The Fusion Categorical Diagonal

We define a Frobenius algebra over fusion categories of the form Rep(G)Rep$(G)$ which generalizes the diagonal subgroup of G× G. This allows us to extend field theoretical constructions which depend on the existence of a diagonal subgroup to non-invertible symmetries. We give explicit calculations for theories with Rep(S₃)Rep(S₃) symmetry, applying the results to gauging topological quantum field theories which carry this non-invertible symmetry. Along the way, we also discuss how Morita equivalence is implemented for algebras in symmetry categories.