$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.1.2.15. Let $S$ be an infinite set of cardinality $\kappa $ and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa ^{+}$-small. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to the nerve of a partially ordered set.
- $(2)$
For every nonempty simplicial set $K$ and every object $X \in \operatorname{\mathcal{C}}$, the constant map
\[ K \rightarrow \{ \operatorname{id}_ X \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \]
exhibits $X$ as a power of itself by $K$.
- $(3)$
Every object $X \in \operatorname{\mathcal{C}}$ admits a power by $S$.
Proof.
The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are immediate from the definitions. We will show that $(3)$ implies $(1)$. Assume that condition $(3)$ is satisfied and fix a pair of objects $X,Y \in \operatorname{\mathcal{C}}$; we wish to show that the morphism space $M = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ is either empty or contractible. Assume otherwise: then there exists a morphism $f: Y \rightarrow X$ in $\operatorname{\mathcal{C}}$ and an integer $n \geq 0$ such that the homotopy set $\pi _{n}(M,f)$ has at least two elements. Using assumption $(3)$, we can choose an object $X' \in \operatorname{\mathcal{C}}$ and a collection of morphisms $\{ g_ s: X' \rightarrow X \} $ which exhibit $X'$ as a power of $X$ by $S$. Choose a morphism $f': Y \rightarrow X$ such that $g_{s} \circ f'$ is homotopic to $f$ for each $s \in S$. Then $\pi _{n}(M', f')$ can be identified with the product ${\prod }_{s \in S} \pi _{n}(M,f)$. This set has cardinality larger than $\kappa $ (Proposition 4.7.2.8), contradicting our assumption that $\operatorname{\mathcal{C}}$ is locally $\kappa ^{+}$-small.
$\square$