model category, model -category
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The term simplicial model category is short for sSet-enriched model category.
A simplicial model category is a model or presentation for an (∞,1)-category that is half way in between a bare model category and a Kan complex-enriched category.
Specifically, a simplicial model category is an sSet-enriched category together with the structure of a model category on its underlying category such that both structures are compatible in a reasonable way.
One important use of simplicial model categories comes from the fact that the full sSet-subcategory on the fibrant-cofibrant objects – which is not just sSet-enriched but actually Kan complex-enriched – is the (∞,1)-category-enhancement of the homotopy category of the model category .
For generalizations of this construction with sSet replaced by another monoidal model category see enriched homotopical category.
The term simplicial model category for the notion described here is entirely standard, but in itself a bit suboptimal. More properly one would speak of simplicially enriched category, which is a proper special case of a simplicial object in Cat (that for which the simplicial set of objects is discrete).
The other caveat is that there are different model category structures on sSet and hence even the term -enriched model category is ambiguous.
For instance the model structure for quasi-categories is an -enriched model category, but not for the standard Quillen model structure on the enriching category: since is a closed monoidal model category it is enriched over itself, hence is a -enriched model category, not an -enriched one. So in the standard terminology, is not a “simplicial model category”.
A simplicial model category is an enriched model category which is enriched over : the category sSet equipped with its standard model structure on simplicial sets.
Spelled out, this means that a simplicial model category is
with the structure of a model category on the underlying category
such that for every cofibration and every fibration in the pullback powering of simplicial sets is a Kan fibration;
Let be a category equipped with the structure of a model category and with that of an sSet-enriched category, which is tensored and cotensored over sSet.
The following conditions – that each make into a simplicial model category – are equivalent:
the tensoring is a left Quillen bifunctor;
for any cofibration and fibration in , the induced morphism
is a fibration, and is in addition a weak equivalence if either of the two morphisms is;
for any cofibration in and fibration in , the induced morphism
is a fibration, and is in addition a weak equivalence if either of the two morphisms is.
This follows directly, by Joyal-Tierney calculus, from the defining properties of tensoring and cotensoring (as in this Prop.).
We list in the following some implications of these equivalent conditions.
Let now be a simplicial model category.
Apply prop. to the case of the cofibration and the fibration , where “” denotes the terminal object. This yields that
is a fibration. But and hence the claim follows.
Similarly we have
If is cofibrant and is fibrant, then is fibrant in sSet, hence is a Kan complex.
Apply prop. to the cofibration , where “” denotes the initial object, and to the fibration to find that
is a fibration. But since is initial and is terminal, all three simplicial sets in the fiber product on the right are the point, hence this is a fibration
For and any two objects and and a cofibrant and fibrant replacement, respectively, is the correct derived hom-space between and (see the discussion there). In particular the full -enriched subcategory on cofibrant fibrant objects is therefore an sSet-enriched category which is fibrant in the model structure on simplicially enriched categories. Its homotopy coherent nerve is a quasi-category. All this are intrinsic incarnatons of the (∞,1)-category that is presented by .
Although need not have the correct homotopy type for the mapping space if is not cofibrant or is not fibrant, simplicial notions of homotopy are still sufficient to detect the model-categorical ones.
If are simplicially homotopic (i.e. in the same connected component of the simplicial set ), then they represent the same map in the homotopy category of . Therefore, any simplicial homotopy equivalence in is a weak equivalence.
This is Lemma 9.5.15 and Proposition 9.5.16 of Hirschhorn.
The classical model structure on simplicial sets is a closed monoidal model category and is hence naturally enriched, as a model category, over itself. This is the archetypical simplicial model category.
The classical model structure on topological spaces for compactly generated topological spaces (here) is similarly enriched over itself. Under geometric realization this also makes it a simplicial model category.
(See also for instance Goerss-Schemmerhorn 06, p. 26)
For any small sSet-enriched category and simplicial combinatorial model category, the global model structure on functors and are themselved simplicial combinatorial model categories. See model structure on simplicial presheaves.
The left Bousfield localization of a combinatorial simplicial model category at any set of morphisms is again a combinatorial simplicial model category. Large classes of examples arise this way.
While many model categories do not admit an -enrichment, for large classes of model categories one can find a Quillen equivalence to a model category that does carry an -enrichment.
These are constructed as Bousfield localization of Reedy model structures on the category of simplicial objects in the given model category.
Let be a
By the discussion at cofibrantly generated model category in the section Presentation and generation, there exists a small set of objects that detect weak equivalences. For some such choice of , let
where denotes the tensoring of over Set, hence denotes the simplicial object consisting of coproducts of as
Write
for the left Bousfield localization of the projective model structure on functors at this set of morphisms.
Similarly, write
for the left Bousfield localization of the Reedy model structure at .
Let be a cofibrantly generated model category.
If is degreewise cofibrant and has all structure maps being weak equivalences, then all coprojections into the homotopy colimit are weak equivalences and hence is a weak equivalence.
This appears as Dugger, prop. 5.4 corollary 5.5.
The model structures from def. have the following properties.
The weak equivalences in both are precisely those morphisms which become weak equivalences under homotopy colimit over .
The fibrant objects in both are precisely those objects that are fibrant in the corresponding unlocalized structures, and such that all the face and degeneracy maps are weak equivalences in .
The colimit/constant adjoint functors
constitute a Quillen equivalence, the identity functors constitute a Quillen equivalence
and the constant/limit adjoint functors (notice that the limit is evaluation in degree 0, since has an initial object) constitute a Quillen equivalence
The canonical sSet-enrichment/tensoring/powering of the category of simplicial objects makes (but not in general ) into a simplicial model category.
This is Dugger, theorem 5.2, theorem 5.7, theorem 6.1.
So in particular every left proper combinatorial model category is Quillen equivalent to a simplicial model category.
We first show that the fibrant objects in are the objectwise fibrant objects all whose structure maps are weak equivalences in . The argument for the fibrant objects in is directly analogous.
By general properties of left Bousfield localization, the fibrant objects in are the projective fibrant objects for which all induced morphisms on derived hom spaces
are weak equivalences. Since is cofibrant in by definition, also is cofibrant in .
So for fibrant, let be a simplicial framing for it. Notice that this means that for all also is a simplicial framing for . This is because
being a weak equivalence means that for all the morphism is a weak equivalence, which means that for all the morphism is a weak equivalence.
being fibrant in means that for all the morphism is a fibration in , hence that for all the morphism is a fibration in , hence that is Reedy fibrant.
Then we find
By assumption on the set , this implies the claim.
Now we show that the weak equivalences in are precisely those morphisms that become weak equivalences under the homotopy colimit.
By functorial cofibrant resolution and two-out-of-three, it is sufficient to show that this holds for morphisms between cofibrant objects.
By lemma , we have weak equivalences
seen by computing the derived homs by simplicial framings.
Now, by properties of left Bousfield localization, is a weak equivalence if for all -local objects the morphism of derived hom-spaces is a weak equivalence. Looking at the diagram
we see that this is the case precisely if the vertical morphism on the right is a weak equivalence for all fibrant , which is the case if is a weak equivalence. Since and here are cofibrant in , the colimits here are indeed homotopy colimits (as discussed there).
Now we discuss that is a Quillen equivalence. First observe that on the global model structure is clearly a right Quillen functor, hence we have a Quillen adjunction on the unlocalized structure. Moreover, by definition and by the above discussion, the derived functor of the left adjoint , namely the homotopy colimit, takes the localizing set to weak equivalences in . Therefore the assumptions of the discussion at Quillen equivalence - Behaviour under localization are met, and hence it follows that descends as a Quillen adjunction also to the localization.
To see that this is a Quillen equivalence, it is sufficient to show that for cofibrant and fibrant, a morphism is a weak equivalence in precisely if the adjunct becomes a weak equivalence under the homotopy colimit.
For this notice that we have a commuting diagram
and so our statement follows (by 2-out-of-3) once we know that the vertical morphisms here are weak equivalences. The left one is because is cofibrant, by assumption, as before. To see that the right one is, too, consider the factorization
of the identity morphism on , for any . By lemma the first morphism is a weak equivalence, and hence so is the morphism in question.
Now we show that the weak equivalences in are the hocolim-equivalences.
By a general result on functoriality of localization, we have that the is at least a Quillen adjunction.
Let then be a morphism in and consider two fibrant replacements
where the first one () is taken in and the second () in .
Assume first that is a hocolim-equivalence. Then so is , because the horizontal morphisms are all objectwise weak equivalences. But and are fibrant in , hence in by construction and at the same time all their structure maps are weak equivalences (use 2-out-of-3), so that they are in fact fibrant in . By general properties of left Bousfield localization, weak equivalences between local fibrant objects are already weak equivalences in the unlocalized structure – so is indeed even an objectwise weak equivalence. It follows then that so is , which is therefore in partiular a weak equivalence in . Finally the left horizontal morphisms are also weak equivalences in , by the above Quillen adjunction. So finally by 2-out-of-3 in it follows that also is a weak equivalence there.
By an analogous diagram chase, one shows the converse implication holds, that being a weak equivalence in implies that it is a hocolim-equivalence.
With this now it is clear that the identity adjunction above is in fact a Quillen equivalence.
Finally we show that is a Quillen equivalence.
First, it is immediate to check that is left Quillen, and since is left Quillen by definition of Bousfield localization, the above is at least a Quillen adjunction.
To see that it is a Quillen equivalence, let be cofibrant and be fibrant – which by the above means that it is a simplicial resolution – and consider a morphism . We need to show that this is a weak equivalence, hence, by the above, that its hocolim is a weak equivalence, precisely if is a weak equivalence in .
To that end, find a cofibrant resolution of in and consider the diagram
The colimits on the right compute the homotopy colimit. By 2-out-of-3 it follows that the right vertical morphism is a weak equivalence precisely if the left vertical morphisms is.
Finally it remains to show that is a simplicially enriched model category. (…)
(uniqueness)
Let be a model category. Then there is a unique model category structure on such that
every morphism that is degreewise a weak equivalence in is a weak equivalence in ;
the cofibrations are those of the Reedy model structure;
the fibrant objects are the Reedy-fibrant objects whose face and degeneracy maps are weak equivalences in .
This is Rezk, Schwede & Shipley, theorem 3.1.
By theorem at least one such model structure exists. By the discussion at model category – Redundancy of the axioms, the classes of cofibrations and fibrant objects already determine a model category structure.
For any left proper combinatorial model category, the derived hom-space between two objects may be computed by
choosing a cofibrant replacement of in ;
choosing a Reedy fibrant replacement of in such that all face and degeneracy maps are weak equivalences,
setting
By theorem we may compute the derived hom space in after the inclusion . Since by that theorem is a simplicial model category, by prop. the derived hom space is given by the simplicial function complex between a cofibrant replacement of and a fibrant replacement of . If is cofibrant, then is already Reedy cofibrant, and by the theorem as stated is a a fibrant resolution of . Finally, the theorem says that the simplicial function complex is given by
There is also a version for stable model categories:
Every proper cofibrantly generated stable model category is Quillen equivalent to a simplicial model category
This is Rezk, Schwede & Shipley, prop 1.3.
A particularly important type of simplicial model categories are those that are also combinatorial model categories.
A combinatorial simplicial model category is precisely a presentation for a locally presentable (∞,1)-category. See there for more details.
simplicial monoidel model category?
The definition originates in
Textbook accounts:
Philip Hirschhorn, Chapter 9 in: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)
Jacob Lurie, section A.3 in Higher Topos Theory (2009)
Garth Waner?: Categorical Homotopy Theory, EPrint Collection, University of Washington (2012) [hdl:1773/19589, pdf, pdf]
Emily Riehl, §3.8 and §11.4 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]
Further review:
On Quillen-equivalent simplicial enhancements of model categories:
Charles Rezk, Stefan Schwede, Brooke Shipley, Simplicial structures on model categories and functors, American Journal of Mathematics, Vol. 123, No. 3 (Jun., 2001), pp. 551-575 (arXiv:0101162, jstor)
Dan Dugger, Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. 353 12 (2001) 5003-5027 [ams:S0002-9947-01-02661-7, pdf]
Daniel Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 1 177-201 [arXiv:math/0007068, doi:10.1006/aima.2001.2015]
(cf. Dugger's theorem)
Last revised on July 27, 2024 at 13:07:28. See the history of this page for a list of all contributions to it.