On the $(\text{Fib} \boxtimes \text{Fib}) \rtimes S_2$ fusion category

There might exist non-rational Virasoro CFTs in two dimensions with a $(\text{Fib} \boxtimes \text{Fib}) \rtimes S_2$ categorical symmetry. We calculate the necessary ingredients for a modular conformal bootstrap analysis of these theories. After reviewing the basics of fusion categories, we present the irreducible representations, the lasso maps that intertwine between different Hilbert spaces, and finally the 22-by-22 modular S matrix. We highlight the peculiarities introduced by the non-invertible nature of the symmetry. This paper is written in a pedagogical manner and can therefore serve as an accessible entry point into the literature.