Kerodon

Proof. The implication $(1) \Rightarrow (2)$ is immediate from the definitions. Assume next that $(2)$ is satisfied; we will prove $(3)$. 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 $\overline{e}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\overline{e}} = \{ \overline{e} \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} ) } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. This follows immediately from assumption $(2)$.

We now complete the proof by showing that $(3)$ 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. Assume first that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Applying Proposition 4.7.9.5, we deduce that $U$ is locally $\kappa $-small. It will therefore suffice to show that, for each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small (Remark 4.7.9.4). Using Corollary 4.5.2.23, we can factor $U$ as a composition $\operatorname{\mathcal{E}}\xrightarrow {\iota } \operatorname{\mathcal{E}}' \xrightarrow {U'} \operatorname{\mathcal{C}}$, where $U'$ is an isofibration and $\iota $ is an equivalence of $\infty $-categories. Then $\operatorname{\mathcal{E}}'$ is essentially $\kappa $-small, so Corollary 4.7.5.16 guarantees that the fiber $\operatorname{\mathcal{E}}'_{C}$ is essentially $\kappa $-small. Remark 4.5.2.24 guarantees that the map of fibers $\operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ is fully faithful, so $\operatorname{\mathcal{E}}_{C}$ is also essentially $\kappa $-small (Corollary 4.7.5.13).

Now suppose that $U$ is an isofibration which is essentially $\kappa $-small; we wish to show that the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Proposition 4.7.9.5 guarantees that $U$ is locally $\kappa $-small. It will therefore suffice to show that the set of isomorphism classes $\pi _0( \operatorname{\mathcal{E}}^{\simeq } )$ is $\kappa $-small. In fact, we will show that the core $\operatorname{\mathcal{E}}^{\simeq }$ is an essentially $\kappa $-small Kan complex. This is a special case of Corollary 4.7.7.2, since $U$ induces a Kan fibration $U^{\simeq }: \operatorname{\mathcal{E}}^{\simeq } \rightarrow \operatorname{\mathcal{C}}^{\simeq }$ whose fibers are essentially $\kappa $-small (see Proposition 4.4.3.7). $\square$

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 4.7.9.5 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 4.7.9.5), so that $V$ is automatically locally $\kappa $-small (by Proposition 4.7.9.5 again). $\square$

Proof. We proceed as in the proof of Corollary 4.7.9.9. Assume first 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 4.7.9.6 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.

Now suppose that $U \circ V$ is essentially $\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 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 particular, we may assume that $\operatorname{\mathcal{C}}$ is an essentially $\kappa $-small $\infty $-category and that the functors $U$ and $U \circ V$ are isofibrations (Example 4.4.1.6). Applying Corollary 4.7.9.6, we deduce that the $\infty $-categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are essentially $\kappa $-small, so that $V$ is also automatically essentially $\kappa $-small (by Corollary 4.7.9.6 again). $\square$

Proof. The “only if” direction is immediate from the definitions. To prove the converse, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex, in which case the desired result follows from Proposition 4.7.9.5. $\square$

Proof. Combine Corollary 4.7.9.11 with Remark 4.7.9.4. $\square$