibrant, every object is injective cofibrant.
But the cofibrant objects in the projective structure are not too nasty either: every object that is degreewise a coproduct of representables is cofibrant. In particular the Čech nerves of any good cover (see below for more details) is a projectively cofibrant object.
A cofibrant replacement functor in the local projective structure is discussed in
Something related to a fibrant replacement functor (“-stackification”) is discussed in section 6.5.3 of
In the injective local model structure on simplicial presheaves all objects are cofibrant. For the projective local structure we have the following useful statement (see also projectively cofibrant diagram).
(see also Low, remark 8.2.3).
A simplicial presheaf is said to have free degeneracies or the degenerate cells split off if in each degree there is a sub-presheaf such that the canonical mophism
is an isomorphism.
So if degenerate cells split off we have in particular that
where is the presheaf of non-degenerate -cells and is a separate presheaf containing all the degenerate cells (and itself a coproduct over separate presheaves for each degree and order of degeneracy).
This is in the proof of lemma 2.7 in section 9 of
If is an acyclic fibration in the local projective model structure with a representable and cofibration in the above way, it is called a split hypercover .
All Čech nerves coming from an open cover have split degeneracies. The condition that the Cech nerve be degreewise a coproduct of representables is a condition akin to that of good open covers (which is precisely the special case for CartSp). This is then a split hypercover of height 0.
A Čech nerve with a weak equivalence in is a good cover if it is degreewise a coproduct of representables.
a useful cofibrant replacement functor for the projective local model structure is discussed.
For an ordinary presheaf (simplicially discrete simplicial presheaf) let be the simplicial presheaf which in degree is
where the range over the representables, i.e. the objects in . The face and degeneracy maps are the obvious ones coming from composing maps and inserting identity maps in the labels over which the coproduct ranges.
For an arbitrary simplicial presheaf let be the diagonal of the bisimplicial presheaf obtained by applying degreewise
For all the object is cofibrant and is weakly equivalent to in .
This is in prop 2.8 of
A local fibration or local weak equivalence of simplicial (pre)sheaves is defined to be one whose lifting property is satisfied after refining to some cover.
Warning. Notice that this is a priori unrelated to equivalences and fibrations with respect to any local model structure.
If the site has enough points, then local fibrations of simplicial presheaves are equivalently those that are stalkwise fibrations of simplicial sets.
This is discussed in (Jardine 96).
Localization and descent
Cech localization at Grothendieck (pre)topologies
We discuss some aspects of the left Bousfield localization of the projective global model structure on simplicial presheaves at Grothendieck topologies and covering families. By the discussion at topological localization these are models for topological localizations leading to (∞,1)-categories of (∞,1)-sheaves.
The central reference is (DuggerHollanderIsaksen) with the central theorem being this one:
Let be a site given by a Grothendieck topology. The left Bousfield localization of and , respectively, at the following classes of morphisms exist and coincide:
the set of all covering sieve subfunctors ;
the set of all morphisms for a covering sieve of ;
the set of all Cech nerve projections for a covering sieve;
the class of all bounded hypercovers ;
the class of morphisms from a simplicial presheaf to the simplicial sheaf obtained by degreewise sheafification.
if the topology is generated from a basis, then: the set of covering sieve subfunctors for each covering family in the basis.
This is theorem A5 in DugHolIsak.
This localization is the Cech localization of with respect to the given Grothendieck topology. It is a presentation of topological localization of an (∞,1)-category of (∞,1)-presheaves to an (∞,1)-category of (∞,1)-sheaves.
The following definition and proposition provides information on what the general morphisms are which become weak equivalences after localization at
Let be a site. A local epimorphism (or generalized cover) in is a morphism of simplicial presheaves with the property that for every representable and every morphism there exists a covering sieve such that for every the composite has a lift through
For a local epimorphism in in the above sense, its Cech nerve projection
is a weak equivalence in .
This is DugHolIsa, corollary A.3.
Cech localization at a coverage
In the literature localization of categories of simplicial presheaves is typically discussed with respect to a Grothendieck topology or a basis for a topology. Here we discuss aspects of localization at a coverage.
Let be a category equipped with a coverage, i.e. a collection of families of morphisms for each object in , called covering families such that for any morphism there exist diagrams
such that is itself a covering family.
Write for the sieve corresponding to a covering family, regarded as a subfunctor of the representable functor , which we both regard as simplicially discrete objects in .
Write for the left Bousfield localization of at these morphisms corresponding to covering families.
A subfunctor inclusion corresponding to a sieve that contains a covering sieve is a weak equivalence in
Write for the set of morphisms in but not in .
Let be a morphism not in . By assumption we can find a covering family such that for all we have commuting diagrams
Consider the commuting diagram
Observe that this is a pushout in , that the top morphism is a cofibration in and hence in , that the left morphism is a local weak equivalence, that by general properties of left Bousfield localization the localization is left proper. Therefore the morphism is a weak equivalence.
Next observe that from the horizontal morphisms of the above commuting diagrams that defined the covers we have an induced morphism , and this exhibits as a retract
By closure of weak equivalences under retracts, this shows that the inclusion is a weak equivalence. By 2-out-of-3 this finally means that is a weak equivalence.
For a covering sieve, its pullback in along any morphism
is also a weak equivalence.
If is the sieve of a covering family and is any sieve such that for every the pullback is a weak equivalence, then becomes an isomorphism in the homotopy category.
First notice that if is a weak equivalence for all , then the pullback of to any element of the sieve is a weak equivalence. Use the co-Yoneda lemma to write
Now consider these objects in the (∞,1)-category of (∞,1)-presheaves that is presented by .
Since that has universal colimits we have the pullback square
and the left vertical morphism is a colimit over morphisms that are weak equivalences in . By the general properties of reflective sub-(∞,1)-categories this means that the total left vertical morphism becomes an isomorphism in the homotopy category of . Also the bottom morphism is an isomorphism there, and hence the right vertical one is.
For values in strict and abelian -groupoids
Many simplicial presheaves appearing in practice are (equivalent) to objects in sub-(∞,1)-categories of of abelian or at least strict ∞-groupoids. These subcategories typically offer convenient and desireable contexts for formulating and proving statements about special cases of general simplicial presheaves.
One well-known such notion is given by the Dold-Kan correspondence. This identifies chain complexes of abelian groups with strict and strictly symmetric monoidal -groupoids.
Dropping the condition on symmetric monoidalness we obtain a more general such inclusion, a kind of non-abelian Dold-Kan correspondence:
the identification of crossed complexes of groupoids as precisely the strict ∞-groupoids. This has been studied in particular in nonabelian algebraic topology.
So we have a sequence of inclusions
of strict -groupoids into all -groupoids. See also the cosmic cube of higher category theory.
Among the special tools for handling -stacks on that factor at some point through the above inclusion are the following:
We state a useful theorem for the computation of descent for presheaves with values in strict ∞-groupoids. Recall the standard terminology for descent, i.e. for the -categorical sheaf-condition:
For a representable, simplicial presheaves and a morphism, we say that satisfies descent along or equivalently that is a -local object if the canonical morphism
is a weak equivalence. Here the first equality is the enriched Yoneda lemma. By the co-Yoneda lemma we may decompose into its cells as
where in the integrand we have the tensoring of over sSet. Using that the enriched hom-functor sends coends to ends, the enriched hom-functor on the right we may equivalently write out as an end
(equality signs denote isomorphisms), where in the second but last line we again used the tensoring of simplicial presheaves over sSet.
In the last line we have the totalization of the cosimplicial simplicial object
sometimes called the descent object of relative to , even though in this case it is really nothing but the hom-object of into . If is fibrant and cofibrant, then is a Kan complex: the descent -groupoid .
Now suppose that is a presheaf with values in strict ∞-groupoids. In the context of strict -groupoids the standard -simplex is given by the th oriental . This allows to perform a construction that looks like a descent object in :
The descent object for relative to is
where the end is taken in .
This objects had been suggested by Ross Street to be the right descent object for strict -category-valued presheaves in Street03
Under the ω-nerve functor this yields a Kan complex . On the other hand, applying the -nerve directly to yields a simplicial presheaf to which the above simplicial descent applies.
The following theorem asserts that under certain conditions both notions coincide.
If and are such that is fibrant in the Reedy model structure , then
is a weak homotopy equivalence of Kan complexes.
This is proven in Verity09.
If is such that is cofibrant in then for we have
If is Reedy cofibrant, then by definition the canonical morphisms
are cofibrations in . Since the latter is an enriched model category and is fibrant, it follows that the hom-functor sends cofibrations to fibrations, so that
is a Kan fibration. But this says that is Reedy fibrant, so that the assumption of Verity’s theorem is met.
For the Cech nerve of a good open cover of a manifold and any we have that
By the above is sufices to note that is cofibrant in if is the Cech nerve of a good open cover. By the assumption of good open cover we have that is degreewise a coproduct of representables and that the inclusion of all degenerate -cells into all -cells is a full inclusion into a coproduct, i.e. an inlusion of the form
induced from an inclusion of subsets . Since all representables are cofibrant in such an inclusion is a cofibration.
In conclusion we find that for determining the -stack condition for strict -Lie groupoids we may equivalently use Street’s formula for strict -groupid valued presheaves. This is sometimes useful for computations in low categorical degree.
The global model structures on simplicial presheaves are all left and right proper model categories. Since left Bousfield localization of model categories preserves left properness (as discussed there), the local model structures are also left proper.
But the local model structures are not in general right proper anymore.
A sufficient condition for an injective or projective local model structure of simplicial presheaves over a site to be right proper is that the weak equivalences are precisely the stalk wise weak equivalences of simplicial sets.
This is true for instance for the injective Jardine model structure when has enough points. (e.g. recalled on p. 12 here).
The key is that forming stalks is, being the inverse image of a geometric morphism
an operation that preserves finite limits.
Let therefore be a stalkwise weak equivalence of simplicial presheaves and let be a fibration. Notice that in all the model structures (injective, projective, global, local) the fibrations are always in particular objectwise fibrations.
Then the pullback in
is a weak equivalence if for all topos points the stalk is a weak equivalence of simplicial sets. But since stalks preserve finite limits, we have a pullback diagram of simplicial sets
It is now sufficient to observe that is a Kan fibration, this implies the result then by the fact that the classical model structure on simplicial sets is right proper.
To see this, notice that is a Kan fibration precisely if for all and the morphism
is an epimorphism of sets. Since stalks commute with finite limits, this is equivalent to
being an epimorphism. Now the morphism in parenthesis is an epimorphism since the fibration is in particular an objectwise Kan fibration, and left adjoint functors such as preserve epimorphisms.
This is mentioned for instance in (Olsson, remark 4.3).
Closed monoidal structure
If the underlying site has finite products, then both the injective and the projective, the global and the local model structure on simplicial presheaves becomes a monoidal model category with respect to the standard closed monoidal structure on presheaves.
See for instance here.
It is sufficient to check that the Cartesian product of presheaves
is a left Quillen bifunctor. As discussed at Quillen bifunctor, since is a cofibrantly generated model category for that it is sufficient to check that satisfies the pushout-prodct axiom on generating (acyclic) cofibrations.
As discussed at model structure on functors, these are those morphisms of the form
for representable and an (acylic) cofibration in . For these morphisms checking the pushout-product axiom amounts to checking it in , where it is evident.
Let be a site with products and let be the left Bousfield localization at the Cech nerve projections.
Then for any cofibrant object, the closed monoidal structure on presheaves-adjunction
is a Quillen adjunction.
The above lemma implies that the left adjoint preserves cofibrations. As discussed in the section on sSet-enriched adjunctions at Quillen adjunction since the adjunction is -enriched and since is a left proper simplicial model category it suffices to check that preserves fibrant objects.
For that let be a covering family and the corresponding Cech nerve. We need to check that if is fibrant, then
is an equivalence of Kan complexes.
Writing and using that the hom-functor preserves ends, this is eqivalent to
being an equivalence. Now we observe that is a local epimorphism in the above sense, namely a morphism such that for every morphism out of a representable, there is a lift
By the above discussion of the Cech-localization of , this is a local morphism, hence does produce an equivalence when hommed into the fibrant object .
Properties of homotopy limits and homotopy colimits of simplicial presheaves are discussed at
Let be a site.
Let be a finite diagram.
Write for any representative of the homotopy limit over computed in the global model structure , well defined up to isomorphism in the homotopy category.
Then presents also the homotopy limit of computed in the local model structure .
By the discussion at (∞,1)-limit the homotopy limit computes the corresponding (∞,1)-limit and (∞,1)-sheafification is a left exact (∞,1)-functor and preserves these finite (∞,1)-limits:
Here is the left derived functor of the identity for the above left Bousfield localization. Since left Bousfield localization does not change the cofibrations and includes the global weak equivalences into the local weak equivalences, the postcomposition of the diagram with is given by cofibrant replacement in the local structure, too. But the homotopy limit of the diagram is invariant, up to equivalence, under cofibrant replacement, and hence a finite homotopy limit diagram in the global structure is also one in the local structure.
Inclusion of chain complexes of sheaves
We discuss how chain complexes of presheaves of abelian groups embed into the model structure on simplicial presheaves. Under passing to the intrinsic cohomology of the (∞,1)-topos presented by by , this realizes traditional abelian sheaf cohomology over and generalizes it to general base objects.
Observe from the discussion at model structure on simplicial abelian groups that the degreewise free functor-forgetful functor adjunction (see algebra over a Lawvere theory for details) induces a Quillen adjunction
between the model structure on simplicial abelian groups and the classical model structure on simplicial sets, which exhibits as the corresponding transferred model structure.
Moreover, the Dold-Kan correspondence constitutes in particular a Quillen equivalence
between the projective model structure on chain complexes of abelian groups in non-negative degree and simplicial abelian groups.
for the composite Quillen adjunction. For any category, postcomposition with induces a Quillen adjunction
between the projective model structure on functors and the global projective model structure on simplicial presheaves, which by convenient abuse of notation we denote by the same symbols.
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
A nice introduction and survey is provided in the notes
Detailed discussion of the injective model structures on simplicial presheaves is in
John F. Jardine, Simplicial presheaves Journal of Pure and Applied Algebra 47 (1987), 35-87 (pdf)
John F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology, homotopy and applications, vol. 3 (2), 2001, pp.361–384
John F. Jardine, Boolean localization, in practice (web)
John F. Jardine, Local homotopy theory (2011) (pdf)
The projective model structure is discussed in
- Benjamin Blander, Local projective model structures on simplicial presheaves, K-Theory, Volume 24, Number 3, November 2001 , pp. 283–301(19) (journal)
A brief review in the context of nonabelian Hodge theory is in section 4 of
- Martin Olsson, Towards non-abelian -adic Hodge theory in the good reduction case (pdf)
A detailed study of descent for simplicial presheaves is given in
A survey of many of the model structures together with a treatment of the left local projective one is in
The characterization of the model category of simplicial presheaves as the canonical presentation of the (hypercompletion of) the (∞,1)-category of (∞,1)-sheaves on a site is in
A set of lecture notes on simplicial presheaves with an eye towrads algebraic sites and derived algebraic geometry is
Last not least, it is noteworthy that the idea of localizing simplicial sheaves at stalkwise weak equivalences is already described and applied in
using instead of a full model category structure the more lightweight one of a Brown category of fibrant objects.
A comparison between Brown-Gersten and Joyal-Jardine approach:
- V. Voevodsky, Homotopy theory of simplicial presheaves in completely decomposable topologies, arxiv/0805.4578
The proposal for descent objects for strict -groupoid-valued presheaves discussed in Descent for strict infinity-groupoids appeared in
The relation to the general descent condition is discussed in
A useful collection of facts is in