  Motivated by the fundamental role that bosonic and fermionic symmetries play
in physics, we study (non-invertible) one-form symmetries in $2 + 1$d
consisting of topological lines with bosonic and fermionic self-statistics. We
refer to these lines as Bose-Fermi-Braided (BFB) symmetries and argue that they
can be classified. Unlike the case of generic anyonic lines, BFB symmetries are
closely related to groups. In particular, when BFB lines are non-invertible,
they are non-intrinsically non-invertible. Moreover, BFB symmetries are, in a
categorical sense, weakly group theoretical. Using this understanding, we study
invariants of renormalization group flows involving non-topological QFTs with
BFB symmetry.
