Kerodon

Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 5.1.2.6 (and does not require the assumption that $q$ is a locally cartesian fibration), and the implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3) \Rightarrow (1)$. Using Remarks 5.1.1.13 and 5.1.3.5, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ n$ is a simplex. By virtue of Corollary 5.1.2.3, it will suffice to show that for each object $X \in \operatorname{\mathcal{C}}$ satisfying $q(X) \leq q(Y)$, the composition map

of Notation 4.6.9.15 is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Since $q$ is a locally cartesian fibration, we can choose a locally $q$-cartesian morphism $f: X \rightarrow Y$ satisfying $q(W) = q(X)$. Using the fact that $\operatorname{\mathcal{C}}$ is an $\infty $-category, we can choose a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ satisfying $d^{2}_0(\sigma ) = g$ and $d^{2}_2(\sigma ) = f$. Set $h= d^{2}_1(\sigma )$, so that we have a commutative diagram

in the $\infty $-category $\operatorname{\mathcal{C}}$. If assumption $(3)$ is satisfied, then $h$ is also a locally $q$-cartesian morphism of $\operatorname{\mathcal{C}}$. Invoking Proposition 4.6.9.12, we conclude that the diagram

commutes (in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). Since $f$ and $h$ are locally $q$-cartesian, the horizontal and left diagonal map in this diagram are isomorphisms (in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$), so the right diagonal map is an isomorphism as well. $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 5.1.4.5, the implication $(2) \Rightarrow (1)$ is immediate, and the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 5.1.5.9. $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 4.4.2.3. To show that $(2) \Rightarrow (3)$, we may assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration (Remark 5.1.5.6), so the desired result follows from Proposition 5.1.4.15. The implication $(3) \Rightarrow (4)$ follows from Corollary 5.1.5.10. If condition $(4)$ is satisfied, then $q$ is a cartesian fibration (Corollary 5.1.5.10), so that $(1)$ follows from Proposition 5.1.4.15. $\square$

Proof. As in the proof of Proposition 5.1.4.18, we factor $q'$ as a composition

where $u$ is a cartesian fibration and every edge of $X_{/f}$ is $u$-cartesian (Proposition 5.1.4.15). The morphism $v$ is a pullback of $q$, and is therefore a locally cartesian fibration (Remark 5.1.5.5). Moreover, an edge of $X \times _{S} S_{/(q \circ f)}$ is locally $v$-cartesian if and only if its image in $X$ is locally $q$-cartesian (Remark 5.1.3.5). Assertions $(1)$ and $(2)$ now follow from Remark 5.1.5.7. $\square$

Proof. Assume that, for every vertex $s \in S$, the $\infty $-category $X_{s}$ is locally $\kappa $-small; we will show that $q$ is locally $\kappa $-small (the reverse implication is immediate from the definitions). By virtue of Proposition 4.7.9.7, we may assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration and we wish to show that $X$ is locally $\kappa $-small. Fix a pair of objects $x,z \in X$; we wish to show that the Kan complex $\operatorname{Hom}_{ X }(x,z)$ is essentially $\kappa $-small. We may assume without loss of generality that $q(x) \leq q(z)$ (otherwise, the Kan complex $\operatorname{Hom}_{X}(x,z)$ is empty). If $q(x) = q(z)$, then the desired result follows from our hypothesis. It will therefore suffice to treat the case where $q(x) = 0$ and $q(z) = 1$. Since $q$ is a locally cartesian fibration, we can choose a $q$-cartesian morphism $f: y \rightarrow z$ of $X$, where $q(y) = 0$. In this case, composition with the homotopy class $[f]$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{X}(x,y) \rightarrow \operatorname{Hom}_{X}(x,z)$ (Corollary 5.1.2.3), so the desired result follows from the local $\kappa $-smallness of the $\infty $-category $X_{0}$. $\square$

Proof. Without loss of generality, we may assume that $S = \Delta ^ n$ is a standard simplex. Using Corollary 4.7.6.17, we can reduce to the case where $\kappa $ is regular. In this case, the result follows by combining Proposition 5.1.5.14 with Remark 4.7.9.6. $\square$

Proof. Combine Corollary 5.1.5.15, Corollary 4.7.9.12, and Example 4.7.9.5. $\square$

Proof. We proceed as in the proof of Proposition 5.1.5.14. Assume that, for each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ is locally $(n-1)$-truncated; we will show that the functor $U$ is essentially $n$-categorical (the reverse implication follows from Proposition 4.8.6.17, and does not require the assumption that $U$ is locally cartesian). By virtue of Proposition 4.8.5.27, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$. Fix a pair of objects $X,Z \in \operatorname{\mathcal{E}}$; we wish to show that the map of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) )$ is $n$-truncated. We may assume that $U(X) \leq U(Z)$ (otherwise, both Kan complexes are empty and there is nothing to prove); in this case, we wish to show that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z)$ is $n$-truncated (Example 3.5.9.4). If $U(X) = U(Z)$, then the desired result follows from our hypothesis on the fibers of $U$. It will therefore suffice to treat the case where $U(X) = 0$ and $U(Z) = 1$. Since $U$ is a locally cartesian fibration, we can choose a $U$-cartesian morphism $f: Y \rightarrow Z$ of $\operatorname{\mathcal{E}}$ satisfying $U(Y) = 0$. In this case, composition with the homotopy calss $[f]$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{E}}}( X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X, Z)$ (Corollary 5.1.2.3). It will therefore suffice to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is $n$-truncated, which follows from our assumption that the fiber $\operatorname{\mathcal{E}}_{0} = U^{-1} \{ 0\} $ is locally $n$-truncated. $\square$

Proof. Combine Variant 5.1.5.17 with Example 4.8.2.4. $\square$

Proof of Proposition 5.1.5.19. Choose a vertex $z \in X$ and an edge $\overline{h}: \overline{x} \rightarrow r(z)$ of the simplicial set $Y$. We wish to prove that there exists a locally $r$-cartesian edge $h: x \rightarrow z$ satisfying $r(h) = \overline{h}$. Since $p$ is a locally cartesian fibration, we can choose a locally $p$-cartesian edge $g: y \rightarrow z$ satisfying $p(g) = q(\overline{h})$. Assumption $(2)$ guarantees that $r(g)$ is locally $q$-cartesian, so we can choose a $2$-simplex $\overline{\sigma }$ of $Y$ satisfying

as indicated in the diagram

Set $s = q( \overline{x} )$, so that $\overline{f}$ can be regarded as an edge of the simplicial set $Y_{s}$. Invoking assumption $(3)$, we conclude that there exists a locally $r_{s}$-cartesian edge $f: x \rightarrow y$ of $X_{s}$ satisfying $r(f) = \overline{f}$. Since $r$ is an inner fibration, we can choose a $2$-simplex $\sigma $ of $X$ satisfying

as depicted in the diagram

We will complete the proof by showing that $h$ is locally $r$-cartesian. To prove this, we can replace $X$ and $Y$ by their pullbacks along the edge $\Delta ^1 \xrightarrow { q(\overline{h})} S$, and thereby reduce to the case $S = \Delta ^1$. In this case, the morphisms $p$ and $q$ are cartesian fibration (Remark 5.1.5.6), so that $g$ is $p$-cartesian and $r(g)$ is $q$-cartesian (Remark 5.1.4.5). Applying Corollary 5.1.2.6, we conclude that $g$ is $r$-cartesian. It follows from Remark 5.1.3.5 that $f$ is locally $r$-cartesian, so that $h$ is locally $r$-cartesian by virtue of Proposition 5.1.3.7. $\square$