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.