For any motivic \(\mathbb{E}_{\infty}\) -ring spectrum A we construct an equivalence ρ between the ∞-category of cellular motivic A-module spectra and modules over an \(\mathbb{E}_{1}\) -algebra Θ in ℤ-graded spectra, under which the motivic grading corresponds to the ℤ-grading. If the base is ℂ or if A admits an \(\mathbb{E}_{\infty}\) -orientation, we refine the \(\mathbb{E}_{1}\) -algebra Θ to an \(\mathbb{E}_{\infty}\) -algebra and ρ to a symmetric monoidal equivalence.
To capture the symmetric monoidal structure in the general situation, we lift ρ to a symmetric monoidal equivalence to modules over an \(\mathbb{E}_{\infty}\) -algebra in \(\cal{J}\) -graded spectra that invert morphisms of J, where J is the diagram category of Sagave–Schlichtkrull [15], a model for Quillen’s localization of the groupoid of finite sets and bijections.
