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K-theory and localizing invariants of large categories

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arxiv 2405.12169 v3 pith:I2ABKE5B submitted 2024-05-20 math.KT math.AGmath.ATmath.CTmath.NT

Alexander I. Efimov This is my paper · ORCID
classification math.KTmath.AGmath.ATmath.CTmath.NT
keywords categoriestheorydualizablestablecontinuousinftylocalizingcategory
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abstract

In this paper we introduce and study the so-called continuous $K$-theory for a certain class of "large" stable $\infty$-categories, more precisely, for dualizable presentable categories. For compactly generated categories, the continuous $K$-theory is simply the usual (non-connective) $K$-theory of the full subcategory of compact objects. More generally, we show that any localizing invariant of small stable $\infty$-categories can be uniquely extended to a localizing invariant of dualizable categories. We compute the continuous $K$-theory for categories of sheaves on locally compact Hausdorff spaces. Using the special case for sheaves on the real line, we give an alternative proof of the theorem of Kasprowski and Winges \cite{KW19} on the commutation of $K$-theory with infinite products for small stable $\infty$-categories. We also study the general theory of dualizable categories. In particular, we give an "explicit" proof of Ramzi's theorem \cite{Ram24a} on the $\omega_1$-presentability of the category of dualizable categories. Among other things, we prove that dualizability is equivalent to "flatness" in the category of presentable stable categories.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dualizable Additive Categories

    math.AT 2026-08 conditional novelty 8.0 of 10

    Dualizable additive categories are characterized intrinsically and via almost modules, and a universal finitary localizing invariant (prestable motives) is constructed whose unit corepresents algebraic K-theory.

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    Spectral sheaves on locally compact Hausdorff spaces are initial among continuous six-functor formalisms, forcing all such formalisms to agree with sheaf (co)homology and to satisfy a universal localizing-invariant formula.

  3. Smashing, Balmer, Zariski spectra: an ideal approach

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    A single frame-theoretic construction, the Zariski frame of radical categorical ideals, unifies Zariski spectra, Balmer spectra, and smashing frames in higher algebra.

  4. Convexity in tensor triangular geometry

    math.CT 2025-06 accept novelty 7.0 of 10

    In locally cohomologically stratified tensor triangular categories with noetherian spectrum, the dualizable localizing ideals are exactly the localizing ideals supported on convex subsets of the Balmer spectrum.

  5. A remark on Continuous K-theory and Fourier-Sato transform

    math.AT 2025-06 conditional novelty 6.0 of 10

    For any conic closed set X in the dual space, the universal localizing invariant of sheaves with microsupport over X equals the compactly supported cohomology of X, and the Tamarkin-category version holds up to suspension.

  6. Wild Betti sheaves

    math.AG 2025-05 conditional novelty 6.0 of 10

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