Kazhdan seminar, Alexander Efimov

Abstract: I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then I will give a formal definition of K-theory and more general localizing invariants of dualizable categories. We will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum. If time permits, I will sketch some applications of this rigidity theorem, such as construction of refined versions of (topological) Hochschild homology.

References:

Part of my results on dualizable categories and their localizing invariants are contained in my preprint
"K-theory and localizing invariants of large categories", arXiv:2405.12169 (to be updated soon).
In particular, it contains the general theory of dualizable categories, the extension of localizing invariants to dualizable categories and the computation of these invariants for categories of sheaves on locally compact spaces.

The more advanced results, including the computation of K-theory for categories of nuclear modules will appear in the forthcoming paper
"Localizing invariants of inverse limits" (in preparation).

The results on the localizing motives, including the rigidity statement and the computations of morphisms will appear in
the paper
"Rigidity of the category of localizing motives" (in preparation).

Originally dualizable categories (and more generally compactly assembled categories) were defined in:
J. Lurie, "Spectral algebraic geometry", available at: https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf
The relevant sections are 21.1.2, D.7.2, D.7.3.

The general theory of accessible and presentable infinity-categories (including the basic results on their limits and colimits), was developed in:
J. Lurie, "Higher topos theory", available at: https://www.math.ias.edu/~lurie/papers/HTT.pdf
The relevant sections are: 5.4, 5.5.

A different proof of presentability of the category of dualizable categories (in the general relative setting), is contained in:
M. Ramzi, "Dualizable presentable $\infty$-categories", arxiv:2410.21537 (preprint).

The study of those triangulated categories which are in fact the homotopy categories of dualizable categories can be found in the earlier papers of Keller and Krause:

B. Keller, "A remark on the generalized smashing conjecture". Manuscripta Math 84, 193-198 (1994).
H. Krause, “Smashing subcategories and the telescope conjecture - an algebraic approach”.
Invent. math. 139, 99-133 (2000).

H. Krause, “Cohomological Quotients and Smashing Localizations”. American Journal of
Mathematics, vol. 127, no. 6, 2005, pp. 1191-1246.

The definition and detailed study of compactly assembled ordinary categories first appeared under the name "continuous categories" in:

P. Johnstone, A. Joyal, “Continuous categories and exponentiable toposes”. Journal of Pure and Applied Algebra, Volume 25, Issue 3, 1982, pp. 255-296.

The notion of a compactly assembled partially ordered set was known earlier under the name "continuous poset" or "continuous lattice" if it has finite joins. They are studied in:

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, “A compendium of continuous lattices”. Springer-Verlag, Berlin, 1980.

B. Banaschewski, R.-E. Hoffmann (eds), “Continuous Lattices”, Lecture Notes in Mathematics 871, pp. 209-248, Springer, 1981.