Higher dualizability and singly-generated Grothendieck categories

TL;DR

This paper characterizes certain dualizable $k$-linear categories as products of vector spaces and classifies Grothendieck categories with a universal copower object, confirming parts of a conjecture and providing new structural insights.

Contribution

It proves that locally presentable, $k$-linear dualizable categories are products of vector spaces, and characterizes Grothendieck categories with a universal copower object as modules over specific rings.

Findings

Dualizable categories are products of vector spaces.

Grothendieck categories with a universal copower are modules over simple, regular, self-injective rings.

Partially confirms a conjecture by Brandenburg, Johnson-Freyd, and the author.

Abstract

Let $k$ be a field. We show that locally presentable, $k$-linear categories $\mathcal{C}$ dualizable in the sense that the identity functor can be recovered as $\coprod_i x_i\otimes f_i$ for objects $x_i\in \mathcal{C}$ and left adjoints $f_i$ from $\mathcal{C}$ to $\mathrm{Vect}_k$ are products of copies of $\mathrm{Vect}_k$. This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object $x$ with the property that every object is a copower of $x$: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III.