Comparison of Motivic Homotopy Theories

TL;DR

This paper constructs and compares functors between motivic homotopy categories and motives, revealing conditions under which these functors are fully faithful, thereby advancing understanding of motivic homotopy theory.

Contribution

It introduces comparison functors linking motivic homotopy categories to localizing motives, including their non-$ ext{A}^1$-invariant versions, and analyzes their properties over fields with resolution of singularities.

Findings

The $ ext{A}^1$-invariant functor is fully faithful over certain fields.

The non-$ ext{A}^1$-invariant functor is not fully faithful in general.

Both functors factor through modules over a dual K-theory spectrum.

Abstract

We construct a comparison functor from the dual category of motivic homotopy category $\mathcal{SH}$ to the category of $\mathbb{A}^1$-invariant localizing motives $\operatorname{Mot}_{\operatorname{loc}}^{\mathbb{A}^1}$ in the sense of Blumberg, Gepner and Tabuada (with $\mathbb{A}^1$-invariance imposed). We as well construct its non-$\mathbb{A}^1$-invariant analogue: a functor from the dual category of Annala-Iwasa-Hoyois's non-$\mathbb{A}^1$-invariant motivic homotopy category $\mathcal{MS}$ to $\operatorname{Mot}_{\operatorname{loc}}$. After the Barr-Beck argument, these functors factor through categories of modules over a dual version of ($\mathbb{A}^1$-invariant) K-theory spectrum $\operatorname{KGL}^{(\mathbb{A}^1)}$. Over a field that admits resolution of singularities, we show that the $\mathbb{A}^1$-invariant factored functor is fully-faithful, while the non-$\mathbb{A}^1$-invariant one is not in general.