  We explore the topological defects of the critical three-state Potts spin
system on the torus, Klein bottle and cylinder. A complete characterization is
obtained by breaking down the Fuchs-Runkel-Schweigert construction of 2d
rational CFT to the lattice setting. This is done by applying the strange
correlator prescription to the recently obtained tensor network descriptions of
string-net ground states in terms of bimodule categories [Lootens, Fuchs,
Haegeman, Schweigert, Verstraete, SciPost Phys. 10, 053 (2021)]. The symmetries
are represented by matrix product operators (MPO), as well as intertwiners
between the diagonal tetracritical Ising model and the non-diagonal three-state
Potts model. Our categorical construction lifts the global transfer matrix
symmetries and intertwiners, previously obtained by solving Yang-Baxter
equations, to MPO symmetries and intertwiners that can be locally deformed,
fused and split. This enables the extraction of conformal characters from
partition functions and yields a comprehensive picture of all boundary
conditions.
