These two volumes, each consisting of two long chapters, represent an important contribution to stable motivic homotopy theory. In chapter 1, the author constructs the Grothendieck four functors f*, f*, f! and f! in this context (this result was earlier announced by Voevodsky but the details of his construction are still being awaited). Chapter 2 is devoted to constructability properties of these functors and the Verdier duality. In chapter 3, the author develops a motivic formalism of nearby and vanishing cycles. Chapter 4 sums up a construction of the stable homotopy category over a given scheme. Both volumes are written very clearly and carefully. They also include a substantial amount of abstract background material used by the author (e.g. 2-categories, properties of triangulated categories, derivators and homotopical algebra).

Reviewer:

jnek