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Comparison of Motivic Homotopy Theories

Tianjian Tan

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.

Comparison of Motivic Homotopy Theories

Abstract

We construct a comparison functor from the dual category of motivic homotopy category to the category of -invariant localizing motives in the sense of Blumberg, Gepner and Tabuada (with -invariance imposed). We as well construct its non--invariant analogue: a functor from the dual category of Annala-Iwasa-Hoyois's non--invariant motivic homotopy category to . After the Barr-Beck argument, these functors factor through categories of modules over a dual version of (-invariant) K-theory spectrum . Over a field that admits resolution of singularities, we show that the -invariant factored functor is fully-faithful, while the non--invariant one is not in general.
Paper Structure (22 sections, 44 theorems, 127 equations)

This paper contains 22 sections, 44 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Categorical preliminaries
  5. Formal inversion of an object
  6. Dualizable objects
  7. A compatibility result of dual operation and formal inversion
  8. Factorization of adjunction pair through category of modules
  9. Category of motives
  10. $\mathcal{SH}$
  11. $\mathcal{MS}$
  12. $\operatorname{Mot}_{\operatorname{loc}}$
  13. Construction of Comparison Functors of Motivic Homotopy Theories
  14. Comparison between $\mathcal{SH}^{\vee}$ and $\operatorname{Mot}_{\operatorname{loc}}^{\mathbb{A}^1}$
  15. Computation of $\mathcal{SH}(S)^{\vee}$
  16. ...and 7 more sections

Key Result

Proposition 1.1

$\mathcal{SH}^\vee$ (resp. $\mathcal{MS}^\vee$) is equivalent to the category constructed from the spectra-valued presheaf category $\mathcal{P}sh((Sm^{\operatorname{fp}})^{op})$, by first localizing it with respect to certain codescent conditions, and then formally inverting a dual version of $\mat

Theorems & Definitions (80)

  • Proposition 1.1
  • Proposition 1.2: Proposition \ref{['formal.inversion.prop']}
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5: Proposition \ref{['comparison.SH']}
  • Proposition 1.6: Proposition \ref{['countability']}
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 70 more