Unstable motivic homotopy categories in Nisnevich and cdh-topologies

One can do the motivic homotopy theory in the context of different motivic homotopy categories. One can vary the topology on the category of schemes used to define the homotopy category or one can vary the category of schemes itself considering only schemes satisfying certain conditions. The category obtained by taking smooth schemes and the Nisnevich topology seems to play a distinguished role in the theory because of the Gluing Theorem (see [7]) and some other, less significant, nice properties. On the other hand, in the parts of the motivic homotopy theory dealing with the motivic cohomology it is often desirable to be able to work with all schemes instead of just the smooth ones. For example, the motivic Eilenberg-MacLane spaces are naturally representable (in characteristic zero) by singular schemes built out of symmetric products of projective spaces but we do not know of any explicit way to represent these spaces by simplicial smooth schemes. The goal of this paper is to show that, under the resolution of singularities assumption, the pointed motivic homotopy category of smooth schemes over a field with respect to the Nisnevich topology is almost equivalent to the pointed motivic homotopy category of all schemes over the same field with