Theorem of the heart for Weibel's homotopy $K$-theory

This paper establishes the theorem of the heart for Weibel's homotopy $K$-theory ($KH$), proving that the realization functor induces an equivalence $KH(\mathcal{C}^{\heartsuit}) \simeq KH(\mathcal{C})$ for small stable $\infty$-categories with bounded $t$-structures, a result derived from a strengthened version of Barwick's theorem that provides precise isomorphism ranges for classical $K$-theory and demonstrates the sharpness of these bounds.

The Gist

Imagine you are trying to understand a massive, complex city (let's call it The Category). This city has skyscrapers, underground tunnels, parks, and chaotic traffic. It's too big to see all at once, and the rules of how things connect are incredibly complicated.

Mathematicians often try to understand these big cities by looking at their "Heart." The Heart is a small, simple neighborhood right in the center where the basic rules are easy to understand. It's like looking at a blueprint of the city's core district.

For a long time, mathematicians had a big question: If we understand the Heart perfectly, do we automatically understand the whole city?

In a specific branch of math called K-Theory (which is like a way of counting and classifying the "shapes" and "holes" in these mathematical cities), there was a famous rule called the "Theorem of the Heart." It said, "Yes, if the city is built nicely, knowing the Heart tells you everything."

However, this rule had a catch. It worked great for "nice" cities, but it seemed to break down for more complex, modern types of cities (called stable $\infty$-categories). When mathematicians tried to apply the rule to these complex cities, they found that the "counting" (K-theory) didn't match up perfectly. There were weird errors in the negative numbers (like $K_{-3}$) that the old rule couldn't explain.

The New Discovery: "Homotopy K-Theory"

Enter Alexander Efimov, the author of this paper. He introduces a new, more flexible way of counting called Homotopy K-Theory (or KH).

Think of standard K-Theory as a rigid ruler. It measures things strictly. If you bend the ruler, the measurement breaks.
Homotopy K-Theory is like a rubber band. It's flexible. It can stretch and bend to fit the shape of the complex city without breaking.

Efimov's main breakthrough is proving that for this flexible rubber band (KH), the old rule actually works again!

The Big Result: If you have a complex mathematical city with a "Heart" (a simple core), and you use the flexible rubber band (KH) to measure it, the measurement of the Heart is exactly the same as the measurement of the whole city.

The "Dual" Analogy: The Mirror World

The paper is fascinating because it feels like looking in a mirror.

• The Old Rule (Dundas-Goodwillie-McCarthy): This rule worked for "Connective" rings (like building a tower from the ground up). It said: "If the bottom layers of your tower are the same, the top layers are the same."
• Efimov's New Rule: This works for "Co-connective" structures (like digging a hole from the top down). It says: "If the top layers of your hole are the same, the bottom layers are the same."

It's a perfect duality. Just as a mirror flips left and right, this math flips "up" and "down." Efimov shows that the logic that works for building towers also works for digging holes, provided you use the right tool (KH).

The "Sharpness" Warning: Where the Rule Breaks

Efimov doesn't just say "it works"; he also draws a very precise line in the sand. He proves that the rule works perfectly up to a certain point, but if you go one step further, it breaks.

Imagine you are climbing a ladder.

• The Rule: "If you know the bottom 3 rungs, you know the whole ladder."
• The Reality: Efimov proves that for the 4th rung down, the rule fails.

He constructs specific, tricky examples (using shapes like a "cuspidal cubic curve," which is like a sharp, pointed loop) to show that if you try to use the rule for $K_{-3}$ (the 3rd rung down), you get the wrong answer. This is a huge deal because it tells mathematicians exactly where the limits of their tools are.

Why Does This Matter?

You might ask, "Who cares about counting shapes in abstract cities?"

1. It Unifies Math: It connects two different worlds of mathematics (algebraic geometry and topology) by showing they follow the same "mirror" logic.
2. It Fixes Broken Tools: It tells us exactly how to fix our counting tools so they work for the most complex, modern mathematical structures we are currently discovering.
3. It Solves Old Mysteries: It explains why certain calculations were failing for decades and provides the correct formula to fix them.

Summary in a Nutshell

• The Problem: A famous math rule about "Hearts" of complex structures stopped working for certain types of complex structures.
• The Solution: The author found a more flexible way of measuring (Homotopy K-Theory) where the rule works again.
• The Twist: The rule works perfectly, but only up to a specific limit. He proved exactly where it stops working, which is a rare and valuable insight in mathematics.
• The Analogy: It's like realizing that while a rigid ruler breaks when measuring a curved snake, a rubber band fits perfectly. But even the rubber band has a limit to how much it can stretch before it snaps.

This paper is a masterclass in finding the right tool for the job and understanding exactly how far that tool can be trusted.

Technical Summary

Here is a detailed technical summary of Alexander I. Efimov's paper, "Theorem of the Heart for Weibel's Homotopy K-Theory."

1. Problem Statement and Context

The paper addresses a fundamental question in algebraic K-theory: under what conditions does the K-theory of the "heart" of a $t$-structure (an abelian category) recover the K-theory of the entire stable $\infty$-category?

• The Classical Context: For a small stable $\infty$-category $\mathcal{C}$ with a bounded $t$-structure, the heart $\mathcal{C}^\heartsuit$ is an abelian category. There is a natural "realization functor" $D^b(\mathcal{C}^\heartsuit) \to \mathcal{C}$.
• The Failure of Classical K-Theory: For standard non-connective K-theory ($K$), the map $K(\mathcal{C}^\heartsuit) \to K(\mathcal{C})$ is generally not an equivalence, even if the $t$-structure is bounded. This is related to the failure of the "Theorem of the Heart" for $K$-theory in non-noetherian settings.
• The Goal: The author investigates whether Weibel's Homotopy K-theory ($KH$), which is designed to be invariant under polynomial extensions (A$^1$-invariance), satisfies a "Theorem of the Heart." Specifically, does the realization functor induce an equivalence $KH(\mathcal{C}^\heartsuit) \simeq KH(\mathcal{C})$?
• Scope: The paper extends these questions beyond small categories to dualizable categories (presentable stable $\infty$-categories that are dualizable in the sense of Gaitsgory-Rozenblyum) and coherently assembled abelian categories (a generalization of locally coherent categories).

2. Methodology and Framework

The author develops a sophisticated framework combining higher category theory, $t$-structures, and localizing invariants.

A. Coherently Assembled Categories

The paper introduces and studies coherently assembled exact/abelian categories. A Grothendieck abelian category $\mathcal{A}$ is coherently assembled if the colimit functor $\text{colim}: \text{Ind}(\mathcal{A}) \to \mathcal{A}$ has an exact left adjoint $\hat{Y}: \mathcal{A} \to \text{Ind}(\mathcal{A})$.

• This condition is equivalent to satisfying Grothendieck's axiom (AB6).
• It allows the definition of continuous K-theory ($K_{\text{cont}}$) and continuous Homotopy K-theory ($KH_{\text{cont}}$) for large categories via their presentable stable envelopes.

B. Compactly Assembled $t$-Structures

The author defines compactly assembled $t$-structures on dualizable categories. A $t$-structure $(\mathcal{C}_{\ge 0}, \mathcal{C}_{\le 0})$ is compactly assembled if the functor $\hat{Y}$ is $t$-exact.

• This ensures the heart $\mathcal{C}^\heartsuit$ is coherently assembled.
• The paper focuses on continuously bounded $t$-structures, where the category is generated by the heart as a localizing subcategory.

C. Refined Theorem of the Heart (Barwick's Extension)

A central technical tool is a strengthened version of Barwick's Theorem of the Heart.

• Barwick's Result: If $\mathcal{C}$ is a small stable category with a bounded $t$-structure, $K(\mathcal{C}^\heartsuit) \to K(\mathcal{C})$ is an equivalence if the $t$-structure is "nice" (e.g., noetherian).
• Efimov's Refinement: The author proves that if the realization functor induces isomorphisms on $\text{Ext}^i$ for $i \le n$, then the map on K-theory spectra is an isomorphism in degrees $j \ge -n-1$ and a monomorphism for $j = -n-2$.
• Sharpness: The paper proves these bounds are sharp, showing that the naive theorem fails for $K_{-3}$ even for dg-categories over a field.

D. Deformed Tensor Algebras and Inductive Limits

To handle the general case, the author uses a sequence of categories constructed via deformed tensor algebras.

• Given a functor $F: \mathcal{C} \to \mathcal{D}$, one constructs a sequence of categories $C_0 \to C_1 \to \dots$ where the monad on $C_n$ is a deformed tensor algebra.
• This sequence converges to $\mathcal{D}$. The author analyzes the coconnectivity of the fibers of the K-theory maps along this sequence, reducing the problem to the "refined heart theorem" for finite steps.

E. Duality with Weight Structures

The paper draws a striking analogy (duality) between the author's results and the Dundas-Goodwillie-McCarthy (DGM) theorem.

• DGM: Relates $K(R)$ and $TC(R)$ for connective $E_1$-rings, showing equivalence modulo nilpotent ideals.
• Analogy: The $t$-structure is dual to a weight structure. The author's result for $KH$ is the "dual" of the DGM theorem, where connectivity estimates on homotopy groups of rings correspond to coconnectivity estimates on K-groups of categories.

3. Key Contributions and Results

Main Theorem (Theorem of the Heart for $KH$)

Theorem 0.1 / 5.1: Let $\mathcal{C}$ be a small stable $\infty$-category (or a dualizable category) with a bounded (compactly assembled continuously bounded) $t$-structure. Then the realization functor induces an equivalence of spectra:
$$KH(\mathcal{C}^\heartsuit) \xrightarrow{\sim} KH(\mathcal{C})$$
This holds even if the heart is non-noetherian and non-artinian.

Devissage Theorem for $KH$

Theorem 0.2 / 6.1: If $F: \mathcal{B} \to \mathcal{A}$ is an exact fully faithful functor between small abelian categories such that $F(\mathcal{B})$ generates $\mathcal{A}$ via extensions, then:
$$KH(\mathcal{B}) \xrightarrow{\sim} KH(\mathcal{A})$$
This generalizes Quillen's Devissage theorem to the homotopy K-theory setting for large categories.

Sharpness of Estimates

The paper proves that the bounds derived in the refined heart theorem are sharp:

• There exist examples (using cuspidal cubic curves and effaceable functors) where the map $K_{-2}(\mathcal{B}) \to K_{-2}(\mathcal{A})$ is not surjective.
• The naive K-theoretic theorem of the heart fails for $K_{-3}$.
• This provides a precise counterexample to the expectation that $K$-theory behaves well for all negative degrees without $KH$.

Continuous K-Theory of Large Categories

The author establishes that for coherently assembled abelian categories, the continuous K-theory $K_{\text{cont}}$ behaves well.

• Counterexample to Schlichting's Vanishing: The paper shows that for the category of sheaves of vector spaces on the real line (a coherently assembled abelian category), $K_{\text{cont}}^{-1} \cong \mathbb{Z}$. This demonstrates that Schlichting's vanishing theorem for $K_{-1}$ of small abelian categories does not generalize to coherently assembled categories.

Examples and Applications

• Sheaves on Locally Compact Spaces: The paper constructs compactly assembled $t$-structures on categories of sheaves $Shv(X; \mathcal{C})$.
• Nuclear Modules: It analyzes the category of nuclear solid modules over $\mathbb{Z}_p$ (Clausen-Scholze), showing it admits a compactly assembled continuously bounded $t$-structure.
• Chase Criterion: A generalization of Chase's criterion for the coherence of rings is proven in the context of dualizable Ab-modules.

4. Significance

1. Resolution of the Heart Conjecture for $KH$: The paper definitively proves that Weibel's $KH$ satisfies the Theorem of the Heart in the broadest possible setting (dualizable categories), resolving a major open problem. This confirms that $KH$ is the correct invariant for studying abelian categories via their derived categories, even in non-noetherian contexts.
2. New Framework for Large Categories: By introducing "coherently assembled" categories and "compactly assembled $t$-structures," the paper provides a robust language to discuss K-theory for large, presentable categories, bridging the gap between small algebraic geometry and modern higher category theory.
3. Sharpness and Counterexamples: The construction of explicit counterexamples for $K_{-3}$ and the non-vanishing of $K_{-1}$ in the continuous setting clarifies the precise limitations of K-theory versus Homotopy K-theory.
4. Duality Insight: The conceptual link drawn between the Theorem of the Heart for $KH$ and the DGM theorem (via weight structures) offers a new perspective on the relationship between K-theory, Topological Cyclic Homology ($TC$), and $t$-structures.
5. Foundational for Future Work: The results on nuclear modules and sheaves on locally compact spaces lay the groundwork for applying K-theoretic methods to analytic geometry and condensed mathematics.

In summary, Efimov's paper is a landmark contribution that establishes the robustness of Homotopy K-theory in the context of $t$-structures, provides sharp bounds for classical K-theory, and develops a powerful machinery for handling large, non-noetherian categories in algebraic K-theory.