Comparison of Motivic Homotopy Theories

1. Problem Statement

The paper addresses the relationship between two major frameworks in modern algebraic geometry and noncommutative geometry:

1. Motivic Homotopy Theories: Specifically, the $\mathbb{A}^1$-invariant stable motivic homotopy category ($SH$) of Morel and Voevodsky, and the non-$\mathbb{A}^1$-invariant variant ($MS$) constructed by Annala, Iwasa, and Hoyois.
2. Noncommutative Motives: The category of localizing motives ($Mot_{loc}$) introduced by Blumberg, Gepner, and Tabuada (BGT), which serves as a universal recipient for localizing invariants (like K-theory).

The central problem is to construct and analyze comparison functors from the dual categories of these motivic homotopy theories ($SH^\vee$ and $MS^\vee$) to the categories of localizing motives ($Mot_{loc}$ and its $\mathbb{A}^1$-invariant version $Mot_{loc}^{\mathbb{A}^1}$). The author seeks to determine under what conditions these functors are fully faithful, thereby establishing a precise equivalence between the duals of motivic spaces and noncommutative motives.

2. Methodology

The paper employs advanced techniques from higher category theory, specifically within the framework of $\infty$-categories and stable presentable categories ($Pr^L_{st}$).

• Dualization in $Pr^L_{st}$: The author utilizes the notion of dualizable objects in the symmetric monoidal category of stable presentable categories. For a category $\mathcal{C}$, its dual $\mathcal{C}^\vee$ is defined as $Fun^L(\mathcal{C}, Sp)$.
• Formal Inversion and Telescopes: The construction relies on formally inverting objects (specifically the projective line $\mathbb{P}^1$) within these categories. The author uses "telescope models" and "c-spectra" (from Annala-Iwasa) to describe these inversions explicitly.
• Compatibility of Duality and Inversion: A crucial technical step is proving that the operation of taking the dual commutes with formal inversion. Specifically, for a compact object $X$, $(\mathcal{C}[X^{-1}])^\vee \simeq \mathcal{C}^\vee[(X^{dual})^{-1}]$.
• Universal Properties: The author characterizes the dual categories $SH^\vee$ and $MS^\vee$ not as abstract duals, but as categories of cosheaves satisfying specific descent (codescent) conditions (Nisnevich, $\mathbb{A}^1$, or elementary blowup excision) and formal inversion of $\mathbb{P}^1$.
• Barr-Beck Theorem: The comparison functors are shown to factor through categories of modules over a specific algebra (a dual version of the K-theory spectrum $KGL$). The fully-faithfulness of the resulting functors is analyzed using the Barr-Beck theorem and conditions on "rigid generation."

3. Key Contributions

A. Construction of Comparison Functors

The paper constructs two primary comparison functors:

1. $\Phi: SH^\vee \to Mot_{loc}^{\mathbb{A}^1}$: A functor from the dual of the $\mathbb{A}^1$-invariant motivic homotopy category to the category of $\mathbb{A}^1$-invariant localizing motives.
2. $\Psi: MS^\vee \to Mot_{loc}$: A functor from the dual of the non-$\mathbb{A}^1$-invariant motivic homotopy category to the category of general localizing motives.

These functors are defined by extending the natural map $X \mapsto U(\text{perf}_X)$ (where $U$ is the universal localizing invariant) from smooth schemes to the dual motivic categories.

B. Characterization of Dual Categories

The author provides a concrete description of the dual categories:

• $SH^\vee$ is equivalent to the category of cosheaves on $(Sm_{fp})^{op}$ satisfying Nisnevich codescent, $\mathbb{A}^1$-codescent, and formally inverted $(\mathbb{P}^1)^{dual}$.
• $MS^\vee$ is equivalent to the category of cosheaves satisfying Nisnevich codescent, elementary blowup codescent, and formally inverted $(\mathbb{P}^1)^{dual}$.

This characterization yields universal properties that allow the extension of functors from smooth schemes to the entire dual categories.

C. Factorization via Module Categories

Using the Barr-Beck argument, the author shows that the comparison functors factor through the category of modules over a specific algebra object:

• $\Phi$ factors through $Mod_{\Phi^*(1)}(SH^\vee)$.
• $\Psi$ factors through $Mod_{\Psi^*(1)}(MS^\vee)$.
  Here, $\Phi^*(1)$ and $\Psi^*(1)$ act as dual versions of the algebraic K-theory spectrum ($KGL$).

4. Main Results

Result 1: Fully-Faithfulness in the $\mathbb{A}^1$-Invariant Case

Theorem: Over a field $k$ that admits resolution of singularities, and after inverting the exponential characteristic $e$ of $k$, the factored functor:
$$e\Phi: Mod_{\Phi^*(1)}(SH(k)[1/e]^\vee) \to Mot(k)_{loc}^{\mathbb{A}^1}[1/e]$$
is fully faithful on the level of mapping spectra.

• Reasoning: The category $SH(k)[1/e]^\vee$ is rigidly generated (generated by compact, dualizable objects). This property, combined with the fact that the unit of $Mot_{loc}^{\mathbb{A}^1}$ is compact, ensures that the projection formula holds and the functor preserves the necessary structure to be fully faithful.

Result 2: Failure of Fully-Faithfulness in the Non-$\mathbb{A}^1$-Invariant Case

Theorem: In the non-$\mathbb{A}^1$-invariant setting, the functor:
$$e\Psi: Mod_{\Psi^*(1)}(MS(k)[1/e]^\vee) \to Mot(k)_{loc}[1/e]$$
is not fully faithful in general.

• Reasoning:
  1. Lack of Rigid Generation: $MS(k)^\vee$ is not rigidly generated.
  2. Cardinality Mismatch: The author constructs a counterexample based on the cardinality of mapping spectra.
     • In the domain (derived from $MS$), the mapping spectra between compact objects are countable (assuming $k$ is a countable field).
     • In the codomain ($Mot_{loc}$), the mapping spectrum between the unit and certain objects relates to the ring of big Witt vectors $W(R)$. For a non-zero ring $R$, $W(R)$ is uncountable.
  • This discrepancy proves that the functor cannot be fully faithful, as it cannot capture the uncountable structure present in the target category.

5. Significance

• Bridging Motives and Noncommutative Geometry: The paper rigorously establishes a bridge between classical motivic homotopy theory and the theory of noncommutative motives. It clarifies that the "dual" of the motivic world is the natural home for noncommutative motives.
• Role of $\mathbb{A}^1$-Invariance: The work highlights a fundamental structural difference between $\mathbb{A}^1$-invariant and non-invariant theories. While $\mathbb{A}^1$-invariance ensures the necessary "rigid generation" for a perfect correspondence with localizing motives, dropping this condition (as in $MS$) introduces complexities (like blow-up excision) that break the full equivalence.
• New Tools for Duality: The paper contributes to the growing literature on duality in presentable $\infty$-categories (building on work by Clausen-Scholze and Efimov), providing explicit models for duals of motivic categories and demonstrating how formal inversion interacts with duality.
• Counterexamples in Motivic Theory: The demonstration of the failure of fully-faithfulness in the non-$\mathbb{A}^1$ case provides a concrete limitation on how far noncommutative motives can model general motivic homotopy theories without additional constraints.

In summary, Tan's thesis successfully constructs a comparison framework showing that while the dual of the $\mathbb{A}^1$-invariant motivic category is fully captured by localizing motives (under mild field assumptions), the non-invariant case exhibits a structural divergence due to the uncountable nature of certain invariants in the noncommutative world.