Kerodon

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

Proposition 1.1.9.8. Let $X$ be a topological space. Then the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a Kan complex.

Proof. Let $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{Sing}_{\bullet }(X)$ be a map of simplicial sets for $n > 0$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex of $X$. Using the geometric realization functor, we can identify $\sigma _0$ with a continuous map of topological spaces $f_0: | \Lambda ^{n}_{i} | \rightarrow X$; we wish to show that $f_0$ factors as a composition

\[ | \Lambda ^{n}_{i} | \rightarrow | \Delta ^{n} | \xrightarrow {f} X. \]

Using Example 1.1.8.13, we can identify $| \Lambda ^{n}_{i} |$ with the subset

\[ \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: t_ j = 0 \text{ for some $j \neq i$} \} \subseteq | \Delta ^{n} |. \]

In this case, we can take $f$ to be the composition $f_0 \circ r$, where $r$ is any continuous retraction of $| \Delta ^{n} |$ onto the subset $| \Lambda ^{n}_{i} |$. For example, we can take $r$ to be the map given by the formula

\[ r( t_0, \ldots , t_ n ) = (t_0 - c, \ldots , t_{i-1} - c, t_{i} + nc, t_{i+1} - c, \ldots , t_ n - c) \]

\[ c = \min \{ t_0, \ldots , t_{i-1}, t_{i+1}, \ldots , t_ n \} . \]
$\square$