Proof.
The implication $(4) \Rightarrow (3)$ is trivial, the implication $(3) \Rightarrow (2)$ follows from Variant 9.1.4.7, and the equivalence $(2) \Leftrightarrow (1)$ is a special case of Proposition 9.1.1.14. We will show that that $(1)$ implies $(4)$. Assume that $\operatorname{\mathcal{C}}$ is filtered and fix an uncountable regular cardinal $\lambda $ such that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small; we wish to show that the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda }) \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ preserves finite limits. We first note that $\varinjlim $ preserves final objects: if $\mathscr {F}$ is a final object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$, then the Kan complex $\varinjlim ( \mathscr {F} )$ is weakly homotopy equivalent to $\operatorname{\mathcal{C}}$ (Example 7.1.2.9) and is therefore contractible (Proposition 9.1.1.13). It will therefore suffice to show that the functor $\varinjlim $ preserves pullbacks (Corollary 7.6.2.30). Using Variant 9.1.4.4, we can reformulate this statement as follows:
- $(\ast )$
Suppose we are given a (levelwise) pullback diagram $\sigma :$
9.1
\begin{equation} \begin{gathered}\label{equation:filtered-via-colimits-precise2} \xymatrix { \mathscr {F}_{01} \ar [r] \ar [d] & \mathscr {F}_{0} \ar [d] \\ \mathscr {F}_{1} \ar [r] & \mathscr {F} } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$. If the colimits $\varinjlim (\mathscr {F} )$, $\varinjlim (\mathscr {F}_0)$, and $\varinjlim ( \mathscr {F}_{1} )$ are contractible, then the colimit $\varinjlim ( \mathscr {F}_{01} )$ is also contractible.
Let us identify $\sigma $ with a functor $\Delta ^1 \times \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$. By virtue of Corollary 5.6.0.6, this functor arises as the covariant transport representation of a left fibration $U: \operatorname{\mathcal{D}}\rightarrow \Delta ^1 \times \Delta ^1 \times \operatorname{\mathcal{C}}$ (which is uniquely determined up to equivalence). For $(i,j) \in \Delta ^1 \times \Delta ^1$, let $\operatorname{\mathcal{D}}_{ij}$ denote the fiber product $\{ (i,j)\} \times _{ \Delta ^1 \times \Delta ^1} \operatorname{\mathcal{E}}$. Then $U$ restricts to a left fibration $U_{ij}: \operatorname{\mathcal{D}}_{ij} \rightarrow \operatorname{\mathcal{C}}$ with covariant transport representation $\mathscr {F}_{ij}$. The hypotheses of $(\ast )$ guarantee that the $\infty $-categories $\operatorname{\mathcal{D}}_{11}$, $\operatorname{\mathcal{D}}_{01}$, and $\operatorname{\mathcal{D}}_{10}$ are weakly contractible (Proposition 7.4.3.6), and we wish to show that the $\infty $-category $\operatorname{\mathcal{D}}_{00}$ is also weakly contractible.
For $(i,j) \in \Delta ^1 \times \Delta ^1$, let $K_{ij}$ denote the full subcategory of $\Delta ^1 \times \Delta ^1$ spanned by those vertices $(i',j')$ satisfying $i' \geq i$ and $j' \geq j$. Let $\operatorname{\mathcal{E}}_{ij}$ denote the fiber product
\[ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( K_{ij}, \operatorname{\mathcal{C}}) } \operatorname{Fun}_{ / \Delta ^1 \times \Delta ^1}( K_{ij} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). \]
The inclusion maps $\{ (i,j) \} \hookrightarrow K_{ij}$ are left anodyne (Example 4.3.7.11), and therefore induce trivial Kan fibrations $\operatorname{\mathcal{E}}_{ij} \rightarrow \operatorname{\mathcal{D}}_{ij}$ (Proposition 4.2.5.4). It follows that the $\infty $-categories $\operatorname{\mathcal{E}}_{11}$, $\operatorname{\mathcal{E}}_{01}$, and $\operatorname{\mathcal{E}}_{10}$ are weakly contractible, and we wish to show that $\operatorname{\mathcal{E}}_{00}$ is also weakly contractible. Since $U$ is a left fibration, each of the projection maps $\operatorname{\mathcal{E}}_{ij} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration (Proposition 4.2.5.1), and the restriction maps
9.2
\begin{equation} \begin{gathered}\label{equation:filtered-via-colimits-precise} \xymatrix { \operatorname{\mathcal{E}}_{00} \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{01} \ar [d] \\ \operatorname{\mathcal{E}}_{10} \ar [r] & \operatorname{\mathcal{E}}_{11} } \end{gathered} \end{equation}
are also left fibrations. Since $\operatorname{\mathcal{C}}$ is filtered, Lemma 9.1.5.4 guarantees that the fiber product $\operatorname{\mathcal{E}}_{01} \times _{ \operatorname{\mathcal{E}}_{11} } \operatorname{\mathcal{E}}_{10}$ is also weakly contractible. To complete the proof, it will suffice to show that the diagram (9.2) induces an equivalence $\theta : \operatorname{\mathcal{E}}_{00} \rightarrow \operatorname{\mathcal{E}}_{01} \times _{ \operatorname{\mathcal{E}}_{11} } \operatorname{\mathcal{E}}_{10}$ of left fibrations over $\operatorname{\mathcal{C}}$. By Corollary 5.1.7.16, this is equivalent to the requirement that for each object $C \in \operatorname{\mathcal{C}}$, $\theta $ restricts to a homotopy equivalence of fibers
\[ \theta _{C}: \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}_{00} \rightarrow \{ C\} \times _{\operatorname{\mathcal{C}}} ( \operatorname{\mathcal{E}}_{01} \times _{\operatorname{\mathcal{E}}_{11}} \operatorname{\mathcal{E}}_{10} ). \]
This is a reformulation of our assumption that (9.1) is a levelwise pullback square (see Proposition 7.4.1.16).
$\square$