6.8.4 Locally Cartesian Fibrations
*** snip
Corollary 6.8.4.1. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets, let $K$ be a simplicial set, and let $q': S \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X) \rightarrow S$ be the projection map onto the first factor. Then:
- $(1)$
The morphism $q'$ is a locally cartesian fibration of simplicial sets.
- $(2)$
Let $e$ be an edge of the simplicial set $S \times _{ \operatorname{Fun}(B,S)} \operatorname{Fun}(B,X)$. Then $e$ is locally $q'$-cartesian if and only if, for every vertex $b \in B$, the image of $e$ under the evaluation functor $S \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}( \{ b\} , X) \simeq X$ is locally $q$-cartesian.
Proof.
By virtue of Remark 6.8.4.6, we may assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration. The desired result now follows by combining Theorem 5.2.1.1 with Remark 5.1.4.6.
$\square$
***
It sometimes convenient to consider a generalization of Definition 5.1.4.1
Definition 6.8.4.2. 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 6.8.4.3. 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.
Warning 6.8.4.8. The collection of locally (co)cartesian fibrations is not closed under composition.
Proposition 6.8.4.9. 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 5.1.2.7 (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 5.1.1.11 and 5.1.3.6, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ n$ is a simplex. By virtue of Corollary 5.1.2.4, 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 5.1.2.3 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 $d_0(\sigma ) = g$ and $d_2(\sigma ) = f$. Set $h= d_1(\sigma )$, 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 4.6.3.12, 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 6.8.4.10. 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 6.8.4.9.
$\square$
Corollary 6.8.4.11. 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 $n \leq 2$.
Proposition 6.8.4.12. 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 5.1.4.16, 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 6.8.4.5). 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 5.1.3.6). Assertions $(1)$ and $(2)$ now follow from Remark 6.8.4.7.
$\square$
One advantage the theory of locally cartesian fibrations holds over the theory of cartesian fibrations is the following “fiberwise” existence criterion:
Proposition 6.8.4.13. 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 6.8.4.14. The analogue of Proposition 6.8.4.13 for cartesian fibrations is false.
Proof of Proposition 6.8.4.13.
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
\[ d_0( \overline{\sigma } ) = r(g) \quad \quad d_1( \overline{\sigma } ) = \overline{h} \quad \quad q( \overline{\sigma } ) = s_0( q(\overline{h} ) ), \]
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
\[ d_0( \sigma ) = g \quad \quad d_2(\sigma ) = f \quad \quad r(\sigma ) = \overline{\sigma }, \]
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 6.8.4.6), so that $g$ is $p$-cartesian and $r(g)$ is $q$-cartesian (Remark 5.1.4.5). Applying Corollary 5.1.2.7, we conclude that $g$ is $r$-cartesian. It follows from Remark 5.1.3.6 that $f$ is locally $r$-cartesian, so that $h$ is locally $r$-cartesian by virtue of Proposition 5.1.3.8.
$\square$
*** work this in ***
Corollary 6.8.4.15. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^-{q} & \operatorname{\mathcal{C}}' \ar [d]^-{q'} \\ \operatorname{\mathcal{D}}\ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}'. } \]
Assume that:
- $(1)$
The functors $q$ and $q'$ are isofibrations.
- $(2)$
The isofibration $q$ is locally cartesian and the functor $F$ carries locally $q$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to locally $q'$-cartesian morphisms of $\operatorname{\mathcal{C}}'$.
- $(3)$
The functor $\overline{F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ is an equivalence of $\infty $-categories.
Then $F$ is an equivalence of $\infty $-categories if and only if, for every object $D \in \operatorname{\mathcal{D}}$ having image $D' = \overline{F}(D) \in \operatorname{\mathcal{D}}'$, the induced map of fibers $F_{D}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}'_{D'}$ is an equivalence of $\infty $-categories. Moreover, if this condition is satisfied, then $q'$ is also a locally cartesian fibration.
Proof.
If $F$ is an equivalence of $\infty $-categories, then Corollary 4.5.4.5 guarantees that each $F_{D}$ is an equivalence of $\infty $-categories. The converse follows by combining Proposition 5.1.5.7.
$\square$