Kerodon

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

Remark 2.3.1.7. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $\{ \mu _{g,f} \} $ be a twisting cochain for $\operatorname{\mathcal{C}}$ (Notation 2.2.6.7), and let $\operatorname{\mathcal{C}}'$ be the twist of $\operatorname{\mathcal{C}}$ with respect to $\{ \mu _{g,f} \} $ (Construction 2.2.6.8). Then the twisting cochain $\{ \mu _{g,f} \} $ defines a strictly unitary isomorphism of $2$-categories $\operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}'$, and therefore induces an isomorphism of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}')$. In other words, the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ cannot detect the difference between $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$. This should be regarded as a feature, rather than a bug. Defining the composition law for $1$-morphisms in a $2$-category $\operatorname{\mathcal{C}}$ often requires certain arbitrary (but ultimately inessential) choices (see Example 2.2.6.13). In such cases, one can often give a more direct description of the simplicial set $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ which avoids such choices. See Example 2.3.1.17 and Remark 2.3.5.9.