On the structure of categorical duality operators

We systematically study categorical duality operators on spin (and anyon) chains with respect to an internal fusion category symmetry C. We parameterize duality operators on the quasi-local algebra in terms of data dependent on the associated quantum cellular automata (QCA) on the symmetric subalgebra $B$. In particular, a QCA $\alpha$ on $B$ defines an invertible C-C bimodule category $M_{\alpha}$, and the duality operators extending $\alpha$ form a simplex, with extreme points in bijective correspondence with the simple object of $M_{\alpha}$. Then we consider the structure of external symmetries generated by a family of duality operators, and show that if the UV models are all defined on tensor product Hilbert spaces, these categories necessarily flow to weakly integral fusion categories in the IR.