Triangulated categories of framed bispectra and framed motives

An alternative approach to the classical Morel–Voevodsky stable motivic homotopy theory S H ( k ) SH(k) is suggested. The triangulated category of framed bispectra S H nis fr ( k ) SH_{\operatorname {nis}}^{\operatorname {fr}}(k) and effective framed bispectra S H nis fr , eff ( k ) 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 S H nis fr ( k ) SH_{\operatorname {nis}}^{\operatorname {fr}}(k) and S H nis fr , eff ( k ) SH_{\operatorname {nis}}^{\operatorname {fr},\operatorname {eff}}(k) recover classical Morel–Voevodsky triangulated categories of bispectra S H ( k ) SH(k) and effective bispectra S H eff ( k ) SH^{\operatorname {eff}}(k) respectively.

Also, S H ( k ) SH(k) and S H eff ( k ) SH^{\operatorname {eff}}(k) are recovered as the triangulated category of framed motivic spectral functors S H S 1 fr [ F r 0 ( k ) ] SH_{S^1}^{\operatorname {fr}}[\mathcal {F}r_0(k)] and the triangulated category of framed motives S H fr ( k ) \mathcal {SH}^{\operatorname {fr}}(k) constructed in the paper.