A characterization of cellular motivic spectra

Abstract

Let $\alpha: \mathcal{C} \to \mathcal{D}$ be a symmetric monoidal functor from a stable presentable symmetric monoidal $\infty$-category $\mathcal{C}$ compactly generated by the tensorunit to a stable presentable symmetric monoidal $\infty$-category $\mathcal{D}$ with compact tensorunit. Let $\beta: \mathcal{D} \to \mathcal{C}$ be a right adjoint of $\alpha$ and $\mathrm{X}: \mathcal{B} \to \mathcal{D}$ a symmetric monoidal functor starting at a small rigid symmetric monoidal $\infty$-category $\mathcal{B}$. We construct a symmetric monoidal equivalence between modules in the $\infty$-category of functors $\mathcal{B} \to \mathcal{C}$ over the $\mathrm{E}_\infty$-algebra $\beta \circ \mathrm{X}$ and the full subcategory of $\mathcal{D}$ compactly generated by the essential image of $\mathrm{X}$. Especially for every motivic $\mathrm{E}_\infty$-ring spectrum $\mathrm{A}$ we obtain a symmetric monoidal equivalence between the $\infty$-category of cellular motivic $\mathrm{A}$-module spectra and modules in the $\infty$-category of functors $\mathrm{QS}$ to spectra over some $\mathrm{E}_\infty$-algebra, where $\mathrm{QS}$ denotes the 0th space of the sphere spectrum