Is there a good way to define a sheaf over a simplicial set - i.e. as a functor from the diagram of the simplicial set to wherever the sheaf takes its values - in a way that while defined on simplex by simplex corresponds in some natural manner to what a sheaf over the geometric realization of the simplicial set would look like?
Edited to add: I'm actually interested in the question in a concrete fashion rather than an abstract one - I'm trying to figure out whether there might be some interesting interpretation of sheaves over the nerve of a category generated by a network; similar to current work by Robert Ghrist that takes a network and views it as a graph, and thus as a topological space (1-dim simplicial complex), and manages to find useful interpretations of sheaves on this particular space in terms of network analysis.
Hence, what I'm really looking for is an interesting definition for, say, the nerve of the category generated by a finite directed graph, or so...
Edited to add: In off-channels, fpqc has clarified his argument in the answer I've accepted. Specifically, $N(C)$ for a category has inherent direction data that is lost in $|N(C)|$. This messes up attempts to formulate an idea of sheaves over $N(C)$ in a way that stays faithful to the definition of sheaves over $|N(C)|$.