Twisted Graded Categories

TL;DR

This paper introduces and classifies twisted graded categories within symmetric monoidal $ty$-categories, analyzing their braiding structures, $ty$-dimension actions, and applications to higher cyclotomic extensions.

Contribution

It generalizes the Day convolution to twisted graded categories, characterizes braiding via symmetric group actions, and computes examples related to higher cyclotomic extensions.

Findings

Braiding characters depend only on the $al T$-equivariant monoidal dimension.

The $al T$-action on invertible objects' dimension is identified with the $al T$-transfer map.

Explicit computations of braiding characters in higher cyclotomic extension examples.

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.