The triangulated categories of framed bispectra and framed motives

TL;DR

This paper introduces new triangulated categories of framed bispectra and framed motives that provide an alternative to the classical motivic homotopy theory, relying solely on Nisnevich local equivalences.

Contribution

The paper constructs and relates new categories of framed bispectra and framed motives to classical motivic homotopy categories, offering an alternative approach.

Findings

Recover classical motivic categories from framed bispectra

New categories use only Nisnevich local equivalences

Establish equivalences with classical motivic homotopy categories

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.