A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical characterization of when functors give covering maps on the topological side (i.e. the map is a local homeomorphism)?



Functors and coverings
Coverings of groupoids are described for example in an ancient book <http://www.maths.ed.ac.uk/~aar/papers/gz.pdf>. 


The condition on a groupoid functor C' -> C is that for any arrow f in C and an object x in C' which is taken to the source or the target of f by the given functor, there is a unique lifting of f in C' with the source or respectively the target x. For functors between categories the condition will be the same. (For groupoids in fact it is sufficient the condition to hold for sources only or for targets only).


I will add that, as mentioned, the geometric realization of a category goes through the nerve functor. The nerve functor is fully faithful, while simplicial sets which arise as nerves are characterized by the Segal condition. The covering functors are those which are taken to coverings of simplicial sets. The latter are charaterized in the Gabriel-Zisman's book. Then the above condition for a functor to be a covering is easily obtained using the Segal condition. 


