# Kerodon

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Corollary 5.4.8.11. Let $\kappa$ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and suppose that $\operatorname{\mathcal{C}}$ is essentially $\kappa$-small. The following conditions are equivalent:

$(1)$

The $\infty$-category $\operatorname{\mathcal{E}}$ is essentially $\kappa$-small.

$(2)$

For every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty$-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa$-small.

Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ is an $\infty$-category. Using Proposition 5.7.7.2, we can write $U$ as the pullback of a cocartesian fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$. Proposition 5.3.6.1 then guarantees that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}'$ is a categorical equivalence of simplicial sets. Since the inclusion $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$ is bijective on vertices, every fiber of $U'$ can also be regarded as a fiber of $U$. We can therefore replace $U$ by $U'$, and thereby reduce to proving Corollary 5.4.8.11 in the special case where $U$ is a cocartesian fibration of $\infty$-categories.

Note that $U$ is an isofibration (Proposition 5.1.4.8). Consequently, for each object $C \in \operatorname{\mathcal{C}}$, the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{C} \ar [r] \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \{ C\} \ar [r] & \operatorname{\mathcal{C}}}$

is a categorical pullback diagram of simplicial sets (Corollary 4.5.2.21). The implication $(1) \Rightarrow (2)$ now follows from Corollary 5.4.5.16 (and does not require the assumption that $\kappa$ is regular). To prove the reverse implication, we first note that the $\infty$-category $\operatorname{\mathcal{C}}$ and each fiber $\operatorname{\mathcal{E}}_{C}$ are locally $\kappa$-small (Example 5.4.8.4). Applying Proposition 5.4.8.7, we see that $\operatorname{\mathcal{E}}$ 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 (Proposition 5.4.8.8). The functor $U$ induces a map $\theta : \pi _0( \operatorname{\mathcal{E}}^{\simeq } ) \rightarrow \pi _0( \operatorname{\mathcal{C}}^{\simeq } )$, whose target is $\kappa$-small. Invoking the regularity of $\kappa$, we are reduced to showing that for every element $[C] \in \pi _0( \operatorname{\mathcal{C}}^{\simeq } )$, the inverse image $\theta ^{-1} \{ [C] \}$ is a $\kappa$-small set. Let us identify $[C]$ with the isomorphism class of an object $C \in \operatorname{\mathcal{C}}$. Then there is a surjective map $\pi _0( \operatorname{\mathcal{E}}_{C}^{\simeq } ) \twoheadrightarrow \theta ^{-1} \{ [C] \}$. Since the $\infty$-category $\operatorname{\mathcal{E}}_{C}$ is essentially $\kappa$-small, the set $\pi _0( \operatorname{\mathcal{E}}_{C}^{\simeq } )$ is $\kappa$-small, so that the quotient $\theta ^{-1} \{ [C] \}$ is also $\kappa$-small (Remark 5.4.3.4). $\square$