Kerodon

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

Proposition 2.4.6.9. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\sigma _0: \operatorname{\partial }\Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be a map of simplicial sets, which we identify with a diagram

\[ \xymatrix { & X_1 \ar [dr]^{ f_{21} } & \\ X_0 \ar [ur]^{ f_{10} } \ar [rr]_{ f_{20} } & & X_2 } \]

as above. Then the construction of Example 2.4.3.10 induces a bijection

\[ \xymatrix { \{ \textnormal{Maps $\sigma : \Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ with $\sigma |_{ \operatorname{\partial }\Delta ^2} = \sigma _0$} \} \ar [d]^{\sim } \\ \{ \textnormal{Homotopies from $f_{21} \circ f_{10}$ to $f_{20}$} \} . } \]

Proof of Proposition 2.4.6.9. Apply Corollary 2.4.6.12 in the case $n = 2$. $\square$