Recent advancements in generalized symmetries have drawn significant attention to gapped phases of matter exhibiting novel symmetries, such as noninvertible symmetries. By leveraging the duality transformations, the classification and construction of gapped phases with noninvertible symmetry can be mapped to those involving conventional group symmetries. We demonstrate this approach by classifying symmetry-protected-topological phases with a broad class of noninvertible symmetries in arbitrary spacetime dimensions. Our results reveal new classifications that extend beyond those based on group symmetries. Additionally, we construct lattice models in $(1+1)D$ and $(2+1)D$ that realize these new phases and explore their anomalous interfaces.

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• Gauge theories
• Spontaneous symmetry breaking
• Symmetry protected topological states
• Topological phases of matter
• Quantum many-body systems
• Quantum spin models
• Cluster models
• Duality
• Lattice gauge theory
• Quantum spin chains
• Spin chains
• Spin lattice models