Example 4.6.5.12 (01L7): Pinched Morphism Spaces in the Duskin Nerve

Example 4.6.5.12 (Pinched Morphism Spaces in the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category (Definition 2.2.1.1). For each integer $n \geq 0$, we can use Remark 2.3.1.8 to identify $(n+1)$-simplices $\sigma $ of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{\mathcal{C}})$ with the following data:

$(0)$: A collection of objects $\{ Z_ i \} _{ 0 \leq i \leq n+1}$ of the $2$-category $\operatorname{\mathcal{C}}$.
$(1)$: A collection of $1$-morphisms $\{ f_{j,i}: Z_ i \rightarrow Z_ j \} _{0 \leq i \leq j \leq n+1 }$ in the $2$-category $\operatorname{\mathcal{C}}$, satisfying $f_{j,i} = \operatorname{id}_{ Z_ i }$ when $i = j$.
$(2)$: A collection of $2$-morphisms $\{ \mu _{k,j,i}: f_{k,j} \circ f_{j,i} \Rightarrow f_{k,i} \} _{0 \leq i \leq j \leq k \leq n+1}$ in the $2$-category $\operatorname{\mathcal{C}}$, satisfying some additional constraints (see $(b)$ and $(c)$ of Proposition 2.3.1.9).

Fix a pair of objects $X$ and $Y$. Then $\sigma $ represents an $n$-simplex of the right pinched morphism space $\operatorname{Hom}^{\mathrm{R}}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y)$ if and only if the above data satisfies the following additional conditions:

For $0 \leq i \leq n$, the object $Z_ i$ is equal to $X$. For $i = n+1$, the object $Z_{i}$ is equal to $Y$.
For $0 \leq i \leq j \leq n$, the $1$-morphism $f_{j,i}$ is equal to the identity $1$-morphism $\operatorname{id}_{X}$.
For $0 \leq i \leq \leq j \leq k \leq n$, the $2$-morphism $\mu _{k,j,i}$ is equal to the unit constraint $\upsilon : \operatorname{id}_{X} \circ \operatorname{id}_{X} \Rightarrow \operatorname{id}_{X}$.

In this case, we can identify $(1)$ with a collection of $1$-morphisms $\{ g_ i: X \rightarrow Y \} _{0 \leq i \leq n}$ given by $g_ i = f_{n+1,i}$, and $(2)$ with a collection of $2$-morphisms $\{ \nu _{j,i}: g_ j \Rightarrow g_ i \} _{ 0 \leq i \leq j \leq n}$, where $\nu _{j,i}$ is given by the composition

\[ g_{j} \xRightarrow {\sim } g_{j} \circ \operatorname{id}_{X} = f_{n+1,j} \circ f_{j,i} \xRightarrow {\mu _{n+1,j,i} } f_{n+1,i} = g_ i. \]

Unwinding the definitions, condition $(b)$ translates to the requirement that $\nu _{j,i}$ is an identity $2$-morphism when $i = j$, and condition $(c)$ translates to the identity $\nu _{k,j} \circ \nu _{j,i} = \nu _{k,i}$ for $0 \leq i \leq j \leq k \leq n$. In this case, we can identify the pair $( \{ g_ i \} _{0 \leq i \leq n}, \{ \nu _{j,i} \} _{0 \leq i \leq j \leq n} )$ with a functor $[n] \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}}$. These identifications depends functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore determine a canonical isomorphism of simplicial sets

\[ \operatorname{Hom}^{\mathrm{R}}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}} ). \]

Using similar reasoning, we obtain an isomorphism of simplicial sets

\[ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ). \]