The triangulated categories of framed bispectra and framed motives

Grigory Garkusha; Ivan Panin

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arXiv · 1809.08006

The triangulated categories of framed bispectra and framed motives

Grigory Garkusha·Ivan Panin

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Abstract

An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $SH_{nis}^{fr}(k)$ and effective framed bispectra $SH_{nis}^{fr,eff}(k)$ are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that $SH_{nis}^{fr}(k)$ and $SH_{nis}^{fr,eff}(k)$ recover the classical Morel-Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $SH^{eff}(k)$ respectively. We also recover $SH(k)$ and $SH^{eff}(k)$ as the triangulated category of framed motivic spectral functors $SH_{S^1}^{fr}[\mathcal Fr_0(k)]$ and the triangulated category of framed motives $\mathcal {SH}^{fr}(k)$ respectively constructed in the paper.

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Grigory Garkusha, Ivan Panin. 2018-09-21. The triangulated categories of framed bispectra and framed motives. https://arxiv.org/abs/1809.08006

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