$\Delta[n] = N([n])$ is the nerve of the linear order category $[n] = (0 \to 1 \to \cdots \to n)$. The unique inner horn filling property is clear: in a linear order, any composable pair has a unique composite (the unique morphism between the endpoints).

The Segal map $\Delta[n]_m \to \Delta[n]_1 \times_{\Delta[n]_0} \cdots \times_{\Delta[n]_0} \Delta[n]_1$ is the bijection between order-preserving maps $[m] \to [n]$ and chains of $m$ composable morphisms in $[n]$.

The simplicial circle $S^1 = \Delta[1]/\partial\Delta[1]$ has one vertex $v$, one non-degenerate edge $\sigma$ with $d_0(\sigma) = d_1(\sigma) = v$, and all higher simplices degenerate. Consider the inner horn $\Lambda^2_1 \to S^1$ with $d_0 = \sigma$ and $d_2 = \sigma$. A filler would need a $2$-simplex $\tau$ with $d_0(\tau) = \sigma$ and $d_2(\tau) = \sigma$. But the only $2$-simplices in $S^1$ are degenerate ones ($s_0(\sigma)$ and $s_1(\sigma)$), and neither satisfies both conditions simultaneously (since $d_0 s_0(\sigma) = \sigma$ but $d_2 s_0(\sigma) = s_0(v)$, which is degenerate).

So $S^1$ does not satisfy the inner horn filling property and is not a nerve of any category.

A simplicial set $X$ is the nerve of a groupoid if and only if every horn $\Lambda^n_k \to X$ (for all $n \geq 1$ and $0 \leq k \leq n$) has a unique filler. This adds the outer horns to the nerve characterization.

Unique outer horn fillers at level $2$ give: for any $g: c_1 \to c_2$ and $h: c_0 \to c_2$, there is a unique $f$ with $g \circ f = h$. Taking $h = \mathrm{id}_{c_1}$ gives $g \circ g^{-1} = \mathrm{id}$, showing $g$ is right-invertible. A dual argument with $\Lambda^2_2$ shows left-invertibility.

A simplicial set $X$ is the nerve of a poset if and only if it is the nerve of a category (unique inner horn fillers) and additionally has at most one $1$-simplex between any two $0$-simplices. Equivalently, the function $X_1 \to X_0 \times X_0$ given by $(d_1, d_0)$ is injective.

For instance, $N(\{0 < 1 < 2\})$ has exactly one edge from $i$ to $j$ when $i \leq j$ and none otherwise.

The Segal map at level $2$ is:

$X_2 \to X_1 \times_{X_0} X_1 = \{(f, g) \in X_1 \times X_1 : d_0(f) = d_1(g)\}$

sending $\sigma \mapsto (d_2(\sigma), d_0(\sigma))$. This is a bijection iff every composable pair determines a unique $2$-simplex.

At level $3$, the Segal map is:

$X_3 \to X_1 \times_{X_0} X_1 \times_{X_0} X_1$

This uses the $3$ consecutive edges of a $3$-simplex. Bijectivity here encodes associativity of composition.

The nerve characterization theorem motivates the definition of quasi-categories:

• Nerve of a category: unique inner horn fillers (strict composition, strict associativity).
• Quasi-category: inner horn fillers exist but need not be unique (composition up to homotopy, associativity up to coherent homotopy).

The passage from "unique" to "existence" is the key weakening that leads from $1$-categories to $(\infty, 1)$-categories. All the coherence data (associativity, Mac Lane pentagon, etc.) is automatically encoded by the higher horn filling conditions.

The nerve characterization generalizes to higher categories. For $2$-categories, the Duskin nerve $N_D(\mathcal{C})$ is a simplicial set with unique inner horn fillers for $n \geq 4$, existence and uniqueness for $\Lambda^3_k$ (inner), and existence (but not uniqueness) for $\Lambda^2_1$. The non-uniqueness at level $2$ reflects the fact that $2$-categories have non-trivial $2$-morphisms between compositions.

More generally, the nerve of an $n$-category has unique inner horn fillers above dimension $n+1$.

A Segal category is a simplicial space $X: \Delta^{\mathrm{op}} \to \mathbf{sSet}$ with $X_0$ discrete and the Segal maps

$X_n \xrightarrow{\;\simeq\;} X_1 \times_{X_0}^h \cdots \times_{X_0}^h X_1$

being weak equivalences (not bijections). This is another model for $(\infty, 1)$-categories, equivalent to quasi-categories by a Quillen equivalence. The Segal condition is thus the common thread linking all models of $\infty$-categories.

For the simplicial circle $S^1$ (one vertex, one non-degenerate edge $\sigma$), the fundamental category $\tau_1(S^1)$ has one object and one generating morphism $\sigma$ with no relations (since there are no non-degenerate $2$-simplices beyond the degenerate ones). This gives the free monoid on one generator, which is $\mathbb{N}$.

Alternatively, using the fundamental groupoid $\tau_{\leq 1}(S^1)$ (which also identifies with $\tau_1$ for Kan complexes), we get $\pi_1(S^1) = \mathbb{Z}$.

If $X = N(\mathcal{C})$, then $\tau_1(N(\mathcal{C})) \cong \mathcal{C}$. This is precisely the statement that the nerve is fully faithful: the left adjoint $\tau_1$ restricted to the essential image of $N$ is an inverse.

For a general simplicial set $X$, $\tau_1(X)$ computes the "best approximation" of $X$ by an ordinary category, discarding homotopy-theoretic information above dimension $1$.