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For any motivic $\mathbb{E}_\infty$-ring spectrum $A$ we construct anequivalence $\rho$ between the $\infty$-category of cellular motivic $A$-modulespectra and modules over an $\mathbb{E}_1$-algebra $\Theta$ in $\mathbb{Z}$-graded spectra, under which the motivic grading corresponds to the$\mathbb{Z}$-grading.  If the base is the complex numbers or if $A$ admits an$\mathbb{E}_\infty$-orientation, we refine the $\mathbb{E}_1$-algebra $\Theta$to an $\mathbb{E}_\infty$-algebra and $\rho$ to a symmetric monoidalequivalence. To capture the symmetric monoidal structure in the generalsituation, we lift $\rho$ to a symmetric monoidal equivalence to modules overan $\mathbb{E}_\infty$-algebra in $\mathcal{J} $-graded spectra that invertmorphisms of $\mathcal{J}$, where $\mathcal{J}$ is the diagram category ofSagave-Schlichtkrull, a model for Quillen's localization of the groupoid offinite sets and bijections.

A topological model for cellular motivic spectra