An introduction is found in Voevodsky’s Seattle lectures, probably on the webpage of Weibel, otherwise in Voevodsky folder.
http://mathoverflow.net/questions/2520/homotopy-theory-of-schemes-examples
http://mathoverflow.net/questions/2694/formalism-of-homotopy-theory-of-schemes
Voevodsky’s Nordfjordeid lecture - supernice (Voevodsky folder). Working over a field unfortunately. Basic constructions, of SH etc. Brief discussion of Thom spaces and homotopy purity. Cohomology theories: the motivic EM spectrum, KGL, MGL, claim that the notions of orientation and FGLs have direct analogs for P1-spectra. The slice filtration (great intro), update on Open problems paper. The zero-th slice of the unit spectrum is HZ, this is known for fields of char zero. Brief discussion of AHSS. Appendix on the Nisnevich topology, Nisnevich descent, and model structures.
Voevodsky ICM talk summary: Starting with a cat C, can use homotopy theory to study it, if we embed it into a bigger nicer cat of spaces, with a notion of weak equiv. Can then define the homotopy cat of spaces, and also the SW cat (naively stabilizing suspension), and better: a stable homotopy cat, using spectra. The second approach gives existence of infinite direct sums, so that rep theorems can be applied. Can carry out this starting with Sm/S for any notherian base, and then define motivic cohom, alg Kth and alg cobord on this cat for any such S. Spaces are Nisnevich sheaves of sets, and there is a notion of space of finite type, which implies categorical compactness. Model structure. Almost fibrant vs quasifibrant: probably EM spaces are the latter only. SW cat and various spheres. MV, Gysin, and blowup loch exact sequences. The Gysin les comes from a closed embedding of smooth schemes, and involves the Thom space of the normal bundle. The blowup sequence here is also for a closed embedding of smooth schemes. Prop 5.5: SH is compactly generated. Specialize to the case of T=P1. Get same long exact seqs. Chapter 6: For any E in SH, can define a corresponding cohomology and homology theory. Any such theory has Gysin, MV, and blowup long exact sequences. Usefulness of a “Brown rep thm” is restricted by the fact that it seems like the statement would be about functors on spaces, or spaces of finite type, and not functors on schemes. Now to EM spaces: Naive idea does not work. Instead, use idea of Suslin: use Dold-Thom thm. The right analogue of Symm is described, using finite correspondences I think. More details and def of the motivic cohom spectrum. Def of BGL, i.e. the alg K-theory spectrum, and comparison statements. Also MGL (alg cobordism). Two future directions: one related to circumventing problems related to RoS and hence restrictions to char zero base field, maybe using techniques in Bloch: Moving lemma for higher Chow groups, from Spivakovsky’s work. The other relating to descriptions of A1-homotopy types: rational case is conjectural, integral case quite unclear.
File: Lectures on motivic cohomology written by Deligne, in Voevodsky folder. Contains discussion of some topics including: some kind of equivariant version, i.e. schemes with action of a finite flat group scheme, and an equivariant analogue of the Nisnevich topology. Notion of solid sheaves, including Thom spaces of VBs. On of the main ideas seem to be the extension of certain functors from schemes to the motivic homotopy category, for example quotienting by a G-action, fixed points, and symmetric products. I think these functors are well-behaved wrt solid sheaves.
Strunk and Herrmann on a model for the motivic homotopy theory: http://front.math.ucdavis.edu/1007.3153
nLab page on Motivic homotopy theory