A topological model for cellular motivic spectra

Abstract For any motivic $$\mathbb{E}_{\infty}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> -ring spectrum A we construct an equivalence ρ between the ∞ -category of cellular motivic A -module spectra and modules over an $$\mathbb{E}_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> -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 $$\mathbb{E}_{\infty}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> -orientation, we refine the $$\mathbb{E}_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> -algebra Θ to an $$\mathbb{E}_{\infty}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> -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 $$\mathbb{E}_{\infty}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> -algebra in $$\cal{J}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> </mml:mrow> </mml:math> -graded spectra that invert morphisms of , where is the diagram category of Sagave–Schlichtkrull [15], a model for Quillen’s localization of the groupoid of finite sets and bijections.