This work classifies twisted graded categories using their braiding and $ ext{T}$-action on dimensions, indicating new structures in monoidal categories.
Given a presentably symmetric monoidal ∞-category C and an E∞-monoid M, we introduce and classify twisted graded categories, which generalize the Day convolution structure on Fun(M, C). These are characterized by a braiding encoded in symmetric group actions on tensor powers, whose character we show depends only on the T-equivariant monoidal dimension. We analyze the T-action on the dimension of invertible objects and identify it with the T-transfer map. Finally, we compute braiding characters in examples arising from higher cyclotomic extensions, such as the (S, n+1)-oriented extension of ModEn^ at all primes and heights, and of the cyclotomic closure of Vectⁿ at low heights.
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Keidar et al. (2025) studied this question.
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