We explore topological manipulations in one spatial dimension, which are defined for a system with a global symmetry and map the system to another one with a dual symmetry. In particular, we discuss fusion category symmetries enhanced by the invariance of the actions of topological manipulations, i.e., self-$G$-alities for topological manipulations. Based on the self-$G$-ality conditions, we provide LSM-type constraints on the ground states of many-body Hamiltonians. We clarify the relationship between different enhanced symmetries and how they are further enhanced when they meet. We explore concrete lattice models for such self-$G$-alities and identify how the self-$G$-ality structures match the IR critical theories.

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• Phase transitions
• Quantum criticality
• Symmetry protected topological states
• Topological phases of matter
• Conformal field theory
• Duality