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 $G\times G$. 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$(S_3)⊠$Rep$(S_3)$ 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.