@TylerLawson thanks for your comment! I'll try to work it out. I think your strategy will show that the simplicial structure given by tensoring with the chains on simplices does not satisfy Quillen's axioms. Let's see if I can also show that there is no other 'exotic' nice choice of a simplicial structure.

@FernandoMuro That would be interesting. The simplicial model structure would make it tensored over simplicial sets, so in particular I think this might mean you should be able to construct a functor (K↦ℤ⊗K) on simplicial sets with a natural isomorphism of ℤ⊗(KxL) with (ℤ⊗K)⊗(ℤ⊗L) (though I don't think you need it to preserve the symmetry isomorphism).

anyone know why the dual of the (2^j)-1'th stiefel whitney class is "obviously" in the image of the homology of BSO?