Classification of fully dualizable linear categories

TL;DR

This paper characterizes fully dualizable linear categories over certain rings as twisted module categories over finite étale algebras, extending to graded and spectral contexts, revealing their structural classification.

Contribution

It provides a classification of fully dualizable $R$-linear categories as twists of module categories over finite étale $R$-algebras, generalizing previous results to broader contexts.

Findings

Fully dualizable $R$-linear categories are equivalent to twists of module categories over finite étale $R$-algebras.

The classification extends to $R$-linear graded categories and $ abla$-categories over connective ring spectra.

Results hold over G-rings and more general rings under compact generation assumptions.

Abstract

We prove that if $R$ is a G-ring then every fully dualizable $R$-linear cocomplete category is equivalent to a twist by a $\mathbb{G}_m$-gerbe of the category of modules over a finite \'etale $R$-algebra. We also show that this holds more generally over an arbitrary commutative ring under an additional compact generation hypothesis. We include variants of these results that apply to $R$-linear graded categories, and to the context of $\infty$-categories linear over connective commutative ring spectra.