The $0$-simplices of $N(\mathcal{C})$ are the objects of $\mathcal{C}$:

$N(\mathcal{C})_0 = \operatorname{Ob}(\mathcal{C}).$

A functor $[0] \to \mathcal{C}$ simply picks out a single object.

The $1$-simplices of $N(\mathcal{C})$ are the morphisms of $\mathcal{C}$:

$N(\mathcal{C})_1 = \operatorname{Mor}(\mathcal{C}).$

A functor $[1] \to \mathcal{C}$ picks out a morphism $c_0 \xrightarrow{f} c_1$.

The face maps are $d_0(f) = c_1$ (target) and $d_1(f) = c_0$ (source). The degeneracy $s_0(c) = \mathrm{id}_c$.

A $2$-simplex of $N(\mathcal{C})$ is a composable pair with its composite:

$c_0 \xrightarrow{f} c_1 \xrightarrow{g} c_2$

together with the implicit composite $g \circ f: c_0 \to c_2$. The faces are $d_0 = g$, $d_1 = g \circ f$, $d_2 = f$.

Crucially, a $2$-simplex witnesses that $d_1 = d_0 \circ d_2$, i.e., it encodes the composition law.

A $3$-simplex of $N(\mathcal{C})$ is a composable triple $c_0 \xrightarrow{f} c_1 \xrightarrow{g} c_2 \xrightarrow{h} c_3$. Its four faces are:

• $d_0$: the chain $c_1 \xrightarrow{g} c_2 \xrightarrow{h} c_3$
• $d_1$: the chain $c_0 \xrightarrow{g \circ f} c_2 \xrightarrow{h} c_3$
• $d_2$: the chain $c_0 \xrightarrow{f} c_1 \xrightarrow{h \circ g} c_3$
• $d_3$: the chain $c_0 \xrightarrow{f} c_1 \xrightarrow{g} c_2$

The fact that such a $3$-simplex exists and is unique (given the composable triple) encodes associativity: $h \circ (g \circ f) = (h \circ g) \circ f$.

A discrete category $\mathcal{C}$ has only identity morphisms. Its nerve $N(\mathcal{C})$ is the constant simplicial set on $\operatorname{Ob}(\mathcal{C})$: all face and degeneracy maps are identities. The geometric realization is a discrete set of points.

A group $G$ can be viewed as a category $BG$ with one object $*$ and $\operatorname{Hom}(*, *) = G$. The nerve $N(BG)$ has:

• One $0$-simplex ($*$)
• $G$ many $1$-simplices (one for each group element)
• $G \times G$ many $2$-simplices (composable pairs $(g_1, g_2)$)
• $G^n$ many $n$-simplices (ordered $n$-tuples)

The geometric realization $|N(BG)|$ is the classifying space $BG$. For $G = \mathbb{Z}/2\mathbb{Z}$, we get $\mathbb{R}P^\infty$. For $G = \mathbb{Z}$, we get $S^1$.

A partially ordered set $(P, \leq)$ is a category where $\operatorname{Hom}(x, y)$ has one element if $x \leq y$ and is empty otherwise. The $n$-simplices of $N(P)$ are chains $x_0 \leq x_1 \leq \cdots \leq x_n$.

For the poset $\{0 < 1 < 2\}$, the nerve has three $0$-simplices, three non-degenerate $1$-simplices, and one non-degenerate $2$-simplex. Its geometric realization is a filled triangle $\Delta^2$.

Non-degenerate $n$-simplices correspond to strict chains $x_0 < x_1 < \cdots < x_n$. So the nerve of a poset is the order complex from combinatorial topology.

Let $\mathcal{G}$ be a groupoid (a category in which every morphism is invertible). The nerve $N(\mathcal{G})$ is a Kan complex (every horn has a filler), because invertibility of morphisms provides the necessary compositions and inverses.

For the groupoid with two objects and a single isomorphism between them (the "walking isomorphism"), the nerve has a geometric realization homeomorphic to $[0,1]$, which is contractible.

The arrow category $[1] = (0 \to 1)$ has nerve $N([1]) = \Delta[1]$: it is precisely the standard $1$-simplex. More generally, $N([n]) = \Delta[n]$ for all $n$, by definition.

This shows that the simplex category $\Delta$ is the "category of finite nonempty linear orders," and the standard simplices are nerves of these linear orders.

Consider the category $\Lambda = \{a \to c \leftarrow b\}$ (three objects, two non-identity morphisms). Its nerve $N(\Lambda)$ is the simplicial set $\Lambda^2_2$ (the inner horn), with three vertices and two edges but no $2$-simplex.

The geometric realization is a "V" shape, not a filled triangle. A functor from $\Lambda$ to $\mathcal{C}$ is precisely a diagram for computing a pullback (or fiber product).

If $F, G: \mathcal{C} \to \mathcal{D}$ are functors and $\alpha: F \Rightarrow G$ is a natural transformation, then $N(F)$ and $N(G)$ are simplicially homotopic. The homotopy $H: N(\mathcal{C}) \times \Delta[1] \to N(\mathcal{D})$ is defined on $(c_0 \to \cdots \to c_n, 0 \leq i_0 \leq \cdots \leq i_m \leq 1)$ by the appropriate composable chain involving $F(f_j)$, $G(f_k)$, and $\alpha_{c_l}$.

In particular, if $F$ and $G$ are naturally isomorphic, then $|N(F)|$ and $|N(G)|$ are homotopic maps. If $\mathcal{C}$ and $\mathcal{D}$ are equivalent categories, their nerves have homotopy equivalent geometric realizations.