Proposition 2.3.3.2. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\sigma $ be a $2$-simplex of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{ g } \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X \ar [ur]^{f } \ar [rr]_{ h } & & Z. } \]
in the $2$-category $\operatorname{\mathcal{C}}$. Assume that the following condition is satisfied:
- $(\ast )$
Let $n \in \{ 3,4\} $ and let $u: \Lambda ^{n}_{1} \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ be a map of simplicial sets such that $u|_{ \Delta ^2} = \sigma $; here we identify $\Delta ^2$ with a simplicial subset of $\Lambda ^{n}_{1} \subseteq \Delta ^{n}$ via the inclusion map $[2] \hookrightarrow [n]$. Then $u$ extends to an $n$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$.
Then $\gamma $ is invertible.
Proof.
Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is strictly unitary (Proposition 2.2.7.7). Applying $(\ast )$ in the case $n = 3$, we can extend $\sigma $ to a $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ which is represented by the pair of diagrams
\[ \xymatrix@C =100pt@R=50pt{ & Y \ar [r]^-{ g } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{\gamma } & Z \ar [dr]^{ \operatorname{id}_ Z } \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{\delta } & \\ X \ar [ur]^{ f} \ar [urr]_{ h } \ar [rrr]^{g \circ f} & & & Z \\ & Y \ar [r]^-{ g } \ar [drr]_{ g } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{ \operatorname{id}_{g \circ f} } & Z \ar [dr]^{ \operatorname{id}_ Z } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\operatorname{id}_ g } & \\ X \ar [ur]^{f} \ar [rrr]^{g \circ f} & & & Z. } \]
It follows that $\gamma $ admits a left inverse, given by the vertical composition $\delta : h \Rightarrow g \circ f$. To show that this composition is also a right inverse, we apply $(\ast )$ in the case $n = 4$ to construct a $4$-simplex $\tau $ of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ whose two-dimensional faces correspond to the $2$-morphisms
\[ \mu _{2,1,0} = \mu _{4,1,0} = \gamma \quad \quad \mu _{3,1,0} = \operatorname{id}_{ g \circ f} \quad \mu _{3,2,0} = \delta \quad \mu _{4,2,0} = \operatorname{id}_ h \]
\[ \mu _{4,3,0} = \gamma \quad \quad \mu _{3,2,1} = \mu _{4,2,1} = \mu _{4,3,1} = \operatorname{id}_ g \quad \quad \mu _{4,3,2} = \operatorname{id}_{\operatorname{id}_ Z}. \]
The $3$-simplex $d^{4}_1(\tau )$ then witnesses the identity
\[ \mu _{4,2,0} (\mu _{4,3,2} \circ \operatorname{id}_ h ) = \mu _{4,3,0} (\operatorname{id}_{ \operatorname{id}_ Z} \circ \mu _{3,2,0}), \]
which shows that $\delta $ is also a right inverse to $\gamma $.
$\square$