This paper explores the use of Symmetry Topological Field Theory (SymTFT) to describe and construct lattice models with non-invertible categorical symmetries. The authors aim to convert SymTFT information into ultraviolet (UV) anyonic chain lattice models that realize gapped and gapless phases, as well as phase transitions, in the infrared (IR) limit. They focus on the general theory and provide detailed constructions for gapped and gapless phases, including the action of non-invertible symmetries on the lattice models. The Hilbert space of the anyonic chain is often tensor product decomposable, allowing it to be realized as a quantum spin-chain Hamiltonian. The paper also discusses operators charged under non-invertible symmetries and their role as order parameters for phases and transitions. The authors illustrate their methods using the symmetry category \(\text{Rep}(S_3)\) and emphasize the importance of understanding the symmetry twisted sectors of the lattice models. The paper concludes with an outlook on extending these methods to higher dimensions and a discussion of specific examples, including Abelian group symmetries and \(\mathbf{Rep}(S_3)\) symmetry.This paper explores the use of Symmetry Topological Field Theory (SymTFT) to describe and construct lattice models with non-invertible categorical symmetries. The authors aim to convert SymTFT information into ultraviolet (UV) anyonic chain lattice models that realize gapped and gapless phases, as well as phase transitions, in the infrared (IR) limit. They focus on the general theory and provide detailed constructions for gapped and gapless phases, including the action of non-invertible symmetries on the lattice models. The Hilbert space of the anyonic chain is often tensor product decomposable, allowing it to be realized as a quantum spin-chain Hamiltonian. The paper also discusses operators charged under non-invertible symmetries and their role as order parameters for phases and transitions. The authors illustrate their methods using the symmetry category \(\text{Rep}(S_3)\) and emphasize the importance of understanding the symmetry twisted sectors of the lattice models. The paper concludes with an outlook on extending these methods to higher dimensions and a discussion of specific examples, including Abelian group symmetries and \(\mathbf{Rep}(S_3)\) symmetry.