Nerve of a category

Summary

The nerve of a small category $\mathcal{C}$ is the simplicial set whose $n$-simplices are the strings of $n$ composable morphisms of $\mathcal{C}$. The construction is a fully faithful functor from small categories to simplicial sets, so no information about $\mathcal{C}$ is lost. Its geometric realization is the classifying space of $\mathcal{C}$; for a group $G$ viewed as a one-object category it gives the classifying space $BG$. Nerves are the basic examples of quasi-categories and, for groupoids, of Kan complexes.