# simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex.

Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) functor from simplicial manifolds to dg-algebras, is homotopical?

For instance: simplicial manifolds naturally embed into the category of simplicial presheaves on the category of manifolds, on which we have the standard local model structure on simplicial presheaves. On dg-algebras there is the standard model structure on dg-algebras.

Is there any literature that discusses explicitly the respect of the simplicial deRham complex operation of the respective weak equivalences?

-

There is a projective model structure on the category of (pre)sheaves with value in any reasonnable model category (e.g. simplicial sets, complexes of abelian groups, commutative k-dg-algebras, where k is some field of char. 0, or k-dg-algebras for any commutative ring k); see for instance def. 4.4.33 and 4.4.40 and cor. 4.4.42 in Ayoub's book (Astérisque 315), whose online version is here (there is also a paper of Barwick which does the job if you want descent à la Lurie (to appear in HHA soon, I think)). If you have a site C and a left Quillen functor F:M->M', you get a left Quillen functor Sh(C,M)->Sh(C,M') between model categories of sheaves. The fibrant objects will always have the good taste of being exactly the termwise fibrant sheaves which satisfy (hyper)descent, and, if you have enough points, the weak equivalences are defined stalk-wise.

If M=SSet and M'=Complexes of R-vector spaces, the usual adjunction SSet<->Comp(R) gives a Quillen adjunction:

Sh(C,Sset) <-> Sh(C,Comp(R))

If you consider the projective model structure on Sh(C,Sset), any simplicial sheaf X such that, for each n, X_n is a sum of representables, is cofibrant (and any cofibrant object is weakly equivalent to such a thing). Hence, if C={differential real manifolds}, if you allow your manifolds to be stable under small sums, the simplicial manifolds are just your favourite cofibrant objects. You can thus apply all the machinery of model categories (e.g. the homotopy category of cofibrant objects is equivalent to the whole Ho(M) for any model category M). As the de Rham complex satisfies descent on manifolds, then it is a fibrant object (for the projective model structure on sheaves on complexes), and using the above Quillen adjunction, you will get that the derived sections of the de Rham complex over a simplicial manifold behaves like maps from a cofibrant object to a fibrant object in any model category: quite well. From there, you should be able to prove that de Rham cohomology of simplicial manifolds is in fact the explicit description of the total left derived functor of the left Quillen functor DR:Sh(manifolds,Sset)->cdga^op.

[Added comments] You can also consider the left Bousfield localization L_R of the projective model category of simplicial sheaves on the category of manifolds which consists to invert the maps XxR->X for any manifold X (where R denotes the real line). The result is a model category which is Quillen equivalent to the model category of simplicial sets. Using the Poincaré lemma (which gives you the homotopy invariance of de Rham cohomology), the functor DR:Sh(manifolds,Sset)->cdga^op induces a functor of shape

Ho(Sset)=Ho(L_R Sh(manifolds,Sset))->Ho(cdga)^op

which is simply (isomorphic to) the functor of Quillen-Sullivan.

-
Thanks for the reply! Recently I wrote up what I thought was the proof that in the projective local struture on sPSh all objects that are degreewise coproducts of representables are cofibrant. But then somebody suggested that there must have been an error in my reasoning and that in general one needs in addition to require a splitness conditions as discussed by Dugger. My notes on this are at ncatlab.org/nlab/show/… From what you said I gather this is right after all. Could you maybe quickly have a glance at that? – Urs Schreiber Oct 28 '09 at 22:35
Here is an argument: consider a simplicial sheaf X, considered as a functor from the opposite category of simplicices to simplicial sheaves. If X is termwise a sum of representables, then X is termwise cofibrant. Hence its homotopy colimit (computed à la Bousfield-Kan) is cofibrant. But what you get then is X itself, so that X is cofibrant (I think this is essentially your proof). – Denis-Charles Cisinski Oct 28 '09 at 23:19
Thanks again! To come back to the original question: this was originally motivated from the notes I have here ncatlab.org/schreiber/show/Chevalley-Eilenberg+algebra I have reworked this now a bit, after te above exchange. But evidently this is (in as far as it is correct at all) still far from the elegant statements you have in mind. I'd be very interested in your opinion. – Urs Schreiber Oct 29 '09 at 10:18
sorry, but I have to ask back concerning your cofibrancy argument. I must be being dense: so I guess you argue that the diagram is Reedy cofibrant such that the BK-map from the hocolim to the ordinary coend is a weak equivalence. But don't we need that this map is a cofibration, too, then? How does that follow? – Urs Schreiber Oct 29 '09 at 15:03
It might be that I am the dense guy, but it seems to me that this coend is the homotopy colimit of X by definition of the Bousfield-Kan construction (because we see X as a functor with value in discrete simplicial sets). – Denis-Charles Cisinski Oct 29 '09 at 17:10