CATEGORIFYING RATIONALIZATION Article Swipe

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Mathematics Equivariant map Derived category Grothendieck group Pure mathematics Cantor set Rationalization (economics) Space (punctuation) Combinatorics Discrete mathematics Linguistics Epistemology Functor Philosophy Abelian group

Clark Barwick , Saul Glasman , Marc Hoyois , Denis Nardin , Jay Shah ·

YOU? · · 2019 · Open Access · · DOI: https://doi.org/10.1017/fms.2019.26 · OA: W3037910407

We construct, 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, $K_{n}(S^{-1}E)\cong S^{-1}K_{n}(E)$ .