Proposition 3.2.1.15. Let $f: (X,x) \rightarrow (Y,y)$ be a morphism of pointed simplicial sets, where $Y$ is a Kan complex. The following conditions are equivalent:
- $(1)$
The morphism $f$ is nullhomotopic as an unpointed map. That is, there exists a vertex $z \in Y$ and a homotopy from $f$ to the constant map $\underline{z}: X \rightarrow Y$ taking the value $z$ (see Definition 3.2.4.5).
- $(2)$
The morphism $f$ is nullhomotopic as a pointed map: that is, there exists a vertex $y \in Y$ and a pointed homotopy from $f$ to the constant map $\underline{y}: X \rightarrow Y$.
Proof.
The implication $(2) \Rightarrow (1)$ is immediate from the definition. To prove the converse, suppose that there exists a a homotopy $h: \Delta ^1 \times X \rightarrow Y$ satisfying $h|_{ \{ 0\} \times X} = f$ and $h |_{ \{ 1\} \times X} = \underline{z}$ for some vertex $z \in Y$. Let $e: y \rightarrow z$ be the edge of $Y$ given by the restriction $h|_{ \Delta ^1 \times \{ x\} }$ and let $\sigma = s^{1}_0(e)$ denote the degenerate $2$-simplex of $Y$ depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{e} & \\ y \ar [ur]^{\operatorname{id}_ y} \ar [rr]^{e} & & z. } \]
Let $\underline{e}: \underline{y} \rightarrow \underline{z}$ denote the image of $e$ in $\operatorname{Fun}(X,Y)$. Since $Y$ is a Kan complex, the restriction map $q: \operatorname{Fun}(X,Y) \rightarrow \operatorname{Fun}(\{ x\} ,Y) \simeq Y$ is a Kan fibration (Corollary 3.1.3.3). It follows that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{2} \ar [r]^-{ (\bullet , h, \underline{e}) } \ar [d] & \operatorname{Fun}(X,Y) \ar [d]^{q} \\ \Delta ^2 \ar [r]^-{\sigma } \ar@ {-->}[ur] & Y, } \]
admits a solution which carries the edge $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Delta ^2$ to a pointed homotopy from $f$ to $\underline{y}$.
$\square$