Higher dualizability and singly-generated Grothendieck categories

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.

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