Example 4.6.6.12 (01L7): Pinched Morphism Spaces in the Duskin Nerve

Example 4.6.6.12 (Pinched Morphism Spaces in the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category (Definition 2.2.1.1) and let $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ denote its Duskin nerve (Construction 2.3.1.1). Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, which we identify with vertices of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Using Example 4.6.1.12, we can identify $n$-simplices $\sigma $ of the simplicial set $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y)$ with commutative diagrams

\[ \xymatrix@R =20pt@C=20pt{ & & & & f_{0,n} & & & & \\ & & & f_{0,n-1} \ar@ {=>}[ur] & & f_{1,n} \ar@ {=>}[ul] & & & \\ & & f_{0,n-2} \ar@ {=>}[ur] & & f_{1,n-1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{2,n} \ar@ {=>}[ul] & & \\ & \cdots \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ur] \ar@ {=>}[ul] & & \cdots \ar@ {=>}[ul] \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ul] & \\ f_{0,0} \ar@ {=>}[ur] & & f_{1,1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{n-1,n-1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{n,n} \ar@ {=>}[ul] } \]

in the category of $1$-morphisms $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. The image of the left-pinch inclusion morphism

\[ \iota ^{\mathrm{L}}_{X,Y}: \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \hookrightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y) \]

consists of those simplices which correspond (under this identification) to commutative diagrams in which each of the leftward pointing $2$-morphisms $f_{i,j} \Rightarrow f_{i-1,j}$ is an identity map. In this case, the entire diagram is determined by the sequence of composable morphisms $f_{0,0} \Rightarrow f_{0,1} \Rightarrow f_{0,2} \Rightarrow \cdots \Rightarrow f_{0,n}$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. Similarly, the image of the right-pinch inclusion morphism

\[ \iota ^{\mathrm{R}}_{X,Y}: \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{R}}( X, Y) \hookrightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y) \]

consists of those simplices which correspond to commutative diagrams in which the rightward pointing $2$-morphisms $f_{i,j} \Rightarrow f_{i,j+1}$ is an identity map, which ensures that the entire diagram is determined by the sequence of composable morphisms $f_{n,n} \Rightarrow f_{n-1,n} \Rightarrow f_{n-2,n} \Rightarrow \cdots \Rightarrow f_{0,n}$ in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. Allowing $[n] \in \operatorname{{\bf \Delta }}$ to vary, we obtain canonical isomorphisms of simplicial sets

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