Unitary categorical symmetries

Abstract 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 C on twisted sector local operators. We propose that the latter transform in * -representations of the tube algebra associated to C, which we introduce and classify using the notion of higher S-matrices of higher braided fusion categories.