We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group $G(d,1,n)$. The construction of the category follows the decomposition of the Fourier matrix as a Kronecker tensor product of exterior powers of the character table $S$ of the cyclic group of order $d$. The representation of the quantum universal enveloping algebra of the general linear Lie algebra $\mathfrak{glm}$, with quantum parameter an even root of unity of order $2d$, provides a categorical interpretation of the matrix $\bigwedge^m S$. We also prove some positivity conjectures of Cuntz at the decategorified level.