Categorifying rationalization

TL;DR

This paper constructs stable $mbda$-categories that categorify the process of inverting primes in algebraic K-theory, solving a problem posed by Khovanov and extending the theory of rationalization.

Contribution

It introduces a method to categorify the localization of algebraic K-theory at a set of primes using stable $mbda$-categories and equivariant sheaves on the Cantor space.

Findings

Constructed triangulated categories with Grothendieck groups equal to localized integers.

Established that K-theory of the constructed categories matches the localization of original K-theory.

Provided a categorification of division by primes in algebraic K-theory.

Abstract

We solve a problem proposed by Khovanov by constructing, for any set of primes $S$, a triangulated category (in fact a stable $\infty$-category) whose Grothendieck group is $S^{-1}\mathbf{Z}$. More generally, for any exact $\infty$-category $E$, we construct an exact $\infty$-category $S^{-1}E$ of equivariant sheaves on the Cantor space with respect to an action of a dense subgroup of the circle. We show that this $\infty$-category is precisely the result of categorifying division by the primes in $S$. In particular, $K_n(S^{-1}E)\cong S^{-1}K_n(E)$.