An abstract characterization of noncommutative projective lines

Let $k$ be a field. We describe necessary and sufficient conditions for a $k$-linear abelian category to be a noncommutative projective line, i.e. a noncommutative ${\mathbb{P}}^1$-bundle over a pair of division rings over $k$. As an application, we prove that ${\mathbb{P}}_n^1$, Piontkovski’s $n$th noncommutative projective line, is the noncommutative projectivization of an $n$-dimensional vector space.