related by the Dold-Kan correspondence
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 .
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).
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”.
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 morphism of simplicial sets is a fibration;
The following conditions – that each make into a simplicial model category – are equivalent:
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.
We list in the following some implications of these equivalent conditions.
Let now be a simplicial model category.
If is fibrant, and is a cofibration in sSet, then
is a fibration in .
is a fibration. But and hence the claim follows.
Similarly we have
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 .
The standard 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.
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.
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 have an -enrichment.
Let be a
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 are weak equivalences.
Hence is a weak equivalence.
This appears as (Dugger, prop. 5.4 corollary 5.5).
The model structures from def. 2 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 .
constitute a Quillen equivalence, the identity functors constitute a Quillen equivalence
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.
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 lemma 1, we have weak equivalences
seen by computing the derived homs by simplicial framings.
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 on , for any . By lemma 1 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.
Let then be a morphism in and consider two fibrant replacements
where the first one () is taken in and the second (\hat A \to \hat B) 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. (…)
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;
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 1 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.
choosing a cofibrant replacement of in ;
choosing a Reedy fibrant replacement of in such that all face and degeneracy maps are weak equivalences,
By theorem 1 we may compute the derived hom space in after the inclusion . Since by that theorem is a simplicial model category, by prop. 2 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:
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 standard textbook reference is section 9.1.5 of
Original results are in
section A.3 in