Introduction to the Category of Derived Motivic Spectra

We formalize an abstraction of Grothendieck’s philosophy of motives and construct a category of derived motivic spectra in the Segal category $\mathbb{R}\underset{‾}{\text{Hom}}(({\text{dSt}}_k{)}_{\mathrm{/}F}^{\text{op}},\text{Top})$ (${\text{dSt}}_k$ the Segal category of derived stacks on s$k$-Alg, Top = $L{\text{Set}}_{\mathrm{\Delta }}$ the Segal category of simplicial sets), thereby providing a starting point for a construction of a stable motivic homotopy category in the Segal setting.