Voevodsky's mixed motives versus Kontsevich's noncommutative mixed motives

We prove that the quotient of Voevodsky's category of geometric mixed motives DMgmDMgm by the endofunctor −⊗Q(1)[2]−⊗Q(1)[2] embeds fully-faithfully into Kontsevich's category of noncommutative mixed motives KMM. We show moreover that this embedding is compatible with the one between Chow motives and noncommutative Chow motives. Along the way, we relate KMM with Morel–Voevodsky's stable A1A1-homotopy category, recover the twisted algebraic K-theory of Kahn–Levine from KMM, and extend Elmendorf–Mandell's foundational work on multicategories to a broader setting.