Motivic Gamma-spaces

Abstract: This is a joint work with Ivan Panin and Paul Arne Østvær. 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 combines Gamma-spaces and framed correspondences into the concept of motivic Gamma-spaces; these are continuous or enriched functors of two variables that take values in motivic spaces and are equipped with a framing. We craft proofs of our main results by imposing further axioms on motivic Gamma-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, A1- and sigma-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. As prime examples we discuss the motivic sphere spectrum, algebraic cobordism, motivic cohomology, and Milnor-Witt motivic cohomology.

algebraic topology

Audience: researchers in the topic

Online algebraic topology seminar

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