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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={manifolds}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)->dga^opDR: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.

1

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={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)->dga^op.