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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.
Robbins et al. (Mon,) studied this question.