Kerodon

Proof. Assume that $(2)$ is satisfied; we will prove $(1)$ (the reverse implication is immediate from the definitions). Fix an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small: that is, that the morphism space $M = \operatorname{Hom}_{\operatorname{\mathcal{E}}_{\sigma }}(X,Y)$ is essentially $\kappa $-small for every pair of objects $X,Y \in \operatorname{\mathcal{E}}_{\sigma }$. Let $i,j \in [n]$ be the images of $X$ and $Y$ under the projection map $\operatorname{\mathcal{E}}_{\sigma } \rightarrow \Delta ^ n$. We may assume without loss of generality that $i \leq j$ (otherwise, the morphism space $M$ is empty). In this case, $M$ can also be identified with a morphism space in the $\infty $-category $\operatorname{\mathcal{E}}_{e}$, where $e$ is the edge of $\operatorname{\mathcal{C}}$ given by the composite map $\Delta ^{1} \xrightarrow { (i,j) } \Delta ^ n \xrightarrow {\sigma } \operatorname{\mathcal{C}}$, so the desired result follows from assumption $(2)$. $\square$

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^ n$ has only finitely many edges. By virtue of Corollary 4.7.6.17, we can reduce to the case where $\kappa $ is regular. In this case, the desired result follows from Proposition 4.7.9.7 and Remark 4.7.9.6. $\square$

Proof. By virtue of Proposition 5.3.8.11, we may assume that the inner fibration $U$ is minimal. Then, for every $n$-simplex $\Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a minimal $\infty $-category which is essentially $\lambda $-small, and is therefore $\lambda $-small (Corollary 4.7.6.12). Applying Remark 4.7.4.9, we conclude that $\operatorname{\mathcal{E}}$ is $\lambda $-small (and therefore essentially $\lambda $-small). $\square$

Proof. As in the proof of Proposition 5.3.8.11, we may assume that $U$ is minimal, which guarantees that $\operatorname{\mathcal{E}}$ is $\lambda $-small. Applying Corollary 4.7.4.19, we conclude that $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is $\lambda $-small. In particular, the simplicial subset $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is $\lambda $-small, and therefore essentially $\lambda $-small. $\square$

Proof. Using Proposition 4.7.6.15, we can choose an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}'$ is minimal. Since $U$ is an isofibration, Corollary 4.5.2.29 guarantees that the $\infty $-category $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is equivalent to $\operatorname{\mathcal{E}}$. We may therefore replace $U$ by the projection map $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$, and thereby reduce to the situation where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}'$ is minimal. In this case, $\operatorname{\mathcal{C}}$ is $\kappa $-small (Corollary 4.7.6.12), so the desired result follows from Proposition 4.7.9.10. $\square$

Proof. Assume first that the inner fibration $U \circ V$ is essentially small, and choose an $n$-simplex $\widetilde{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$; we wish to show that the fiber product $\Delta ^ n \times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. To prove this, we are free to replace $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ by $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, respectively, and thereby reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$. In this case, the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are essentially $\kappa $-small, so the desired result follows from Example 4.7.9.5.

We now prove the converse. Assume that $\kappa $ is regular, that $V$ is an essentially $\kappa $-small isofibration, and choose an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. This follows by applying Corollary 4.7.9.12 to the isofibration of $\infty $-categories $(\operatorname{id}\times V): \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, which is essentially $\kappa $-small by virtue of Remark 4.7.9.3. $\square$

Proof. Assume first that $(1)$ is satisfied; we will prove $(2)$. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{E}}$, and set $\overline{X} = U(X)$ and $\overline{Y} = U(Y)$. We wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. By virtue of Proposition 4.6.1.21, the functor $U$ induces a Kan fibration $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$. Our assumption that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small guarantees that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$ is essentially $\kappa $-small. By virtue of Corollary 4.7.7.2, it will suffice to show that for every morphism $e: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{e} = \{ e \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} ) } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. This follows immediately from the local $\kappa $-smallness of the $\infty $-category $\operatorname{\mathcal{E}}_{e} = \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

We now show that $(2)$ implies $(1)$. Assume that $\operatorname{\mathcal{E}}$ is locally $\kappa $-small and choose a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we will show that the $\infty $-category $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small. Fix a pair of objects $\widetilde{X}, \widetilde{Y} \in \operatorname{\mathcal{E}}_{\sigma }$; we wish to show that the Kan complex $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{\sigma } }( \widetilde{X}, \widetilde{Y} )$ is essentially $\kappa $-small. Let $X$ and $Y$ denote the images of $\widetilde{X}$ and $\widetilde{Y}$ in the $\infty $-category $\operatorname{\mathcal{E}}$, and set $\overline{X} = U(X)$ and $\overline{Y} = U(Y)$ as above. If the Kan complex $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{\sigma } }( \widetilde{X}, \widetilde{Y} )$ is nonempty, then it can be identified with a fiber of the Kan fibration $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$, which is essentially $\kappa $-small by virtue of Corollary 4.7.7.2. $\square$

Proof. Suppose first that $V$ is locally $\kappa $-small. Choose an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small. This follows by applying Proposition 4.7.9.15 to the inner fibration of $\infty $-categories $(\operatorname{id}\times V): \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, which is locally $\kappa $-small by virtue of Remark 4.7.9.3.

Now suppose that $U \circ V$ is locally $\kappa $-small, and choose an $n$-simplex $\widetilde{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$; we wish to show that the fiber product $\Delta ^ n \times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small. To prove this, we are free to replace $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ by $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and $\Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, respectively, and thereby reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$ is a locally $\kappa $-small $\infty $-category. In this case, our assumptions on $U$ and $U \circ V$ guarantee that the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are also locally $\kappa $-small (Proposition 4.7.9.15), so that $V$ is automatically locally $\kappa $-small (by Proposition 4.7.9.15 again). $\square$