Note on the projective limit on small categories

Abstract:

Let X be a small category and let ${\mathbf {A}}{{\mathbf {b}}^X}$ be the category of covariant functors on X with values in Ab. Consider the projective limit functor proj ${\lim _X}:{\mathbf {A}}{{\mathbf {b}}^X} \to {\mathbf {Ab}}$. The categories X for which proj ${\lim _X}$ is exact are characterized, proving a conjecture of Oberst.