A finite group $G$ can be considered as a category with one object. Taking its nerve $NG$, and then geometrically realizing we get $BG$ the classifying space of $G$, which classifies principal $G$ bundles.


Instead starting with any category $C$, what does $NC$ classify? (Either before or after taking realization.) Does it classify something reasonable?



What does the classifying space of a category classify?
Ieke Moerdijk has written a small Springer Lecture Notes tome addressing this question:["Classifying Spaces and Classifying Topoi" SLNM 1616](http://openlibrary.org/books/OL800793M/Classifying_spaces_and_classifying_topoi).


Roughly the answer is: A $G$-bundle is a map whose fibers have a $G$-action, i.e. are $G$-sets (if they are discrete), i.e. they are functors from $G$ seen as a category to $\mathsf{Sets}$. Likewise a $\mathcal C$-bundle for a category $\mathcal C$ is a map whose fibers are functors from $\mathcal C$ to $\mathsf{Sets}$, or, if you want, a disjoint union of sets (one for each object of $\mathcal C$) and an action by the morphisms of $\mathcal C$ — a morphism $A \to B$ in $\mathcal C$ takes elements of the set corresponding to $A$ to elements of the set corresponding to $B$.


There is a completely analogous version for topological categories also.


