These notes record some elementary observations about continuous K-theory (also called Efimov K-theory), largely following [sheaves-on-manifolds] and [bonn-k] .

K-theory as a universal homology theory

For an ordinary finite set $S$ we measure its size by its cardinality $|S|$. The familiar identity

\[ |S \cup T| = |S| + |T| - |S \cap T| \]

hides a non-trivial fact: once we fix $|\{*\}| = 1$, the operation $|-|$ is uniquely determined. Every finite set is built from a point by finitely many disjoint unions and quotientings along subsets, and the identity above simply records that $|-|$ is compatible with these assembly operations.

In homotopy theory we routinely derive such excision properties. The derived analogue of the category of sets $\mathsf{Set}$ is the category of anima $\mathsf{An}$, and finite anima play the role of derived finite sets. To make the analogy precise we introduce the notion of a homology theory: a functor

\[ H\colon \mathsf{An}^{\mathrm{fin}} \to \mathcal{C} \]

sending homotopy pushouts of finite anima to homotopy pushouts in $\mathcal{C}$,

\[ H(A \cup_C B) \simeq H(A) \cup_{H(C)} H(B). \]

Every finite anima is built from the point by finitely many pushouts, so $H$ is determined by its value $H(*) \in \mathcal{C}$ — exactly as cardinality is determined by $|\{*\}| = 1$.

One way to summarise this: a homology theory is a form of counting that respects excision. The specific shape of excision depends on what we are counting and how.

From this perspective, K-theory quantifies algebraic and geometric objects, but its counting values are no longer integers — they are anima. Anima carry far richer algebraic structure than numbers, and that extra structure is what lets one propagate computations from known cases to new ones.

Algebraic K-theory

This principle extends to stable categories. Every idempotent-complete small stable category $\mathcal{C}$ has an algebraic K-theory spectrum attached to it. Writing $\mathsf{Cat}^{\mathrm{perf}}$ for the category of such small stable categories, K-theory is a functor

\[ K\colon \mathsf{Cat}^{\mathrm{perf}} \to \mathsf{Sp}. \]

For a ring $R$ we set $K(R) \coloneqq K(\mathsf{Mod}^{\mathrm{perf}}(R))$, the K-theory of the category of perfect complexes over $R$.

In practice, however, we frequently care more about dualizable stable categories. Every dualizable stable category is presentable — that is, big. The Ind-completion functor

\[ \mathsf{Ind}\colon \mathsf{Cat}^{\mathrm{perf}} \hookrightarrow \mathsf{Cat}^{\mathrm{dual}} \]

sends a small $\mathcal{C}$ to the category $\mathsf{Ind}(\mathcal{C})$ obtained by freely adjoining filtered colimits (equivalently, arbitrary direct sums). It is fully faithful. But trying to count a big category produces a familiar obstruction:

Example 1 (Eilenberg swindle).
Let $\mathcal{C}$ be a small category with finite colimits. If there is a finite-colimit-preserving endofunctor $F\colon \mathcal{C} \to \mathcal{C}$ with $F \sqcup \mathrm{id}_{\mathcal{C}} \simeq F$, then $K(\mathcal{C}) \simeq 0$.

The reason is transparent. The map $f$ induced by $F$ on K-theory satisfies $f + \mathrm{id} = f$, hence $\mathrm{id} = 0$. For a presentable stable $\mathcal{C}$ all small colimits exist, so one can take $F = \bigoplus_{\mathbb{N}} \mathrm{id}_{\mathcal{C}}$; this $F$ satisfies $F \sqcup \mathrm{id} \simeq F$, and the swindle says $K(\mathcal{C}) = 0$.

Continuous K-theory

And yet in practice we constantly meet categories that are big but carry a topological continuity — sheaves of spectra on a locally compact Hausdorff space, nuclear modules in the sense of Clausen–Scholze, and so on. These categories sit far outside the reach of classical K-theory.

This is the setting for continuous (Efimov) K-theory. The central idea is to extend K-theory from small categories to dualizable stable categories; in the stable setting these are exactly the compactly assembled categories. A compactly assembled category need not have many compact objects, but it always has enough compact morphisms: every object is an $\mathbb{N}$-colimit along compact morphisms — in other words, every object is compactly exhaustible. Examples abound in geometry (sheaves on a locally compact Hausdorff space), and this class is the natural stage on which algebra meets topology.

What does continuous K-theory count? Classical K-theory counts compact objects of a category (for a ring: finitely generated projective modules). The Eilenberg swindle forbids that direct approach for big categories. The trick is to count not compact objects themselves but the residue they leave behind inside a Verdier cofibre sequence

\[ \mathcal{C} \xrightarrow{\;\hat{Y}\;} \mathrm{Ind}(\mathcal{C}^{\omega_1}) \longrightarrow \mathrm{Ind}\!\left(\mathrm{Calk}^{\mathrm{cont}}(\mathcal{C})\right), \]

where $\hat{Y}$ is a fully faithful embedding and the continuous Calkin category $\mathrm{Calk}^{\mathrm{cont}}(\mathcal{C})$ is the category of compact objects of the Verdier quotient $\mathrm{Ind}(\mathcal{C}^{\omega_1}) / \mathcal{C}$ — the world of $\omega_1$-compact objects with the original category quotiented out. At the level of morphisms this quotient divides by compactly assembled maps,

\[ \mathrm{hom}_{\mathrm{Calk}^{\mathrm{cont}}(\mathcal{C})}(pX, pY) \simeq \mathrm{hom}_{\mathcal{C}}(X, Y) \big/ \mathrm{hom}_{\mathcal{C}}^{\mathrm{ca}}(X, Y), \]

in strict analogy with the Calkin algebra $\mathcal{B}(H) / \mathcal{K}(H)$ in functional analysis: bounded operators modulo compact operators. Since $\mathcal{C}^{\omega_1}$ has countable colimits, the Eilenberg swindle forces the middle term’s K-theory to vanish, and continuous K-theory is determined by the quotient:

\[ K^{\mathrm{cont}}(\mathcal{C}) \simeq \Omega K\!\left(\mathrm{Calk}^{\mathrm{cont}}(\mathcal{C})\right). \]

This relative counting neatly evades the swindle and produces a non-trivial invariant.

The excision picture here is more striking than in the classical case: it becomes a genuinely geometric property. For a locally compact Hausdorff space $X$ and a compactly assembled coefficient category $\mathcal{C}$,

\[ K^{\mathrm{cont}}\!\left(\mathsf{Shv}(X;\mathcal{C})\right) \simeq \Gamma_c\!\left(X,\, K^{\mathrm{cont}}(\mathcal{C})\right). \]

The topology of $X$ is fully encoded: however one decomposes $X$ by open covers, the K-theory spectrum glues sheaf-theoretically along that decomposition. Continuous K-theory is, on the nose, a cosheaf on $X$. Specialising to $X = \mathbb{R}^n$,

\[ K^{\mathrm{cont}}\!\left(\mathsf{Shv}(\mathbb{R}^n;\mathcal{C})\right) \simeq \Omega^n K^{\mathrm{cont}}(\mathcal{C}), \]

so the topology of the base is faithfully reflected in the K-theory spectrum.

The two questions we opened with — what are we counting, and how — thus have clean answers in the continuous setting: the residue of a compactly assembled category relative to its compact objects, with the counting rule dictated by the topology of the underlying space.

More deeply, Efimov proved the following universal property: restriction along $\mathsf{Ind}$ is an equivalence

\[ \mathrm{Loc}^{\mathrm{cont}}(D) \xrightarrow{\;\sim\;} \mathrm{Loc}(D), \]

so every classical localizing invariant (K-theory, topological Hochschild homology, etc.) extends uniquely to a continuous localizing invariant on compactly assembled categories. This grants continuous K-theory the status of a universal homology theory, echoing the philosophical discussion at the start of this page.

The plan of these notes is to follow this thread: we begin with classical connective and non-connective algebraic K-theory and build up to Efimov’s central result — the universal property of continuous K-theory and the sheaf formula.

Contents

Pages

  • Algebraic K-theory
    Abstract
    Connective and non-connective algebraic K-theory via cospans and the Waldhausen S-construction; Verdier sequences in stable and non-stable settings; the non-connective extension via iterated Calkin categories.
  • Compactly assembled categories
    Abstract
    Dualizable stable categories from the compactly assembled viewpoint: R-linear categories, dualizability in $\mathsf{Pr}_{\mathrm{st}}^L$, compactly exhaustible objects, the Lurie–Clausen characterisation, and the symmetric monoidal structure of $\mathsf{Pr}^L_{\mathrm{ca}}$.
  • Continuous (Efimov) K-theory
    Abstract
    Extending algebraic K-theory from $\mathsf{Cat}^{\mathrm{rex}}$ to $\mathsf{Pr}^L_{\mathrm{ca}}$: the continuous Calkin category, Verdier cofibre sequences in the large setting, Efimov’s definition of continuous K-theory, its basic properties, and a sketch of the universal property of $K^{\mathrm{cont}}$.

References

  • A. Krause, T. Nikolaus, P. Pützstück. Sheaves on Manifolds. 2024. PDF.
  • A. I. Efimov. K-theory and localizing invariants of large categories. 2025. arXiv:2405.12169.
  • M. Ramzi. Dualizable presentable $\infty$-categories. 2024. arXiv:2410.21537.
  • K. Hilman, J. McCandless. Lecture Notes on Algebraic K-Theory. 2024. Page.
  • J. Lurie. Higher Topos Theory. Ann. Math. Stud. 170, Princeton Univ. Press, 2009. PDF.
  • J. Lurie. Higher Algebra. PDF.
  • J. Lurie. Kerodon. Online, 2018–. kerodon.net.