  Non-invertible categorical symmetries have emerged as a powerful tool to
uncover new beyond-Landau phases of matter, both gapped and gapless, along with
second order phase transitions between them. The general theory of such phases
in (1+1)d has been studied using the Symmetry Topological Field Theory
(SymTFT), also known as topological holography. This has unearthed the infrared
(IR) structure of these phases and transitions. In this paper, we describe how
the SymTFT information can be converted into an ultraviolet (UV) anyonic chain
lattice model realizing in the IR limit these phases and transitions. In many
cases, the Hilbert space of the anyonic chain is tensor product decomposable
and the model can be realized as a quantum spin-chain Hamiltonian. We also
describe operators acting on the lattice models that are charged under
non-invertible symmetries and act as order parameters for the phases and
transitions. In order to fully describe the action of non-invertible
symmetries, it is crucial to understand the symmetry twisted sectors of the
lattice models, which we describe in detail. Throughout the paper, we
illustrate the general concepts using the symmetry category $\mathsf{Rep}(S_3)$
formed by representations of the permutation group $S_3$, but our procedure can
be applied to any fusion category symmetry.
