Kerodon

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1.1.7 Kan Complexes

We close this section by articulating an important property enjoyed by simplicial sets of the form $\operatorname{Sing}_{\bullet }(X)$.

Definition 1.1.7.1. Let $S_{\bullet }$ be a simplicial set. We will say that $S_{\bullet }$ is a Kan complex if it satisfies the following condition:

$(\ast )$

For $0 \leq i \leq n$, any map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ can be extended to a map $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$. Here $\Lambda ^{n}_{i} \subseteq \Delta ^ n$ denotes the $i$th horn (see Construction 1.1.2.9).

Exercise 1.1.7.2. Show that for $n > 0$, the standard simplex $\Delta ^{n}$ is not a Kan complex (for a more general statement, see Proposition 1.2.4.2).

Proposition 1.1.7.3. 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; 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 1.1.6.13, 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

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

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