A Generalized Crystalline Equivalence Principle

We prove a general version of the crystalline equivalence principle which gives an equivalence of categories between a category of TQFTs defined on a generic space with $G$-symmetry, and a category of TQFTs with internal symmetry. We give a definition and classification of anomalies associated to TQFTs in the presence of spatial symmetry, which we then generalize to a definition of an anomaly for a categorical symmetry.

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