  It is known that the asymptotic density of states of a 2d CFT in an
irreducible representation $\rho$ of a finite symmetry group $G$ is
proportional to $(\dim\rho)^2$. We show how this statement can be generalized
when the symmetry can be non-invertible and is described by a fusion category
$\mathcal{C}$. Along the way, we explain what plays the role of a
representation of a group in the case of a fusion category symmetry; the answer
to this question is already available in the broader mathematical physics
literature but not yet widely known in hep-th. This understanding immediately
implies a selection rule on the correlation functions, and also allows us to
derive the asymptotic density.
