An increasing amount of evidence indicates that symmetries of quantum fields are in correspondence with topological membranes of various dimensionality, encoding the conserved quantum numbers of local and extended operators. One feature of topological membranes is that they can form intricated networks of topological junctions, which give generalized symmetries a higher structure. The higher structure has physical implications, for example in the form of selection rules, anomalies, and generalized Ward identities. In this talk, after introducing these ideas, I will discuss some recent progress in characterizing the higher structure of the symmetries of axion-Maxwell theory as a concrete example. This seminar is based on an upcoming work with Matteo Dell’Acqua, Shani Nadir Meynet and Elias Riedel-Gårding.