Twisted Graded Categories

Given a presentably symmetric monoidal $\mathrm{\infty }$-category $\mathcal{C}$ and an ${\mathbb{E}}_{\mathrm{\infty }}$-monoid $M$, we introduce and classify twisted graded categories, which generalize the Day convolution structure on $\mathrm{F}\mathrm{u}\mathrm{n}(M,\mathcal{C})$. These are characterized by a braiding encoded in symmetric group actions on tensor powers, whose character we show depends only on the $\mathbb{T}$-equivariant monoidal dimension. We analyze the $\mathbb{T}$-action on the dimension of invertible objects and identify it with the $\mathbb{T}$-transfer map. Finally, we compute braiding characters in examples arising from higher cyclotomic extensions, such as the $(\mathbb{S},n+1)$-oriented extension of ${\mathrm{M}\mathrm{o}\mathrm{d}}_{En}^{\wedge }$ at all primes and heights, and of the cyclotomic closure of ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}^n$ at low heights.