  We summarize several results about the regular coverings and the fundamental
groupoids of Alexandroff spaces. In particular, we show that the fundamental
groupoid of an Alexandroff space $X$ is naturally isomorphic to the
localization, at its set of morphisms, of the thin category associated to the
set $X$ considered as a preordered set with the specialization preorder. We
also show that the regular coverings of an Alexandroff space $X$ are
represented by certain morphism-inverting functors with domain $X$, extending a
result of E. Minian and J. Barmak about the regular coverings of locally finite
T$_0$ spaces.
