Higher Dualizability and Singly-Generated Grothendieck Categories

Let k be a field. We showthat locally presentable, k-linear categories C dualizable in the sense that the identity functor can be recovered as coproduct(i) x(i) circle times f(i) for objects x(i) is an element of 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.