  We provide a generalization of the Symmetry Topological Field Theory (SymTFT)
framework to characterize phase transitions and gapless phases with categorical
symmetries. The central tool is the club sandwich, which extends the SymTFT
setup to include an interface between two topological orders: there is a
symmetry boundary, which is gapped, and a physical boundary that may be
gapless, but in addition, there is also a gapped interface in the middle. The
club sandwich generalizes so-called Kennedy-Tasaki (KT) transformations.
Building on the results in [1, 2] on gapped phases with categorical symmetries,
we construct gapless theories describing phase transitions with non-invertible
symmetries by applying suitable KT transformations on known phase transitions
provided by the critical Ising model and the 3-state Potts model. We also
describe in detail the order parameters in these gapless theories
characterizing the phase transitions, which are generally mixtures of
conventional and string-type order parameters mixed together by the action of
categorical symmetries. Additionally, removing the physical boundary from the
club sandwiches results in club quiches, which characterize all possible gapped
boundary phases with (possibly non-invertible) symmetries that can arise on the
boundary of a bulk gapped phase. We also provide a mathematical
characterization of gapped boundary phases with symmetries as pivotal tensor
functors whose targets are pivotal multi-fusion categories.
