In Rezk's paper "A model for the homotopy theory of homotopy theory" numerous references to simplicial covering maps are made. It's first appearance being at the bottom of page 8. Unfortunately no definition is provided in the paper and I was wondering if there is a purely combinatorial definition for this concept or whether we have to pass to the geometric realization.

Maps of simplicial sets already match cells of the same dimension (roughly speaking), but it is the evenly covered concept that requires some work (I imagine).

Any help would be appreciated.

Simplicial objects in algebraic topology, or Chapter V of Goerss-Jardine. Such bundles have "structure group" which is a simplical group $G$; the special case where $G$ is actually discrete gives you a theory of covering space. In which case you can reformulate as Tom describes. – Charles Rezk Aug 24 '10 at 19:33Calculus of fractions and homotopy theory, though I don't have a copy hand to check exactly what they say. – Charles Rezk Aug 24 '10 at 19:52