Kazhdan seminar, Alexander Efimov

"Dualizable categories and localizing invariants"

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.