Example 5.5.6.17 (0216): $2$-Simplices of $\operatorname{ \pmb {\mathcal{QC}} }

Example 5.5.6.17 ($2$-Simplices of $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$). By virtue of Example 5.5.6.12, a morphism of simplicial sets $\sigma _0: \operatorname{\partial \Delta }^2 \rightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ can be identified with the following data:

A collection of $\infty $-categories $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ equipped with distinguished objects $C \in \operatorname{\mathcal{C}}$, $D \in \operatorname{\mathcal{D}}$, and $E \in \operatorname{\mathcal{E}}$.
A collection of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, and $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$.
A collection of morphisms $\alpha : F(C) \rightarrow D$, $\beta : G(D) \rightarrow E$, and $\gamma : H(C) \rightarrow E$ in the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$.

Unwinding the definitions, we see that extending $\sigma _0$ to a $2$-simplex $\sigma $ of $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ is equivalent to choosing a natural transformation of functors $\mu : (G \circ F) \rightarrow H$ and a morphism of simplicial sets $\theta : \operatorname{\raise {0.1ex}{\square }}^{2} \rightarrow \operatorname{\mathcal{E}}$ whose restriction to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ (G \circ F)(C) \ar [r]^-{ \mu (C) } \ar [d]^-{G(\alpha )} & H(C) \ar [d]^-{\gamma } \\ G(D) \ar [r]^-{ \beta } & E. } \]

Moreover:

The $2$-simplex $\sigma $ belongs to the image of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}} \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ if and only if $\mu : G \circ F \rightarrow H$ is an isomorphism in the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
The $2$-simplex $\sigma $ belongs to the image of $\operatorname{\mathcal{QC}}_{\ast } \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ if and only if $\mu $, $\alpha $, $\beta $, and $\gamma $ are all isomorphisms.
The $2$-simplex $\sigma $ belongs to the image of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } ) \hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ if and only if $\mu $, $\alpha $, $\beta $, and $\gamma $ are identity morphisms (so that $H = G \circ F$, $D = F(C)$, and $E = G(D)$) and the morphism $\theta : \operatorname{\raise {0.1ex}{\square }}^{2} \rightarrow \operatorname{\mathcal{E}}$ is constant.