Kerodon

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Example 5.3.6.13. To get a feeling for the content of Proposition 5.3.6.11, let us consider the special case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}')$ is the Duskin nerve of a strict $2$-category $\operatorname{\mathcal{C}}'$. Let $\sigma $ be a $4$-simplex of $\operatorname{\mathcal{C}}$, which we identify with a collection of objects $\{ X_ i \} _{0 \leq i \leq 4}$, $1$-morphisms $\{ f_{ji}: X_ j \rightarrow X_ i \} _{0 \leq i < j \leq 4}$, and $2$-morphisms $\{ \mu _{kji}: f_{kj} \circ f_{ji} \Rightarrow f_{ki} \} _{0 \leq i < j < k \leq 4}$ of $\operatorname{\mathcal{C}}$ satisfying the condition described in Proposition 2.3.1.9. Proposition 5.3.6.11 asserts that if the $2$-morphisms $\mu _{310}$, $\mu _{420}$, $\mu _{321}$, and $\mu _{432}$ are invertible, then the $2$-morphism $\mu _{430}$ is also invertible. This follows by inspecting the cubical diagram

\[ \xymatrix@R =50pt@C=40pt{ f_{43} \circ f_{32} \circ f_{21} \circ f_{10} \ar@ {=>}[rr]^{ \mu _{210} } \ar@ {=>}[dr]^{ \mu _{321} }_-{\sim } \ar@ {=>}[dd]^{ \mu _{432} } & & f_{43} \circ f_{32} \circ f_{20} \ar@ {=>}[dr]^{\mu _{320} } \ar@ {=>}[dd]^(.6){\mu _{432}}_(.6){\sim } & \\ & f_{43} \circ f_{31} \circ f_{10} \ar@ {=>}[dd]^(.6){ \mu _{431} } \ar@ {=>}[rr]^(.4){ \mu _{310} }_(.4){\sim } & & f_{43} \circ f_{30} \ar@ {=>}[dd]^{ \mu _{430} } \\ f_{42} \circ f_{21} \circ f_{10} \ar@ {=>}[rr]^(.4){ \mu _{210} } \ar@ {=>}[dr]^{ \mu _{421} } & & f_{42} \circ f_{20} \ar@ {=>}[dr]^{\mu _{420}}_-{\sim } & \\ & f_{41} \circ f_{10} \ar@ {=>}[rr]^{ \mu _{410} } & & f_{40} } \]

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X_0, X_4)$ and applying the two-out-of-six property to the chain of $2$-morphisms

\[ f_{43} \circ f_{32} \circ f_{21} \circ f_{10} \xRightarrow {\mu _{210} } f_{43} \circ f_{32} \circ f_{20} \xRightarrow { \mu _{320} } f_{43} \circ f_{30} \xRightarrow { \mu _{430} } f_{40}. \]