Nerve of a category

Remarks

A functor $[n] \to \mathcal{C}$ is the same as a string $c_0 \xrightarrow{f_1} c_1 \to \cdots \xrightarrow{f_n} c_n$ of composable morphisms. So $N\mathcal{C}_0$ is the set of objects and $N\mathcal{C}_1$ the set of morphisms. The face map $d_i$ for $0 < i < n$ composes $f_{i+1} \circ f_i$, while $d_0$ drops $f_1$ and $d_n$ drops $f_n$; the degeneracy $s_i$ inserts an identity at $c_i$. Example: if $\mathcal{C}$ has one object and only the identity morphism, then $N\mathcal{C}_n$ has one element for each $n$, and $N\mathcal{C}$ is the point $\Delta^0$. Non-example: the boundary $\partial \Delta^2$ consists of three edges forming a triangle, with no 2-simplex. It is not the nerve of any category, because in a nerve each pair of composable edges is filled by a 2-simplex. A simplicial set is isomorphic to a nerve exactly when every inner horn $\Lambda^n_k \to X$ with $0 < k < n$ has a unique filler.