Kerodon

Proof. Let $e: C \rightarrow C'$ be an edge of $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{E}}_{e}$ be the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, and regard $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}_{C'}$ as full subcategories of $\operatorname{\mathcal{E}}_{e}$. Since the projection map $\operatorname{\mathcal{E}}_{e} \rightarrow \Delta ^1$ is a cartesian fibration, the full subcategory $\operatorname{\mathcal{E}}_{C} \subseteq \operatorname{\mathcal{E}}_{e}$ is coreflective (Corollary 6.2.5.2) and therefore closed under the formation of colimits (Variant 7.1.4.26). By virtue of Proposition 9.4.5.11, it will suffice to show that the following conditions are equivalent:

$(b_ e)$

Every $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{E}}_{C}$ is also $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}_{e}$.

$(b'_{e})$

The contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-finitary.

Fix a regular cardinal $\mu \geq \lambda $ such that $\operatorname{\mathcal{E}}_{e}$ is locally $\mu $-small and let $X$ be a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{E}}_{C}$, and let $h^{X}: \operatorname{\mathcal{E}}_{e} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ be the functor corepresented by $X$. Using Remark 9.2.9.13, we see that $X$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}_{e}$ if and only if the restriction $h^{X}|_{ \operatorname{\mathcal{E}}_{C'} }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is $(\kappa ,\lambda )$-finitary. Identifying $h^{X}|_{ \operatorname{\mathcal{E}}_{C'} }$ with the composite functor $\operatorname{\mathcal{E}}_{C'} \xrightarrow { e^{\ast } } \operatorname{\mathcal{E}}_{C} \xrightarrow { h^{X}|_{ \operatorname{\mathcal{E}}_{C}} } \operatorname{\mathcal{S}}_{< \mu }$, the equivalence $(b_ e) \Leftrightarrow (b'_{e} )$ now follows from Corollary 9.4.1.17. $\square$

Proof. Let $e: C \rightarrow C'$ be an edge of $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{E}}_{e}$ be the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, and regard $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}_{C'}$ as full subcategories of $\operatorname{\mathcal{E}}_{e}$. Without loss of generality, we may assume that the cocartesian fibration $U$ is $(\kappa ,\lambda )$-cocomplete, so that the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ is $(\kappa ,\lambda )$-finitary (see Remark 9.2.9.8). By virtue of Proposition 9.4.5.11, it will suffice to show that the following conditions are equivalent:

$(b_{e} )$

Every $(\kappa ,\lambda )$-compact object $X \in \operatorname{\mathcal{E}}_{C}$ is also $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}_{e}$.

$(b'_{e} )$

For every $(\kappa ,\lambda )$-compact object $X \in \operatorname{\mathcal{E}}_{C}$, the image $e_{!}(X)$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}_{C'}$.

Fix a regular cardinal $\mu \geq \lambda $ such that $\operatorname{\mathcal{E}}_{e}$ is locally $\mu $-small and let $h^{X}: \operatorname{\mathcal{E}}_{e} \rightarrow \operatorname{\mathcal{S}}_{ < \mu }$ be the functor corepresented by $X$. Since the restriction $h^{X}|_{ \operatorname{\mathcal{E}}_{C'} }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is corepresented by the object $e_{!}(X)$, the equivalence $(b_{e}) \Leftrightarrow (b'_{e} )$ is a special case of Remark 9.2.9.13. $\square$

Proof. The implication $(2) \Rightarrow (1)$ follows immediately from Proposition 9.4.1.11. To prove the converse, let $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$ be the full simplicial subset of $\operatorname{\mathcal{E}}$ spanned by those vertices $X$ which are $(\kappa ,\lambda )$-compact when viewed as objects of $\{ U(X) \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. We wish to show that, if $U$ is $(\kappa ,\lambda )$-compactly generated, then the inclusion map $\operatorname{\mathcal{E}}_0 \hookrightarrow \operatorname{\mathcal{E}}$ exhibits $U$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of the inner fibration $U_0 = U|_{ \operatorname{\mathcal{E}}_0 }$. Fix an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}_0$. Replacing $\operatorname{\mathcal{C}}$ by $\Delta ^ n$, we can reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. In this case, we wish to show that $\operatorname{\mathcal{E}}$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of the full subcategory $\operatorname{\mathcal{E}}_0$. Since $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compactly generated, it will suffice to show that $\operatorname{\mathcal{E}}_0$ is the full subcategory of $\operatorname{\mathcal{E}}$ spanned by the $(\kappa ,\lambda )$-compact objects (Proposition 9.4.1.11), which was established in the proof of Proposition 9.4.5.11. $\square$

Proof. Assume first that the inner fibration $U$ is $(\kappa ,\lambda )$-compactly generated. Condition $(1)$ then follows from Example 9.4.5.6 and Remark 9.4.5.7. To verify condition $(2)$, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$ and that $e$ is the nondegenerate edge of $\operatorname{\mathcal{C}}$, so that $\operatorname{\mathcal{E}}\simeq \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compactly generated. It follows from Remark 9.2.9.13 that the inclusion map $\operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-finitary: that is, $\operatorname{\mathcal{E}}_{C}$ is closed under the formation of $\lambda $-small $\kappa $-filtered colimits in $\operatorname{\mathcal{E}}$. To verify condition $(2)$, it will suffice to show that every $(\kappa ,\lambda )$-compact object $X \in \operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{E}}$. Since $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compactly generated, we can realize $X$ as the colimit of a $\lambda $-small $\kappa $-filtered diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{E}}$. Since $X$ belongs to $\operatorname{\mathcal{E}}_{C}$, the diagram $F$ automatically factors through $\operatorname{\mathcal{E}}_{C}$. Our assumption that $X$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}_{C}$ then guarantees that it is a retract of $F(K)$ for some object $K \in \operatorname{\mathcal{K}}$. Since $F(K)$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}$ then guarantees that $X$ is also $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}$ (Remark 9.2.5.18).

We now prove the converse. Assume that $U$ satisfies conditions $(1)$ and $(2)$; we wish to show that $U$ is $(\kappa ,\lambda )$-compactly generated. Fix a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the fiber product $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compactly generated. Replacing $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{E}}_{\sigma }$, we may assume that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex, and we wish to show that $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compactly generated. Invoking Proposition 9.2.9.7, we deduce that the $\infty $-category $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-cocomplete, and that the inclusion functor $\operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-finitary for each $C \in \operatorname{\mathcal{C}}$. Fix an object $X \in \operatorname{\mathcal{E}}$ having image $C = U(X)$ in $\operatorname{\mathcal{C}}$. Our assumption that $\operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-compactly generated guarantees that we can realize $X$ as the colimit of a $\lambda $-small $\kappa $-filtered diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{E}}_{C}$ having the property that for each $K \in \operatorname{\mathcal{K}}$, $F(K)$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{E}}_{C}$. To complete the proof, it will suffice to show that $F(K)$ is also $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{E}}$. Fix a regular cardinal $\mu \geq \lambda $ such that $\operatorname{\mathcal{E}}$ is locally $\mu $-small, and let $h^{X}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ be the functor corepresented by $X$; we wish to show that $h^{X}$ is $(\kappa ,\lambda )$-finitary. By virtue of Remark 9.2.9.13, it will suffice to show that the restriction $h^{X}|_{ \operatorname{\mathcal{E}}_{C'} }$ for every object $C' \in \operatorname{\mathcal{C}}$. If $C \leq C'$ (as elements of the partially ordered set $[n] = \{ 0, 1, \cdots , n \} $), this follows from assumption $(2)$. Otherwise, $h^{X}|_{ \operatorname{\mathcal{E}}_{C'} }$ is the constant functor taking the value $\emptyset $, and the result follows from Example 9.2.2.12. $\square$

Proof. Assertion $(1)$ is a special case of Proposition 7.6.6.41, assertion $(2)$ follows from Proposition 9.2.9.16, and assertion $(3)$ is a reformulation of $(2)$ (Proposition 9.3.2.1). $\square$

Proof. We proceed as in the proof of Proposition 9.2.8.9. We say that a simplicial set $\operatorname{\mathcal{C}}'$ equipped with a map $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ is good if the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ is $(\kappa ,\lambda )$-compactly generated, and an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$ is $(\kappa ,\lambda )$-compact if and only if $F(C) \in \operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-compact for each vertex $C \in \operatorname{\mathcal{C}}'$. We will complete the proof by showing that if $\operatorname{\mathcal{C}}'$ is $\kappa $-small, then it 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). It follows from Corollary 9.4.4.7 that if each $\operatorname{sk}_{n}(\operatorname{\mathcal{C}}')$ is good, then $\operatorname{\mathcal{C}}'$ is good. 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). Our inductive hypothesis guarantees that $\operatorname{\mathcal{C}}''$ and $\coprod _{\sigma \in S} \operatorname{\partial \Delta }^ n$ are good. By virtue of Theorem 9.4.4.4, to prove that $\operatorname{\mathcal{C}}'$ is good, it will suffice to show that the disjoint union $\coprod _{\sigma \in S} \Delta ^ n$ is good. Using Proposition 9.4.3.13, can further reduce to the case where $\operatorname{\mathcal{C}}' = \Delta ^ n$ is the standard $n$-simplex.

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. For $n \geq 2$, this follows from our inductive hypothesis. 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 our assumption that the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-compactly generated. We may therefore assume that $n = 1$. Replacing $U$ by the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}'$, we are reduced to proving Theorem 9.4.5.13 in the special case where $\operatorname{\mathcal{C}}= \Delta ^1$ is the standard $1$-simplex. Set $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_{1} = \{ 1\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Then $\operatorname{Fun}_{ / \Delta ^1}( \Delta ^1, \operatorname{\mathcal{C}})$ can be identified with the oriented fiber product $\operatorname{\mathcal{E}}_0 \vec{\times }_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_1$, and the desired result follows from Theorem 9.4.3.4. $\square$

Proof. Apply Theorem 9.4.5.13 to the projection map $\operatorname{\mathcal{C}}\times K \rightarrow K$. $\square$

Proof. Choose a morphism of $\lambda $-smalll simplicial sets $T: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. If $T$ is surjective on edges, then we have a categorical pullback diagram of $\infty $-categories

Applying Theorems 9.4.4.4 and 9.4.5.13, we are reduced to proving Corollary 9.4.5.16 after replacing $U$ by the projection map $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$. We can therefore reduce to the case where the simplicial set $\operatorname{\mathcal{C}}$ is a $\kappa $-small disjoint union of edges. Using Proposition 9.4.3.13, we can further reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^1$ is the standard $1$-simplex. In this case, the evaluation functor $\operatorname{ev}_0: \operatorname{Fun}_{/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is an equivalence of $\infty $-categories (Proposition 5.3.1.21), which immediately implies $(1)$. To prove $(2)$, we must show that if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is a cocartesian section of $U$ for which $F(0) \in \{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compact, then $F(1) \in \{ 1\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is also $(\kappa ,\lambda )$-compact. This follows from the observation that the covariant transport functor $\{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \{ 1\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compact (see Corollary 9.4.5.9). $\square$

Proof. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration with covariant transport representation $\mathscr {F}$, so that we can identify the limit $\varprojlim (\mathscr {F})$ with the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$ (Proposition 7.4.4.1). The desired result is now a reformulation of Corollary 9.4.5.16. $\square$

Proof. We define a subcategory $\operatorname{\mathcal{Q}}\subseteq \operatorname{\mathcal{QC}}_{< \mu }$ as follows:

• A $\mu $-small $\infty $-category $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{Q}}$ if it is $(\kappa ,\lambda )$-compactly generated and $\operatorname{\mathcal{C}}_{< \kappa }$ is $\kappa _0$-sequentially cocomplete, where $\operatorname{\mathcal{C}}_{< \kappa }$ denotes the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $(\kappa ,\lambda )$-compact objects.

• If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$ are objects of $\operatorname{\mathcal{Q}}$, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ is a morphism of $\operatorname{\mathcal{Q}}$ if it is $(\kappa ,\lambda )$-compact and the induced map $\operatorname{\mathcal{C}}_{< \kappa } \rightarrow \operatorname{\mathcal{C}}'_{< \kappa '}$ preserves $\kappa _0$-sequential colimits.

It follows from Remark 9.4.2.16 that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ induces an equivalence from $\operatorname{\mathcal{QC}}^{ \kappa _0-\mathrm{seq} }_{< \mu }$ to $\operatorname{\mathcal{Q}}$. It will therefore suffice to show that the inclusion functor $\operatorname{\mathcal{Q}}\hookrightarrow \operatorname{\mathcal{QC}}_{< \mu }$ preserves $\kappa $-small limits, which is a reformulation of Corollary 9.4.5.17 (see Remark 9.4.5.19). $\square$