Framed motivic $\Gamma$-spaces

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Abstract

We combine several mini miracles to achieve an elementary understandingof infinite loop spaces and very effective spectra in the algebro-geometricsetting of motivic homotopy theory. Our approach combines $\Gamma$-spacesand Voevodsky's framed correspondences into the concept of framed motivic$\Gamma$-spaces; these are continuous or enriched functors of two variablesthat take values in framed motivic spaces. We craft proofs of our mainresults by imposing further axioms on framed motivic $\Gamma$-spacessuch 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 pointsof view on the $\mathbb{A}^1$-homotopy theory of algebraic varieties.

About the authors

Grigory Anatolevich Garkusha

Swansea University

Doctor of physico-mathematical sciences, Professor

Ivan Alexandrovich Panin

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences; University of Oslo

Email: paniniv@gmail.com
Doctor of physico-mathematical sciences

Paul Arne Østvær

Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano; University of Oslo

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Copyright (c) 2023 Гаркуша Г.A., Панин И.A., Østvær P.A.

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