on dg-algebras/on dg-coalgebras and on on cosimplicial rings (related by monoidal Dold-Kan correspondence)
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.
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.
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 morphism of simplicial sets is a fibration;
and such that this fibration is an acyclic fibration whenever either or are acyclic.
Let be a simplicial model category
From the axioms of enriched model category it follows that for cofibrant and fibrant, the simplicial set given by the -enrichement is fibrant in the standard model structure on simplicial sets, hence is a Kan complex. In fact, for and any two objects and and a cofibrant and fibrant replacement, respectively, is the correct derived hom-space between and . In particular the full -enriched subcategory on cofibrant fibrant objects is therefore a -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 .
The category is a monoidal model category and is hence naturally enriched, as a model category, over itself. This is the archetypical simplicial model category.
For any small sSet-enriched category and simplicial combinatorial model category, the global model structure on functors and are themselved simplicial combinatorial model categories.
The left Bousfield localization of these at any set of morphisms is again a combinatorial simplicial model category. Large clases of examples arise this way. In particular for itself this yields the model structure on simplicial presheaves.
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.
While some model categories do not admit an -enrichment, for large classes of model categories one can find an Quillen equivalence to a model cateory that does have an -enrichment.
Every left proper combinatorial model category is Quillen equivalent to a simplicial model category.
More precisely: let be a
model category. Then the category of simplicial objects in carries the structure of a simplicial model category with respect to its canonical -enrichment, such that the functor that sends a simplicial object to its degree 0 piece exhibits a Quillen equivalence
In we have
the weak equivalences are those morphisms that induce weak equivalences in under homotopy colimit over ;
the cofibrations are cofibrations in the Reedy model structure on .
the fibrant objects are precisely the Reedy-fibrant objects all whose face and degenarcy maps are week equivalences in .
This is theorem 1.2 of
The simplicial enrichment is theorem 6.1
The charaxcterization of the fibrant objects is theorem 5.7.
There is a similar model structure on where instead the cofibrations are the cofibrations with respect to the projective global model structure on functors on .
In this the fibrant objects are precisely the simplicial objects that are degreewise fibrant in , and for which all face and degeneracy maps are weak equivalences in .
The identity functor established a Quillen equivalence between this model structure and the one discussed above
This is theorem 1.3 in the above article.
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 prop 1.3
(uniqueness)
Let be a model category. Then there is at most one model category structure on such that
every morphism that is degreewise a weak equivalence in is a weak equivalence;
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 theorm 3.1 in
The term simplicial model category for the notion described here is entirely standard, but in itself a bit subobtimal. More properly one would speak of simplicially enriched category, which is different (though not much) from a simplicial object in Cat.
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 Andre Joyal-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 standrt terminology, is not a “simplicial model category”.
section 9.1.5 of
section A.3 in