$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 3.5.1.12. Let $X$ be a Kan complex and let $n$ be a nonnegative integer. The following conditions are equivalent:
- $(1)$
The Kan complex $X$ is $n$-connective.
- $(2)$
For every integer $0 \leq m \leq n$, every morphism $\operatorname{\partial \Delta }^{m} \rightarrow X$ can be extended to an $m$-simplex of $X$.
- $(3)$
Let $B$ be a simplicial set of dimension $\leq n$ and let $A \subseteq B$ be a simplicial subset. Then every morphism $f_0: A \rightarrow X$ admits an extension $f: B \rightarrow X$.
- $(4)$
Let $A$ be a simplicial set of dimension $< n$. Then every morphism $f: A \rightarrow X$ is nullhomotopic.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows from Lemma 3.2.4.13 (and Variant 3.2.4.14), the implication $(2) \Rightarrow (3)$ follows from Proposition 1.1.4.12, and the implication $(4) \Rightarrow (2)$ follows from Variant 3.2.4.12. We complete the proof by showing that $(3)$ implies $(4)$. Applying assumption $(3)$ to the inclusion map $\emptyset \subseteq \Delta ^0$, we deduce that there exists a vertex $x \in X$. It will therefore suffice to show that if $A$ is a simplicial set of dimension $< n$, then every pair of morphisms $f_0, f_1: A \rightarrow X$ are homotopic (in particular, $f_0$ is homotopic to the constant map $A \rightarrow \{ x\} $). This follows by applying $(3)$ to the inclusion map $\operatorname{\partial \Delta }^1 \times A \hookrightarrow \Delta ^1 \times A$ (see Proposition 1.1.3.6).
$\square$