TRIANGULATED CATEGORIES OF FRAMED BISPECTRA AND FRAMED MOTIVES

An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory SH(k) is suggested. The triangulated category of framed bispectra SHfrnis(k) and effective framed bispectra SHfr,eff nis (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 SHfrnis(k) and SHfr,eff nis (k) recover classical Morel-Voevodsky triangulated categories of bispectra SH(k) and effective bispectra SHeff (k) respectively. Also, SH(k) and SHeff (k) are recovered as the triangulated category of framed motivic spectral functors SHSfr1 [Fr0(k)] and the triangulated category of framed motives SHfr(k) constructed in the paper.