Proposition 1.3.4.1. Let $S$ be a simplicial set. Then $S$ is isomorphic to the nerve of a category if and only if it satisfies the following condition:
For every pair of integers $0 < i < n$ and every map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S$, there exists a unique map $\sigma : \Delta ^{n} \rightarrow S$ such that $\sigma _0 = \sigma |_{ \Lambda ^{n}_{i} }$.
Proof of Proposition 1.3.4.1. Let $S$ be a simplicial set satisfying condition $(\ast ')$ of Proposition 1.3.4.1; we will show that there is a category $\operatorname{\mathcal{C}}$ and an isomorphism of simplicial sets $u: S \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (the converse follows from Lemma 1.3.4.2). It follows from Proposition 1.3.3.1 that the category $\operatorname{\mathcal{C}}$ is uniquely determined (up to isomorphism), and from the proof of Proposition 1.3.3.1 we can extract an explicit construction of $\operatorname{\mathcal{C}}$:
The objects of $\operatorname{\mathcal{C}}$ are the vertices of $S$.
Given a pair of objects $C, D \in \operatorname{\mathcal{C}}$, we let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ denote the collection of edges $e$ of $S$ having source $C$ and target $D$.
For each object $C \in \operatorname{\mathcal{C}}$, we define the identity morphism $\operatorname{id}_{C} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,C)$ to be the degenerate edge $s^{0}_0(C)$.
Given a triple of objects $C,D,E \in \operatorname{\mathcal{C}}$ and a pair of morphisms $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ and $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( D, E)$, we can apply hypothesis $(\ast ')$ (in the special case $n = 2$ and $i = 1$) to conclude that there is a unique $2$-simplex $\sigma $ of $S_{\bullet }$ satisfying $d^{2}_2(\sigma ) = f$ and $d^{2}_0(\sigma ) = g$. We define the composition $g \circ f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E)$ to be the edge $d^{2}_1(\sigma )$.
We claim that $\operatorname{\mathcal{C}}$ is a category. For this, we must check the following:
The composition law on $\operatorname{\mathcal{C}}$ is unital: for every morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, we have equalities
Let us verify the identity on the left; the proof in the other case is similar. For this, we must construct a $2$-simplex $\sigma $ of $S$ such that $d^{2}_0(\sigma ) = \operatorname{id}_{D}$ and $d^{2}_1(\sigma ) = d^{2}_2(\sigma ) = f$. The degenerate $2$-simplex $s^{1}_1(f)$ has these properties.
The composition law on $\operatorname{\mathcal{C}}$ is associative. That is, for every triple of composable morphisms
in $\operatorname{\mathcal{C}}$, we have an identity $h \circ (g \circ f) = (h \circ g) \circ f$ in $\operatorname{\mathcal{C}}$. Applying condition $(\ast ')$ repeatedly, we deduce the following:
There is a unique $2$-simplex $\sigma _0$ of $S$ satisfying $d^{2}_0(\sigma _0) = h$ and $d^{2}_2(\sigma _0) = g$ (it follows that $d^{2}_1(\sigma _0) = h \circ g$).
There is a unique $2$-simplex $\sigma _3$ of $S$ satisfying $d^{2}_0(\sigma _3) = g$ and $d^{2}_2(\sigma _3) = f$ (it follows that $d^{2}_1(\sigma _3) = g \circ f$).
There is a unique $2$-simplex $\sigma _2$ of $S$ satisfying $d^{2}_0( \sigma _2) = h \circ g$ and $d^{2}_2( \sigma _2) = f$ (it follows that $d^{2}_1( \sigma _2) = (h \circ g) \circ f$).
There is a unique $3$-simplex $\tau $ of $S$ satisfying $d^{3}_0( \tau ) = \sigma _0$, $d^{3}_2(\tau ) = \sigma _2$, and $d^{3}_3(\tau ) = \sigma _3$ (this follows by applying $(\ast ')$ to the horn inclusion $\Lambda ^{3}_{1} \hookrightarrow \Delta ^3$).
The $3$-simplex $\tau $ can be depicted in the following diagram
Set $\sigma _1 = d^{3}_1( \tau )$. Then $\sigma _1$ is a $2$-simplex of $S$ satisfying $d^{2}_0( \sigma _1) = h$, $d^{2}_1( \sigma _1) = (h \circ g) \circ f$, and $d^{2}_2(\sigma _1) = g \circ f$. It follows that $\sigma _1$ “witnesses” the identity $h \circ (g \circ f) = (h \circ g) \circ f$.
Note that every $n$-simplex $\sigma : \Delta ^{n} \rightarrow S$ determines a functor $[n] \rightarrow \operatorname{\mathcal{C}}$, given on objects by the values of $\sigma $ on the vertices of $\Delta ^ n$ and on morphisms by the values of $\sigma $ on the edges of $\Delta ^ n$. This construction determines a map of simplicial sets $u: S \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which is bijective on simplices of dimension $\leq 1$. Since the simplicial sets $S$ and $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ both satisfy condition $(\ast ')$ (Lemma 1.3.4.2), it follows from Lemma 1.3.4.3 that $u$ is an isomorphism. $\square$