Framed motivic $\Gamma$-spaces

Grigory Garkusha; Ivan Panin; Paul Arne Østvær

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arXiv · 1907.00433

Framed motivic $\Gamma$-spaces

Grigory Garkusha·Ivan Panin·Paul Arne Østvær

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Abstract

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 framed 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 framed motivic $\Gamma$-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, ${\mathbb A}^{1}$- and $\sigma$-invariance, Nisnevich excision, Suslin contractibility, and grouplikeness. This adds to the discussion in the literature on coexisting points of view on the ${\mathbb A}^{1}$-homotopy theory of algebraic varieties.

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BibTeX RIS

Grigory Garkusha, Ivan Panin, Paul Arne Østvær. 2019-06-30. Framed motivic $\Gamma$-spaces. https://arxiv.org/abs/1907.00433

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