Kerodon

Proof. Apply Proposition 9.2.9.1 in the special case where $\kappa = \aleph _0$ and $\lambda = \operatorname{\Omega }$ is a fixed strongly inaccessible cardinal. $\square$

Proof. The implication $(1) \Rightarrow (2)$ is immediate from the definition. We next show that $(2) \Rightarrow (3)$. Assume that condition $(2)$ is satisfied. For each vertex $C \in \operatorname{\mathcal{C}}$, we can apply condition $(2)$ to the degenerate edge $\operatorname{id}_{C}$ to guarantee that the $\infty $-category $\Delta ^1 \times \operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-cocomplete. It follows immediately that the factor $\operatorname{\mathcal{E}}_{C}$ is also $(\kappa ,\lambda )$-cocomplete (see Example 7.1.3.11): that is, every $\lambda $-small $\kappa $-filtered diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{E}}_{C}$ can be extended to a colimit diagram $\overline{F}: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$. We wish to show that in this case, $\overline{F}$ is an edgewise $U$-colimit diagram. To prove this, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$ and that $C \in \operatorname{\mathcal{C}}$ is the initial vertex. In this case, condition $(2)$ guarantees that the $\infty $-category $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-cocomplete, so that $f$ extends to a colimit diagram $\overline{F}': \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$. Let us identify $\overline{F}$ and $\overline{F}'$ with objects $X,X' \in \operatorname{\mathcal{E}}_{F/}$. Our assumption that $\overline{F}'$ is a colimit diagram guarantees that $X'$ is an initial object of $\operatorname{\mathcal{E}}_{F/}$, so that there exists a morphism $X' \rightarrow X$. Then $\overline{F}'$ factors through the full subcategory $\operatorname{\mathcal{E}}_{C} \subseteq \operatorname{\mathcal{E}}$, and is therefore also a colimit diagram in $\operatorname{\mathcal{E}}_{C}$. It follows that $\overline{F}$ is isomorphic to $\overline{F}'$ (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{\mathcal{E}}_{C} ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{\mathcal{E}})$), and is therefore also a colimit diagram in $\operatorname{\mathcal{E}}$.

We now complete the proof by showing that $(3) \Rightarrow (1)$. Assume that condition $(3)$ is satisfied; we wish to show that for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-cocomplete. Replacing $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{E}}_{\sigma }$, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex; in this case, we wish to show that the $\infty $-category $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-cocomplete. Let $\operatorname{\mathcal{K}}$ be an $\infty $-category which is $\lambda $-small and $\kappa $-filtered; we must show that every diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{E}}$ admits a colimit. For $0 \leq i \leq n$, let $\operatorname{\mathcal{K}}_{i} \subseteq \operatorname{\mathcal{K}}$ be the full subcategory spanned by those objects $K$ satisfying $(U \circ F)(X) = i$. Since $\operatorname{\mathcal{K}}$ is nonempty, there is some largest integer $0 \leq i \leq n$ for which $\operatorname{\mathcal{K}}_{i}$ is nonempty. Applying Corollary 9.1.5.17, we see that $\operatorname{\mathcal{K}}_ i$ is $\kappa $-filtered and the inclusion $\operatorname{\mathcal{K}}_ i \hookrightarrow \operatorname{\mathcal{K}}$ is right cofinal. Using Corollary 7.2.2.10, we can replace $F$ by $F|_{ \operatorname{\mathcal{K}}_{i} }$ and thereby reduce to the case where the diagram $F$ takes values in the $\infty $-category $\operatorname{\mathcal{E}}_{i}$. In this case, assumption $(3)$ guarantees that $F$ can be extended to a colimit diagram $\overline{F}: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{i}$ which is also a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$ (Proposition 7.1.7.14). Since $U \circ \overline{F}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}= \Delta ^ n$, it follows that $\overline{F}$ is also a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$ (Corollary 7.1.6.12). $\square$

Proof. The $\infty $-category $\operatorname{Fun}_{ / \Delta ^1}( \Delta ^1, \operatorname{\mathcal{E}})$ can be identified with the oriented fiber product $\operatorname{\mathcal{E}}_0 \vec{\times }_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_1$. The desired result now follows from Proposition 9.2.8.3. $\square$

Proof of Proposition 9.2.9.16. We proceed as in the proof of Proposition 9.2.8.9. For every morphism of simplicial sets $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, Remark 9.2.9.12 guarantees that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ is $(\kappa ,\lambda )$-cocomplete, and that the evaluation functors $\operatorname{ev}_{C}: \operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_ C$ are $(\kappa ,\lambda )$-finitary for each $C \in \operatorname{\mathcal{C}}'$. Let us say that $\operatorname{\mathcal{C}}'$ is good if the restriction $F|_{\operatorname{\mathcal{C}}'}$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$. We will complete the proof by showing that if $\operatorname{\mathcal{C}}'$ is $\kappa $-small, then $\operatorname{\mathcal{C}}'$ is good.

Note that $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ can be realized as the inverse limit of a tower of $\infty $-categories

where the transition maps are isofibrations (Corollary 4.4.5.12). Each of the $\infty $-categories appearing in this tower is $(\kappa ,\lambda )$-cocomplete and each of the transition functors is $(\kappa ,\lambda )$-finitary. It will therefore suffice to show that each of the skeleta $\operatorname{sk}_{n}(\operatorname{\mathcal{C}}')$ is good: this is immediate if $\kappa = \aleph _0$ (the assumption that $\operatorname{\mathcal{C}}'$ is finite guarantees that it coincides with $\operatorname{sk}_{n}(\operatorname{\mathcal{C}}')$ for $n \gg 0$), and follows from Corollary 9.2.8.7 when $\kappa > \aleph _0$. We may therefore replace $\operatorname{\mathcal{C}}'$ by $\operatorname{sk}_{n}(\operatorname{\mathcal{C}}')$ and thereby reduce to the case where the simplicial set $\operatorname{\mathcal{C}}$ has dimension $\leq n$, for some integer $n \geq -1$.

We now proceed by induction on $n$. If $n = -1$, then the simplicial set $\operatorname{\mathcal{C}}'$ is empty and the result is clear. Otherwise, let $\operatorname{\mathcal{C}}'' = \operatorname{sk}_{n-1}(\operatorname{\mathcal{C}}')$ be the $(n-1)$-skeleton of $\operatorname{\mathcal{C}}'$ and let $S$ be the collection of all nondegenerate $n$-simplices of $\operatorname{\mathcal{C}}$. Applying Proposition 1.1.4.12, we obtain a pullback diagram of $\infty $-categories

Since the horizontal maps in this diagram are isofibrations (Corollary 4.4.5.12), it is also a categorical pullback square (Corollary 4.5.2.28). Moreover, each of the $\infty $-categories in the diagram is $(\kappa ,\lambda )$-cocomplete of the transition functors is $(\kappa ,\lambda )$-finitary. Combining our inductive hypothesis with Corollary 9.2.8.5 and Proposition 9.2.8.6, we can reduce to the case where $\operatorname{\mathcal{C}}' = \Delta ^ n$ is a standard simplex.

If $n = 0$, then the map $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ can be identified with a vertex $C \in \operatorname{\mathcal{C}}$, and the desired result follows from the observation that $F(C)$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ (this is a special case of assumption $(\ast )$, applied to the degenerate edge $\operatorname{id}_{C}: C \rightarrow C$). If $n = 1$, then the desired result is a reformulation of Proposition 9.2.9.14. We may therefore assume that $n \geq 2$. Let $\operatorname{Spine}[n] \subseteq \Delta ^ n$ be as in Example 1.5.7.7. The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne and therefore induces a trivial Kan fibration $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \Delta ^ n, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Spine}[n], \operatorname{\mathcal{E}})$. Consequently, to show that $\Delta ^ n$ is good, it will suffice to show that $\operatorname{Spine}[n]$ is good. This follows from our inductive hypothesis, since the simplicial set $\operatorname{Spine}[n]$ has dimension $1$. $\square$

Proof. It follows from Corollary None that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ admits $\lambda $-small $\kappa $-filtered colimits, which are preserved by the evaluation functors $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$ for $C \in \operatorname{\mathcal{C}}$. Using Remark 9.2.9.8 we see that the full subcategory $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is closed under the formation of $\lambda $-small $\kappa $-filtered colimits. Consequently, to show that $F$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, it will suffice to show that it is $(\kappa ,\lambda )$-compact when viewed as an object of the larger $\infty $-category $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. This follows from Proposition 9.2.9.16 (together with Warning 9.2.9.17). $\square$

Proof of Proposition 9.2.9.1. Let $\kappa \leq \lambda $ and $\mu $ be regular cardinals, where $\mu $ is uncountable, and let $\operatorname{\mathcal{D}}$ be the limit of a $\kappa $-small diagram

Then $\mathscr {F}$ is the covariant transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which is essentially $\mu $-small and $(\kappa ,\lambda )$-cocomplete (Example 7.6.6.33). Using Proposition 7.4.4.1, we can identify $\operatorname{\mathcal{D}}$ with the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$. Under this identification, each of the projection maps $\operatorname{\mathcal{D}}\rightarrow \mathscr {F}(C)$ corresponds to the functor

given by evaluation at $C$. Applying Corollary 9.2.9.19, we see that if an object $D \in \operatorname{\mathcal{D}}$ has $(\kappa ,\lambda )$-compact image in each $\mathscr {F}(C)$, then $D$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{D}}$. $\square$