Diagramatics for cyclic pointed fusion categories
Agustina Czenky
TL;DR
The paper tackles the classification of cyclic pointed fusion categories by parametrizing $\mathrm{Vec}_{\mathbb{Z}_n}^{\zeta}$ with an $n$-th root of unity $\zeta$, and provides a diagrammatic presentation via generators and relations. It proves a universal property that describes tensor functors out of $\mathrm{Vec}_{\mathbb{Z}_n}^{\zeta}$ in terms of objects with an $n$-fold tensor to the unit and a compatibility condition tied to $\zeta$. A diagrammatic category $\mathcal{D}_{\zeta,n}$ is constructed and shown to be monoidally equivalent to $\langle \delta_1 \rangle$, yielding a concrete presentation of $\mathrm{Vec}_{\mathbb{Z}_n}^{\zeta}$ by generators and relations. The work also computes the equivalence classes of cyclic pointed fusion categories (via a counting formula $c(n)$) and characterizes the automorphism 2-group $\mathrm{Aut}_{\otimes}(\mathrm{Vec}_{\mathbb{Z}_n}^{\zeta})$, including its $\,\pi_0$ and $\pi_1$ invariants and morphism spaces. Overall, the results provide a practical framework for working with cyclic pointed fusion categories and understanding their symmetries through a diagrammatic, universal-property approach.
Abstract
We give a parametrization of cyclic pointed categories associated to the cyclic group of order $n$ in terms of $n$-th roots of unity. We also provide a diagramatic description of these categories by generators and relations, and use it to characterize their $2$-group of automorphisms.