Construction 2.3.1.1 (009U): The Duskin Nerve

Construction 2.3.1.1 (The Duskin Nerve). Let $n$ be a nonnegative integer and let $[n]$ denote the linearly ordered set $\{ 0 < 1 < 2 < \cdots < n \} $. We will regard $[n]$ as a category, hence also as a $2$-category having only identity $2$-morphisms (Example 2.2.0.6). For any $2$-category $\operatorname{\mathcal{C}}$, we let $\operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{C}})$ denote the set of all strictly unitary lax functors from $[n]$ to $\operatorname{\mathcal{C}}$ (Definition 2.2.4.17). The construction $[n] \mapsto \operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{C}})$ determines a simplicial set, given as a functor by the composition

\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \hookrightarrow \operatorname{Cat}^{\operatorname{op}} \hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}^{\operatorname{op}} \xrightarrow { \operatorname{Hom}_{ \operatorname{2Cat}_{\operatorname{ULax}} }( \bullet , \operatorname{\mathcal{C}}) } \operatorname{Set}. \]

We will denote this simplicial set by $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ and refer to it as the Duskin nerve of the $2$-category $\operatorname{\mathcal{C}}$.