  We propose a general framework to characterize gapped infra-red (IR) phases
of theories with non-invertible (or categorical) symmetries. In this paper we
focus on (1+1)d gapped phases with fusion category symmetries. The approach
that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key
input: associated to a field theory in d spacetime dimensions, the SymTFT lives
in one dimension higher and admits a gapped boundary, which realizes the
categorical symmetries. It also admits a second, physical, boundary, which is
generically not gapped. Upon interval compactification of the SymTFT by
colliding the gapped and physical boundaries, we regain the original theory. In
this paper, we realize gapped symmetric phases by choosing the physical
boundary to be a gapped boundary condition as well. This set-up provides
computational power to determine the number of vacua, the symmetry breaking
pattern, and the action of the symmetry on the vacua. The SymTFT also
manifestly encodes the order parameters for these gapped phases, thus providing
a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find
that for non-invertible symmetries the order parameters involve multiplets
containing both untwisted and twisted sector local operators, and hence can be
interpreted as mixtures of conventional and string order parameters. We also
observe that spontaneous breaking of non-invertible symmetries can lead to
vacua that are physically distinguishable: unlike the standard symmetries
described by groups, non-invertible symmetries can have different actions on
different vacua of an irreducible gapped phase. This leads to the presence of
relative Euler terms between physically distinct vacua. We also provide a
mathematical description of symmetric gapped phases as 2-functors from
delooping of fusion category characterizing the symmetry to Euler completion of
2-vector spaces.
