Nerve of a category

Given a category $C$, its nerve $N(C)$ is the simplicial set constructed as follows. The set $N(C{)}_k$ of $k$-simplices is the set of diagrams

$$A_0\to A_1\to \cdots \to A_k$$

of objects and morphisms from $C$. The face maps ${\partial }_i:N(C{)}_k\to N(C{)}_{k-1}$ are given by composition of morphisms at the $i$-th node in the diagram (or dropping the first or last arrow if $i=0$ or $k$ respectively), and the degeneracy maps are given by inserting identity morphisms. The intuition here is that a $k$-simplex in $N(C)$ is precisely a commutative diagram in $C$ with the shape of a $k$-simplex. If $C$ is a topological category, we can enrich $N(C)$ to be a simplicial space.