Triangulated categories of framed bispectra and framed motives

An alternative approach to classical Morel–Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $SH_{\mathrm{nis}}^{\mathrm{fr}}(k)$ and effective framed bispectra $SH_{\mathrm{nis}}^{\mathrm{fr},\mathrm{eff}}(k)$ 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 $SH_{\mathrm{nis}}^{\mathrm{fr}}(k)$ and $SH_{\mathrm{nis}}^{\mathrm{fr},\mathrm{eff}}(k)$ recover classical Morel–Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $SH^{\mathrm{eff}}(k)$ respectively.
Also, $SH(k)$ and $SH^{\mathrm{eff}}(k)$ are recovered as the triangulated category of framed motivic spectral functors $SH_{S^1}^{\mathrm{fr}}[\mathcal{F}r_0(k)]$ and the triangulated category of framed motives $\mathcal{SH}^{\mathrm{fr}}(k)$ constructed in the paper.