Kerodon

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

Example 3.1.4.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories and suppose we are given a pair of functors $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, which we identify with morphisms of simplicial sets $\operatorname{N}_{\bullet }(F), \operatorname{N}_{\bullet }(G): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$. By definition, a homotopy from $\operatorname{N}_{\bullet }(F)$ to $\operatorname{N}_{\bullet }(G)$ is a map of simplicial sets

\[ h: \Delta ^1 \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }( [1] \times \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]

satisfying $h|_{ \{ 0\} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } = \operatorname{N}_{\bullet }(F)$ and $h|_{ \{ 1\} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} = \operatorname{N}_{\bullet }(G)$. By virtue of Proposition 1.2.2.1, this is equivalent to the datum of a functor $H: [1] \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $H|_{\{ 0\} \times \operatorname{\mathcal{C}}} = F$ and $H|_{ \{ 1\} \times \operatorname{\mathcal{C}}} = G$. In other words, we have a canonical bijection

\[ \xymatrix { \{ \text{Natural transformations from $F$ to $G$} \} \ar [d]^{\sim } \\ \{ \text{Homotopies from $\operatorname{N}_{\bullet }(F)$ to $\operatorname{N}_{\bullet }(G)$} \} }. \]

In particular, if there exists a natural transformation from $F$ to $G$, then $\operatorname{N}_{\bullet }(F)$ and $\operatorname{N}_{\bullet }(G)$ are homotopic.