Higher Dualizability and Singly-Generated Grothendieck Categories

Let k be a field. We show that locally presentable, k-linear categories C dualizable in the sense that the identity functor can be recovered as ∐ _ix_i⊗ f_i for objects x_i∈ C and left adjoints f_i from C to Vect _k are products of copies of 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.