Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.5.2.9. Let $S$ be a finite subset of the unit interval $[0,1]$ which contains $0$ and $1$. Then the cosimplicial set

\[ | \Delta ^{\bullet } |_{S}: \operatorname{{\bf \Delta }}\rightarrow \operatorname{Set}\quad \quad [n] \mapsto | \Delta ^ n |_{S} \]

is a corepresentable functor. More precisely, there exists a functorial bijection $| \Delta ^ n |_{S} \simeq \operatorname{Hom}_{ \operatorname{Lin}_{\neq \emptyset } }( \pi _0( [0,1] \setminus S), [n] )$.

Proof. Let $S = \{ 0 = s_0 < s_1 < \cdots < s_ k = 1 \} $ be a finite subset of the unit interval $[0,1]$ which contains $0$ and $1$. Let $n$ be a nonnegative integer and let $(t_0, \ldots , t_ n)$ be a point of $| \Delta ^ n |_{S}$. For every real number $u \in [0,1] \setminus S$, there exists a unique integer $0 \leq i \leq n$ satisfying

\[ t_0 + t_1 + \cdots + t_{i-1} < u < t_0 + t_1 + \cdots + t_{i}. \]

The construction $u \mapsto i$ defines a continuous nondecreasing function $([0,1] \setminus S) \rightarrow [n]$. This observation induces a bijection

\begin{eqnarray*} | \Delta ^ n |_{S} & \simeq & \{ \text{Continuous nondecreasing functions $f: [0,1] \setminus S \rightarrow [n]$} \} \\ & \simeq & \operatorname{Hom}_{ \operatorname{Lin}_{\neq \emptyset } }( \pi _0( [0,1] \setminus S), [n] ). \end{eqnarray*}

Explicitly, the inverse bijection carries a continuous nondecreasing function $f: [0,1] \setminus S \rightarrow [n]$ to the sequence

\[ ( \mu ( f^{-1} \{ 0\} ), \mu ( f^{-1} \{ 1\} ), \cdots , \mu ( f^{-1} \{ n\} ) ), \]

where

\[ \mu ( f^{-1} \{ i\} ) = \sum _{ (s_{j-1}, s_ j) \subseteq f^{-1} \{ i\} } (s_ j - s_{j-1} ) \]

denotes the measure of the inverse image $f^{-1} \{ i \} $. $\square$