### 5.1.5 Locally Cartesian Fibrations

It will sometimes be convenient to consider a generalization of Definition 5.1.4.1

Definition 5.1.5.1. Let $q: X \rightarrow S$ be a morphism of simplicial sets. We say that $q$ is a *locally cartesian fibration* if the following conditions are satisfied:

- $(1)$
The morphism $q$ is an inner fibration.

- $(2)$
For every edge $\overline{e}: s \rightarrow t$ of the simplicial set $S$ and every vertex $z \in X$ satisfying $q(z) = t$, there exists a locally $q$-cartesian edge $e: y \rightarrow z$ of $X$ satisfying $q( e) = \overline{e}$.

We say that $q$ is a *locally cocartesian fibration* if it satisfies condition $(1)$ together with the following dual version of $(2)$:

- $(2')$
For every edge $e: s \rightarrow t$ of the simplicial set $S$ and every vertex $y \in X$ satisfying $q(y) = s$, there exists a locally $q$-cocartesian edge $e: y \rightarrow z$ of $X$ satisfying $q( e ) = \overline{e}$.

Example 5.1.5.2. Let $q: X \rightarrow S$ be a morphism of simplicial sets. If $q$ is a cartesian fibration, then it is a locally cartesian fibration. If $q$ is a cocartesian fibration, then it is a locally cocartesian fibration.

Exercise 5.1.5.3. Let $Q$ be a partially ordered set, let $\operatorname{Chain}[Q]$ denote the collection of all finite nonempty linearly ordered subsets of $Q$ (Notation 3.3.2.1), and let $\mathrm{Max}: \operatorname{Chain}[Q] \rightarrow Q$ be the map which carries each element $S \in \operatorname{Chain}[Q]$ to the largest element of $S$.

Show that the induced map of nerves $\operatorname{N}_{\bullet }(\mathrm{Max}): \operatorname{N}_{\bullet }(\operatorname{Chain}[Q]) \rightarrow \operatorname{N}_{\bullet }(Q)$ is a locally cocartesian fibration.

Show that, if $Q = [n]$ for $n \geq 2$, the functor $\operatorname{N}_{\bullet }(\mathrm{Max}): \operatorname{N}_{\bullet }(\operatorname{Chain}[Q]) \rightarrow \operatorname{N}_{\bullet }(Q)$ is not a cocartesian fibration.

Warning 5.1.5.8. The collection of locally (co)cartesian fibrations is not closed under composition.

Proposition 5.1.5.9. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a locally cartesian fibration of simplicial sets and let $g: Y \rightarrow Z$ be an edge of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

The edge $g$ is $q$-cartesian.

For every $2$-simplex $\sigma $

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]

of $\operatorname{\mathcal{C}}$, the edge $f$ is locally $q$-cartesian if and only if the edge $h$ is locally $q$-cartesian.

For every $2$-simplex $\sigma $

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]

of $\operatorname{\mathcal{C}}$, if $f$ is locally $q$-cartesian, then $h$ is locally $q$-cartesian.

**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 5.1.1.12 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

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { [g] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

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

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]

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

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [dr]^{ [g] \circ } & \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \ar [rr]^{ [h] \circ } \ar [ur]^{ [f] \circ } & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) } \]

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.
$\square$

Corollary 5.1.5.10. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets. The following conditions are equivalent:

The morphism $q$ is a cartesian fibration.

Every locally $q$-cartesian edge of $X$ is $q$-cartesian.

For every $2$-simplex $\sigma $:

\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]_{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z } \]

of the simplicial set $X$, if $f$ and $g$ are locally $q$-cartesian, then $h$ is locally $q$-cartesian.

**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.
$\square$

Corollary 5.1.5.11. Let $p: X \rightarrow S$ be an inner fibration of simplicial sets. Then $p$ is a cartesian fibration if and only if every pullback $X \times _{S} \Delta ^ n \rightarrow \Delta ^ n$ is a cartesian fibration for $n \leq 2$.

Corollary 5.1.5.12. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets. The following conditions are equivalent:

- $(1)$
The morphism $q$ is a right fibration.

- $(2)$
For every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is a Kan complex.

- $(3)$
Every edge of $X$ is locally $q$-cartesian.

- $(4)$
Every edge of $X$ is $q$-cartesian.

**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 5.1.4.14. 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 5.1.4.14.
$\square$

Proposition 5.1.5.13. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets and let $f: K \rightarrow X$ be any morphism of simplicial sets. Then:

- $(1)$
The induced map $q': X_{/f} \rightarrow S_{/(q \circ f)}$ is a locally cartesian fibration of simplicial sets.

- $(2)$
An edge $e$ of $X_{/f}$ is locally $q'$-cartesian if and only if its image in $X$ is locally $q$-cartesian.

**Proof.**
As in the proof of Proposition 5.1.4.17, we factor $q'$ as a composition

\[ X_{/f} \xrightarrow {u} X \times _{S} S_{/(q \circ f)} \xrightarrow {v} S_{ /(q \circ f)}, \]

where $u$ is a cartesian fibration and every edge of $X_{/f}$ is $u$-cartesian (Proposition 5.1.4.14). 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.
$\square$

Proposition 5.1.5.14. Let $\kappa $ be an uncountable regular cardinal and let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets. Then $q$ is locally $\kappa $-small (in the sense of Definition 4.7.9.1) if and only if, for every vertex $s \in S$, the $\infty $-category $X_{s} = \{ s\} \times _{S} X$ is locally $\kappa $-small.

**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 Corollary 4.7.9.11, 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 $X_{0}$.
$\square$

Corollary 5.1.5.15. Let $\kappa $ be an uncountable regular cardinal and let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets. Then $q$ is essentially $\kappa $-small if and only if, for every vertex $s \in S$, the $\infty $-category $X_{s} = \{ s\} \times _{S} X$ is essentially $\kappa $-small.

**Proof.**
Combine Proposition 5.1.5.14 with Remark 4.7.9.4.
$\square$

Corollary 5.1.5.16. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of $\infty $-categories. 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.**
Combine Corollaries 5.1.5.15 and 4.7.9.6.
$\square$

Variant 5.1.5.17. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of $\infty $-categories and let $n \geq -1$ be an integer. Then $U$ is essentially $n$-categorical (in the sense of Definition 4.8.6.1) if and only if, for each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $(n-1)$-truncated (in the sense of Definition 4.8.2.1).

**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.
$\square$

Corollary 5.1.5.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories and let $n \geq -2$ be an integer. Then $U$ is locally $n$-truncated if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{\mathcal{E}}_{C}$ is $(n+1)$-truncated.

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

One advantage the theory of locally cartesian fibrations holds over the theory of cartesian fibrations is the following “fiberwise” existence criterion:

Proposition 5.1.5.19. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [dr]_{p} \ar [rr]^{r} & & Y \ar [dl]^{q} \\ & S & } \]

satisfying the following conditions:

The maps $p$ and $q$ are locally cartesian fibrations, and $r$ is an inner fibration.

The map $r$ carries locally $p$-cartesian edges of $X$ to locally $q$-cartesian edges of $Y$.

For every vertex $s$ of $S$, the induced map $r_{s}: X_{s} \rightarrow Y_{s}$ is a locally cartesian fibration.

Then $r$ is a locally cartesian fibration.

Warning 5.1.5.20. The analogue of Proposition 5.1.5.19 for cartesian fibrations is false.

**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

\[ d^{2}_0( \overline{\sigma } ) = r(g) \quad \quad d^{2}_1( \overline{\sigma } ) = \overline{h} \quad \quad q( \overline{\sigma } ) = s^{1}_0( q(\overline{h} ) ), \]

as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & r(y) \ar [dr]^{ r(g) } & \\ \overline{x} \ar [ur]^{\overline{f}} \ar [rr]^{\overline{h} } & & r(z). } \]

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

\[ d^{2}_0( \sigma ) = g \quad \quad d^{2}_2(\sigma ) = f \quad \quad r(\sigma ) = \overline{\sigma }, \]

as depicted in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{ g } & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z. } \]

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.
$\square$