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 $SH_{\operatorname {nis}}^{\operatorname {fr}}(k)$ and effective framed bispectra $SH_{\operatorname {nis}}^{\operatorname {fr},\operatorname {eff}}(k)$ are introduced in the paper. Both triangulated categories only involve Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that $SH_{\operatorname {nis}}^{\operatorname {fr}}(k)$ and $SH_{\operatorname {nis}}^{\operatorname {fr},\operatorname {eff}}(k)$ recover classical Morel–Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $SH^{\operatorname {eff}}(k)$ respectively.

Also, $SH(k)$ and $SH^{\operatorname {eff}}(k)$ are recovered as the triangulated category of framed motivic spectral functors $SH_{S^1}^{\operatorname {fr}}[\mathcal {F}r_0(k)]$ and the triangulated category of framed motives $\mathcal {SH}^{\operatorname {fr}}(k)$ constructed in the paper.