Higher structure of non-invertible symmetries from Lagrangian descriptions

The symmetry structure of a quantum field theory is determined not only by the topological defects that implement the symmetry and their fusion rules, but also by the topological networks they can form, which is referred to as the higher structure of the symmetry. In this paper, we consider theories with non-invertible symmetries that have an explicit Lagrangian description, and use it to study their higher structure. Starting with the 2d free compact boson theory and its non-invertible duality defects, we will find Lagrangian descriptions of networks of defects and use them to recover all the $F$-symbols of the familiar Tambara-Yamagami fusion category $\operatorname{TY}(\mathbb{Z}_N,+1)$. We will then use the same approach in 4d Maxwell theory to compute $F$-symbols associated with its non-invertible duality and triality defects, which are 2d topological field theories. In addition, we will also compute some of the $F$-symbols using a different (group theoretical) approach that is not based on the Lagrangian description, and find that they take the expected form.