Algebra of local symmetric operators and braided fusion $n$-category -- symmetry is a shadow of topological order

Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In particular, isomorphic operator algebras correspond to equivalent symmetries. In this paper, we pointed out that the algebra of local symmetry operators actually contains extended string-like, membrane-like, {\it etc} operators. The algebra of those extended operators in $n$-dimensional space gives rise to a non-degenerate braided fusion $n$-category, which happens to describe a topological order in one higher dimension. This allows us to show that the equivalent classes of finite symmetries actually correspond to topological orders in one higher dimension. Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, non-invertible (higher) symmetries (also known as algebraic higher symmetries), and (invertible and non-invertible) gravitational anomalies, in a unified and systematic way. We demonstrate this unified holographic framework via some simple examples of (higher and/or anomalous) symmetries, as well as a symmetry that is neither anomalous nor anomaly-free. We also show the equivalence between $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry with mixed anomaly and $\mathbb{Z}_4$ symmetry, as well as between many other symmetries, in 1-dimensional space.Comment: 35 pages, 21 figure