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…