Kerodon

Proof of Lemma 9.1.5.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ denote the projection onto the first factor. It follows from Remark 5.1.4.7 that $U$ is a cocartesian fibration. We will show that $U$ satisfies condition $(3)$ of Proposition 9.1.3.8. Suppose we are given an object $C \in \operatorname{\mathcal{C}}$ and a diagram $e: K \rightarrow \operatorname{\mathcal{E}}_{C}$, where $K$ is a finite simplicial set. For each $i \in I$, our assumption that $\operatorname{\mathcal{E}}_ i$ is $\kappa $-filtered guarantees that we can choose a morphism $f_ i: C \rightarrow D_ i$ of $\operatorname{\mathcal{C}}$ for which the the composite diagram

admits an extension to $K^{\triangleright }$. Using condition $(\ast _{I})$, we can choose a morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ which factors through each of the morphisms $f_{i}$. Then the diagram

admits an extension to $K^{\triangleright }$. $\square$

Proof. It follows from Theorem 9.1.3.2 that the $\infty $-categories $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_{0}$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are filtered. Applying Lemma 9.1.5.2, we conclude that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}_0 \times _{ \widetilde{\operatorname{\mathcal{C}}} } \widetilde{\operatorname{\mathcal{C}}}_{1}$ is also filtered, and therefore weakly contractible (Proposition 9.1.1.13). $\square$

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

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

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

This is a reformulation of our assumption that (9.1) is a levelwise pullback square (see Proposition 7.4.1.16). $\square$

Proof. As in the proof of Proposition 9.1.5.5, the implications $(4) \Rightarrow (3) \Rightarrow (2) \Leftrightarrow (1)$ follow from Variant 9.1.4.7 and Proposition 9.1.1.14. The implication $(1) \Rightarrow (5)$ follows from Remarks 9.1.1.8 and 9.1.5.3. We will complete the proof by showing that $(5) \Rightarrow (4)$. Assume that condition $(5)$ is satisfied and fix uncountable regular cardinal $\lambda $ of exponential cofinality $\geq \kappa $ such that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small; we wish to show that the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$ preserves $\kappa $-small limits. Since $\operatorname{\mathcal{C}}$ is filtered, the functor $\varinjlim $ preserves finite limits (Proposition 9.1.5.5). It will therefore suffice to show that $\varinjlim $ reserves $I$-indexed products, for every $\kappa $-small set $I$ (Exercise 7.6.6.11). Using Variant 9.1.4.4, we can reformulate this statement as follows:

$(\ast )$

Let $\{ \mathscr {F}_ i: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda } \} _{i \in I}$ be an $I$-indexed collection of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{S}}^{< \lambda }$, and let $\mathscr {F} = \prod _{i \in I} \mathscr {F}_{i}$ be their product. If each colimit $\varinjlim ( \mathscr {F}_{i} )$ is contractible, then $\varinjlim (\mathscr {F} )$ is also contractible.

For each $i \in I$, we can assume that $\mathscr {F}_{i}$ arises as the covariant transport representation of a left fibration $U_ i: \operatorname{\mathcal{E}}_ i \rightarrow \operatorname{\mathcal{C}}$. Our assumption that $\varinjlim (\mathscr {F}_ i)$ is contractible then guarantees that the $\infty $-category $\operatorname{\mathcal{E}}_{i}$ is weakly contractible (Proposition 7.4.3.6) and therefore filtered (Theorem 9.1.3.2). Applying Lemma 9.1.5.2, we conclude that the $\infty $-category $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(I, \operatorname{\mathcal{C}}) } (\prod _{i \in I} \operatorname{\mathcal{E}}_ i )$ is also filtered, and therefore weakly contractible (Proposition 9.1.1.13). Since $\mathscr {F}$ is a covariant transport representation for the projection map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, this is equivalent to the contractibility of $\varinjlim (\mathscr {F} )$ (Proposition 7.4.3.6). $\square$

Proof. Fix an uncountable regular cardinal $\lambda $ of exponential cofinality $\geq \kappa ^{+}$ such that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small. It follows from Proposition 9.1.5.8, the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ preserves $\kappa $-small limits, and we wish to show that it preserves $\kappa ^{+}$-small limits. By virtue of Corollary 7.6.6.11, it will suffice to show that if $I$ is set of cardinality $\kappa $, then $\varinjlim $ preserves $I$-indexed products. Our assumption that $\kappa $ is singular guarantees that we can decompose $I$ as a disjoint union $\coprod _{j \in J} I_{j}$, where $J$ is $\kappa $-small and each $I_{j}$ is $\kappa $-small. The desired result now follows from Corollary 7.6.1.22. $\square$