We define the notion of the (G,Γ)-crossed center of a (G,Γ)-crossed tensor category in the sense of Natale. We show that the (G, Γ)-crossed center is a (G Γ, G × Γ)-braided tensor category. This construction generalizes the graded center construction for graded tensor categories and the equivariant center construction for tensor categories with group actions.
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Mizuki Oikawa (2024) studied this question.
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