The triangulated categories of framed bispectra and framed motives

Abstract

An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $S{H}_{nis}^{fr}(k)$ and effective framed bispectra $S{H}_{nis}^{fr,eff}(k)$ 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 $S{H}_{nis}^{fr}(k)$ and $S{H}_{nis}^{fr,eff}(k)$ recover the classical Morel-Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $S{H}^{eff}(k)$ respectively. We also recover $SH(k)$ and $S{H}^{eff}(k)$ as the triangulated category of framed motivic spectral functors $S{H}_{S^1}^{fr}[\mathcal{F}{r}_0(k)]$ and the triangulated category of framed motives ${\mathcal{S}\mathcal{H}}^{fr}(k)$ respectively constructed in the paper.

The final revised version