Framed motivic Γ-spaces

We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach com-bines Γ-spaces and Voevodsky’s framed correspondences into the concept of framed motivic Γ-spaces; these are continuous or enriched functors of two variables that take values in framed motivic spaces. We craft proofs of our main results by imposing further axioms on framed motivic Γ-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, A1-and σ-invariance, Nisnevich excision, Suslin contractibility, and grouplikeness. This adds to the discussion in the literature on coexisting points of view on the A1-homotopy theory of algebraic varieties.