[Paper Review] A characterization of cellular motivic spectra

Hadrian Heine|arXiv (Cornell University)|Dec 1, 2017

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This paper establishes a symmetric monoidal equivalence between cellular motivic A-module spectra and modules over an E∞-algebra in the ∞-category of functors from the 0th space of the sphere spectrum to spectra. By leveraging adjoint functors and symmetric monoidal structures in stable ∞-categories, it provides a new algebraic characterization of cellular motivic spectra via functor categories, generalizing to all motivic E∞-ring spectra.

ABSTRACT

Let $ \alpha: \mathcal{C} o \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 $\beta: \mathcal{D} o \mathcal{C}$ be a right adjoint of $\alpha$ and $ \mathrm{X}: \mathcal{B} o \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} o \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.

Motivation & Objective

• To characterize cellular motivic spectra in terms of functor categories over E∞-algebras.
• To generalize the algebraic structure of motivic module spectra using symmetric monoidal adjunctions.
• To provide a universal description of cellular motivic A-module spectra via the 0th space of the sphere spectrum.
• To establish a framework applicable to all motivic E∞-ring spectra through functorial constructions.

Proposed method

• Utilizes a symmetric monoidal functor α: C → D between stable presentable symmetric monoidal ∞-categories with compact tensor units.
• Constructs a right adjoint β: D → C to α, enabling passage from D to C while preserving monoidal structure.
• Applies a symmetric monoidal functor X: B → D from a small rigid symmetric monoidal ∞-category B to D.
• Establishes a symmetric monoidal equivalence between modules over β∘X in Fun(B, C) and the full subcategory of D compactly generated by the essential image of X.
• Applies the construction to motivic E∞-ring spectra A, identifying cellular A-module spectra with modules over an E∞-algebra in Fun(QS, Sp).
• Relies on ∞-categorical techniques, including compact generation, adjunctions, and E∞-algebra structures in functor categories.

Experimental results

Research questions

• RQ1 How can cellular motivic A-module spectra be algebraically characterized using functor categories?
• RQ2 What symmetric monoidal structure arises from composing adjoint functors α and β in the context of motivic spectra?
• RQ3 Can the category of cellular motivic A-modules be equivalent to a category of modules over an E∞-algebra in a functor ∞-category?
• RQ4 What role does the 0th space of the sphere spectrum play in classifying motivic modules?
• RQ5 How does compact generation interact with symmetric monoidal structures in this construction?

Key findings

• A symmetric monoidal equivalence is established between the ∞-category of cellular motivic A-module spectra and the ∞-category of modules over an E∞-algebra in Fun(QS, Sp).
• The construction applies universally to all motivic E∞-ring spectra A, providing a uniform characterization.
• The essential image of X generates a full subcategory of D that is equivalent to the category of modules over β∘X in Fun(B, C).
• The method relies on the compact generation of the tensor unit in C and D, ensuring the existence of well-behaved adjunctions.
• The equivalence preserves symmetric monoidal structures, making it suitable for higher algebraic applications.
• The framework generalizes to any symmetric monoidal functor α with right adjoint β and rigid source category B.