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Construction (Relative Homotopy $n$-Categories). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $n \geq 0$ be an integer. For every $m$-simplex $\sigma $ of $\operatorname{\mathcal{D}}$, let $\operatorname{\mathcal{C}}_{\sigma }$ denote the fiber product $\Delta ^ m \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. We let $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}_{m}$ denote the collection of pairs $(\sigma , \tau )$, where $\sigma $ is an $m$-simplex of $\operatorname{\mathcal{D}}$ and $\tau $ is a section of the projection map

\[ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_{\sigma })} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\Delta ^ m)} \simeq \Delta ^{m}. \]

If $f: [m'] \rightarrow [m]$ is a nondecreasing function, we let $f^{\ast }: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}_{m} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}_{m'}$ denote the map given by $f^{\ast }( \sigma , \tau ) = (\sigma ', \tau ' )$, where $\sigma '$ is the composite map $\Delta ^{m'} \xrightarrow {f} \Delta ^{m} \xrightarrow {\sigma } \operatorname{\mathcal{D}}$ and $\tau '$ is given by the composition

\begin{eqnarray*} \Delta ^{m'} & \simeq & \Delta ^{m'} \times _{ \Delta ^{m} } \Delta ^{m} \\ & \xrightarrow {(\operatorname{id}, \tau )} & \Delta ^{m'} \times _{ \Delta ^ m } \mathrm{h}_{\mathit{\leq n}}\mathit{( \operatorname{\mathcal{C}}_{\sigma } )} \\ & \simeq & \mathrm{h}_{\mathit{\leq n}}\mathit{( \Delta ^{m'} \times _{\Delta ^{m}} \operatorname{\mathcal{C}}_{\sigma } )} \\ & \simeq & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_{\sigma '} )}, \end{eqnarray*}

where the second isomorphism is provided by Proposition By means of this construction, we can view the assignment $[m] \mapsto \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}_{m}$ as a simplicial set, which we will denote by $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$. Note that the construction $(\sigma , \tau ) \mapsto \sigma $ determines a comparison map of simplicial sets $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$.

It will be useful to extend this construction to the case where $n < 0$. If $n = -1$, we define $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ to be the full simplicial subset of $\operatorname{\mathcal{D}}$ whose vertices belong to the image of $F$, and we take $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \hookrightarrow \operatorname{\mathcal{D}}$ to be the inclusion map. If $n \leq -2$, we define $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ to be the simplicial set $\operatorname{\mathcal{D}}$, and $G$ to be the identity morphism $\operatorname{id}_{\operatorname{\mathcal{D}}}$.