Lemma 9.4.4.2 (05M3)

Lemma 9.4.4.2. Let $\mathbb {K}$ be a collection of simplicial sets, let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, and and suppose we are given a commutative diagram

9.46

\begin{equation} \begin{gathered}\label{equation:cocompletion-of-full-subcategory-revisited} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [rr]^-{H'} \ar [dr]_{U'} & & \widehat{\operatorname{\mathcal{E}}}' \ar [dl]^{ \widehat{U}' } \\ & \operatorname{\mathcal{C}}& } \end{gathered} \end{equation}

which exhibits $\widehat{\operatorname{\mathcal{E}}}'$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}'$. Let $\operatorname{\mathcal{E}}$ be a full simplicial subset of $\operatorname{\mathcal{E}}'$ and let $\widehat{\operatorname{\mathcal{E}}} \subseteq \widehat{\operatorname{\mathcal{E}}}'$ be the full simplicial subset spanned by those vertices $Y$ with the following property:

If $C = \widehat{U}'(Y)$, then $Y$ belongs to the smallest full subcategory of $\widehat{\operatorname{\mathcal{E}}}'_{C}$ which contains the essential image of $H|_{ \operatorname{\mathcal{E}}_{C} }$ and is closed under $K$-indexed colimits, for each $K \in \mathbb {K}$.

Set $H = H'|_{\operatorname{\mathcal{E}}}$, $U = U'|_{\operatorname{\mathcal{E}}}$, and $\widehat{U} = \widehat{U}'|_{\widehat{\operatorname{\mathcal{E}}}}$. Then the diagram

9.47

\begin{equation} \begin{gathered}\label{equation:cocompletion-of-full-subcategory-revisited2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U}} \\ & \operatorname{\mathcal{C}}& } \end{gathered} \end{equation}

exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$; in this case, we must show that the diagram (9.47) satisfies conditions $(1)$ through $(4)$ of Definition 9.4.1.8. Condition $(1)$ follows from Corollary 8.4.6.10, and the remaining conditions follow immediately from the corresponding conditions on the diagram (9.46) $\square$