Unitary Categorical Symmetries

Global invertible symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to non-invertible symmetries by considering unitary actions of higher fusion category symmetries $\mathcal{C}$ on twisted sector local operators. We propose that the latter transform in $\ast$-representations of the tube algebra associated to $\mathcal{C}$, which we introduce and classify using the notion of higher $S$-matrices of higher braided fusion categories.