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We characterize discrete (anti-)unitary symmetries and their non-invertiblegeneralizations in $2+1$d topological quantum field theories (TQFTs) throughtheir actions on line operators and fusion spaces. We explain all possiblesources of non-invertibility that can arise in this context. Our approach givesa simple $2+1$d proof that non-invertible generalizations of unitary symmetriesexist if and only if a bosonic TQFT contains condensable bosonic line operators(i.e., these non-invertible symmetries are necessarily "non-intrinsic"). Movingbeyond unitary symmetries and their non-invertible cousins, we define anon-invertible generalization of time-reversal symmetries and derive variousproperties of TQFTs with such symmetries. Finally, using recent results on2-categories, we extend our results to corresponding statements in $2+1$dquantum field theories that are not necessarily topological.

Invertibility of Condensation Defects and Symmetries of 2 + 1d QFTs