To make things clear: Yes, this is a God of War trailer. No, it doesn’t feature Kratos in any shape or form.
Sony Santa Monica showcased the competitive (and cooperative, if you team with friends) aspect of God of War: Ascension. Play 4 vs 4, be it straight out Free-for-All fights or objective-based games, but make sure you choose an alliance to a Greek god beforehand. Most likely this will affect the magic one can use.
Overall, the game looks pretty exciting, if only because we like the idea of Greek Mythology meets Assault.