Kerodon

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

Warning 2.3.1.11. Let $\operatorname{\mathcal{C}}$ be a $2$-category. By virtue of Proposition 2.3.1.9, we can identify $n$-simplices of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ with triples

\[ ( \{ X_ i \} _{0 \leq i \leq n}, \{ f_{j,i} \} _{0 \leq i < j \leq n}, \{ \mu _{k,j,i} \} _{0 \leq i < j < k \leq n} ) \]

satisfying condition $(c')$ of Proposition 2.3.1.9. This gives a description of $\operatorname{N}_{n}^{\operatorname{D}}(\operatorname{\mathcal{C}})$ which makes no reference to the identity $1$-morphisms of $\operatorname{\mathcal{C}}$ or the left and right unit constraints of $\operatorname{\mathcal{C}}$. The resulting identification is functorial with respect to injective maps of linearly ordered sets $[m] \rightarrow [n]$. In other words, we can construct the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as a semisimplicial set (see ยง) without knowing the left and right unit constraints of $\operatorname{\mathcal{C}}$. However, the left and right unit constraints of $\operatorname{\mathcal{C}}$ are needed to define the degeneracy operators on the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$.