A topological model for cellular motivic spectra

Abstract

For any motivic ${\mathbb{E}}_{\infty }$-ring spectrum $A$ we construct an equivalence $\ddot{I}$ between the $\infty$-category of cellular motivic $A$-module spectra and modules over an ${\mathbb{E}}_1$-algebra $\hat{I}˜$ 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 $\hat{I}˜$ to an ${\mathbb{E}}_{\infty }$-algebra and $\ddot{I}$ to a symmetric monoidal equivalence. To capture the symmetric monoidal structure in the general situation, we lift $\ddot{I}$ to a symmetric monoidal equivalence to modules over an ${\mathbb{E}}_{\infty }$-algebra in $\mathcal{J}$-graded spectra that invert morphisms of $\mathcal{J}$, where $\mathcal{J}$ is the diagram category of Sagave-Schlichtkrull, a model for Quillen's localization of the groupoid of finite sets and bijections.