Framed motivic $\Gamma$-spaces
Izvestiya. Mathematics , Tome 87 (2023) no. 1, pp. 1-28
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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 Voevodsky's framed correspondences into the concept of framed motivic
$\Gamma$-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 $\Gamma$-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.
Keywords:
framed correspondences, motivic spaces, framed motivic
$\Gamma$-spaces, connective and very effective motivic spectra,
infinite motivic
loop spaces.
Mots-clés : $\Gamma$-spaces
Mots-clés : $\Gamma$-spaces
@article{IM2_2023_87_1_a0,
author = {G. A. Garkusha and I. A. Panin and P. {\O}stv{\ae}r},
title = {Framed motivic $\Gamma$-spaces},
journal = {Izvestiya. Mathematics },
pages = {1--28},
publisher = {mathdoc},
volume = {87},
number = {1},
year = {2023},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_2023_87_1_a0/}
}
G. A. Garkusha; I. A. Panin; P. Østvær. Framed motivic $\Gamma$-spaces. Izvestiya. Mathematics , Tome 87 (2023) no. 1, pp. 1-28. http://geodesic.mathdoc.fr/item/IM2_2023_87_1_a0/