Twisted Graded Categories

Abstract

Given a presentably symmetric monoidal $\infty$-category $\mathcal{C}$ and an ${\mathbb{E}}_{\infty }$-monoid $M$, we introduce and classify twisted graded categories, which generalize the Day convolution structure on $\mathrm{Fun}(M,\mathcal{C})$. 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 ${\mathrm{Mod}}_{En}^{\wedge }$ at all primes and heights, and of the cyclotomic closure of ${\mathrm{Vect}}^n$ at low heights.

Added a subsection on "homotopy graded categories". Typos and small mistakes fixed. 82 pages, Comments are welcome!