Kerodon

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

Example 2.3.1.16 ($3$-Simplices of the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Using Proposition 2.3.1.9, we see that a map of simplicial sets $\partial \Delta ^3 \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with the following data:

  • A collection of objects $\{ X_ i \} _{0 \leq i \leq 3}$ of the $2$-category $\operatorname{\mathcal{C}}$.

  • A collection of $1$-morphisms $\{ f_{j,i}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq 3}$.

  • A quadruple of $2$-morphisms

    \[ \mu _{2,1,0}: f_{2,1} \circ f_{1,0} \Rightarrow f_{2,0} \quad \quad \mu _{3,2,1}: f_{3,2} \circ f_{2,1} \Rightarrow f_{3,1} \]
    \[ \mu _{3,1,0}: f_{3,1} \circ f_{1,0} \Rightarrow f_{3,0} \quad \quad \mu _{3,2,0}: f_{3,2} \circ f_{2,0} \Rightarrow f_{3,0}. \]

This data can be conveniently visualized as a pair of diagrams

\[ \xymatrix@C =100pt@R=50pt{ & X_1 \ar [r]^{ f_{2,1} } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{\mu _{2,1,0}} & X_2 \ar [dr]^{ f_{3,2}} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{\mu _{3,2,0}} & \\ X_0 \ar [ur]^{ f_{1,0}} \ar [urr]_{ f_{2,0} } \ar [rrr]^{f_{3,0}} & & & X_3 \\ & X_1 \ar [r]^{ f_{2,1} } \ar [drr]_{ f_{3,1} } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{\mu _{3,1,0}} & X_2 \ar [dr]^{ f_{3,2} } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\mu _{3,2,1}} & \\ X_0 \ar [ur]^{f_{1,0}} \ar [rrr]^{f_{3,0}} & & & X_3, } \]

representing “front” and “back” perspectives of the boundary of a $3$-simplex. A $3$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ can be identified with a map $\operatorname{\partial }\Delta ^3 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as above which satisfies an additional compatibility condition: namely, the commutativity of the diagram

\[ \xymatrix { & f_{3,2} \circ (f_{2,1} \circ f_{1,0} ) \ar@ {=>}[dl]_-{ \operatorname{id}_{ f_{3,2} } \circ \mu _{2,1,0} } \ar@ {=>}[rr]^{\alpha _{ f_{3,2}, f_{2,1}, f_{1,0} } } & & (f_{3,2} \circ f_{2,1} ) \circ f_{1,0} \ar@ {=>}[dr]^-{ \mu _{3,2,1} \circ \operatorname{id}_{f_{1,0}}} & \\ f_{3,2} \circ f_{2,0} \ar@ {=>}[drr]^{ \mu _{3,2,0} } & & & & f_{3,1} \circ f_{1,0} \ar@ {=>}[dll]_{ \mu _{3,1,0} } \\ & & f_{3,0} & & } \]

in the ordinary category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X_0, X_3)$.