Remark 8.1.8.8 (03L3)

Remark 8.1.8.8. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of objects $X$ and $Y$. Then we have a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{\rho _{+} } \ar [d]^{\sim } & \operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \ar [d]^{\sim } & \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}} \ar [l]_-{ \rho _{-} } \ar [d]^{\sim } \\ \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y) \ar [r]^-{ \iota ^{\mathrm{L}} } & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }(X,Y) & \operatorname{Hom}^{\mathrm{R}}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y) \ar [l]_-{ \iota ^{\mathrm{R}} } } \]

where the upper horizontal maps are the inclusions of Construction 8.1.7.1 and Variant 8.1.7.14, the lower horizontal maps are the pinch inclusion maps of Construction 4.6.5.7, the outer vertical maps are the isomorphisms of Example 4.6.5.13, and the inner vertical map is the isomorphism of Corollary 8.1.8.6.

Stated more concretely, Corollary 8.1.8.6 asserts that we can identify $n$-simplices of the simplicial set $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y)$ with commutative diagrams

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

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

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

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

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

consists of those simplices which correspond to commutative diagrams in which the rightward pointing $2$-morphisms $f_{i,j} \Rightarrow f_{i,j+1}$ are identity maps, 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 $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.