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5.2.6 Locally Cartesian Fibrations

It sometimes convenient to consider a generalization of Definition 5.2.4.1

Definition 5.2.6.1. 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 5.2.6.2. 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.

Example 5.2.6.3. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $q$ is a locally cartesian fibration (in the sense of Definition 5.1.4.20) if and only if the induced morphism of simplicial sets $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a locally cartesian fibration (in the sense of Definition 5.2.6.1). Similarly, $q$ is a locally cocartesian fibration if and only if $\operatorname{N}_{\bullet }(q)$ is a cocartesian fibration of simplicial sets. See Corollary 5.2.2.2.

Remark 5.2.6.4. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is a locally cartesian fibration if and only if the opposite morphism $q^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ is a locally cocartesian fibration.

Remark 5.2.6.5. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d]^{q'} & X \ar [d]^{q} \\ S' \ar [r] & S. } \]

If $q$ is a locally cartesian fibration, then $q'$ is also a locally cartesian fibration (see Remark 5.2.1.9). If $q$ is a locally cocartesian fibration, then $q'$ is also a locally cocartesian fibration.

Remark 5.2.6.6. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets. The following conditions are equivalent:

  • The morphism $q$ is a locally cartesian fibration.

  • For every pullback diagram

    \[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{q'} \ar [r] & X \ar [d] \\ \Delta ^1 \ar [r] & S, } \]

    the morphism $q'$ is a locally cartesian fibration.

  • For every pullback diagram

    \[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{q'} \ar [r] & X \ar [d] \\ \Delta ^1 \ar [r] & S, } \]

    the morphism $q'$ is a cartesian fibration.

Remark 5.2.6.7. Let $p: X \rightarrow Y$ and $q: Y \rightarrow Z$ be morphisms of simplicial sets. If $p$ is a cartesian fibration and $q$ is a locally cartesian fibration, then the composition $q \circ p$ is a locally cartesian fibration. Moreover, an edge $e$ of $X$ is locally $(q \circ p)$-cartesian if and only if it is $p$-cartesian and $p(e)$ is locally $q$-cartesian of $Y$. To prove this, we can assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration (Remark 5.2.6.6), so the desired result follows from Proposition 5.2.4.11.

Warning 5.2.6.8. The collection of locally (co)cartesian fibrations is not closed under composition.

Proposition 5.2.6.9. 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 5.2.2.7 (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 5.2.1.10 and 5.2.3.6, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ n$ is a simplex. By virtue of Corollary 5.2.2.4, 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 5.2.2.3 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 5.2.6.10. 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 5.2.4.5, the implication $(2) \Rightarrow (1)$ is immediate, and the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 5.2.6.9. $\square$

Corollary 5.2.6.11. 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 5.2.6.12. 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 5.2.4.14, 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 5.2.4.12). The morphism $v$ is a pullback of $q$, and is therefore a locally cartesian fibration (Remark 5.2.6.5). 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 5.2.3.6). Assertions $(1)$ and $(2)$ now follow from Remark 5.2.6.7. $\square$

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

Proposition 5.2.6.13. 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.

Proof of Proposition 5.2.6.13. 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 5.2.6.6), so that $g$ is $p$-cartesian and $r(g)$ is $q$-cartesian (Remark 5.2.4.5). Applying Corollary 5.2.2.7, we conclude that $g$ is $r$-cartesian. It follows from Remark 5.2.3.6 that $f$ is locally $r$-cartesian, so that $h$ is locally $r$-cartesian by virtue of Proposition 5.2.3.8. $\square$

Proposition 5.2.6.15. 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 locally cartesian fibrations.

$(2)$

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 fully faithful.

Then $F$ is fully faithful 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 fully faithful.

Proof. The “only if” direction follows from Proposition 4.6.2.6. For the converse, assume that each of the functors $F_{D}$ is fully faithful; we will show that $F$ is fully faithful. Let $X$ and $Z$ be objects of $\operatorname{\mathcal{C}}$ having images $\overline{X}, \overline{Z} \in \operatorname{\mathcal{D}}$; we wish to show that the upper horizontal map in the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( F(X), F(Z) ) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Z} ) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}'}( \overline{F}( \overline{X} ), \overline{F}( \overline{Z} )) } \]

is a homotopy equivalence. Since $q$ and $q'$ are inner fibrations, the vertical maps are Kan fibrations (Proposition 4.6.1.19). Assumption $(3)$ guarantees that the lower horizontal map is a homotopy equivalence. By virtue of Proposition 3.2.7.1, it will suffice to show that for every morphism $\overline{e}: \overline{X} \rightarrow \overline{Z}$ in $\operatorname{\mathcal{D}}$, the induced map of fibers

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\overline{e}} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( F(X), F(Z) )_{ \overline{F}(\overline{e} ) } \]

is a homotopy equivalence.

Let $[\theta ]$ denote the homotopy class of $\theta $, regarded as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Since $q$ is a locally cartesian fibration, there exists a locally $q$-cartesian morphism $g: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}$ satisfying $q(g) = \overline{e}$. We then have a commutative diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{ \overline{X} }}(X,Y) \ar [d]^{ [g] \circ } \ar [r] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}'_{ \overline{F}(\overline{X})} }( F(X), F(Y) ) \ar [d]^{ [F(g)] \circ } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{ \overline{e} } \ar [r]^-{ [ \theta ] } & \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( F(X), F(Z) )_{ \overline{F}( \overline{e} ) } } \]

in $\mathrm{h} \mathit{\operatorname{Kan}}$, where the vertical maps are given by the composition law of Notation 5.2.3.11. Assumption $(2)$ guarantees that $F(g)$ is locally $q'$-cartesian, so that the vertical maps in this diagram are isomorphisms in $\mathrm{h} \mathit{\operatorname{Kan}}$ (Proposition 5.2.3.12). It will therefore suffice to show that the functor $F_{\overline{X}}$ induces a homotopy equivalence of mapping spaces $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{ \overline{X} }}(X,Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{C}}'_{ \overline{F}(\overline{X})} }( F(X), F(Y) )$, which follows from our assumption that $F_{ \overline{X} }$ is fully faithful. $\square$

Corollary 5.2.6.16. 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 locally cartesian isofibrations.

$(2)$

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.

Proof. If $F$ is an equivalence of $\infty $-categories, then Corollary 4.5.4.4 guarantees that each $F_{D}$ is an equivalence of $\infty $-categories. The converse follows by combining Proposition 5.2.6.15, Remark 4.6.2.16, and Theorem 4.6.2.17. $\square$

Corollary 5.2.6.17. 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 cartesian fibrations.

$(2)$

The functor $F$ carries $q$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to $q'$-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.

Proof. This is a special case of Corollary 5.2.6.16, since every cartesian fibration is a locally cartesian isofibration (Example 5.2.6.2 and Proposition 5.2.4.7). $\square$

Corollary 5.2.6.18. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{q} & \operatorname{\mathcal{C}}' \ar [d]^{q'} \\ \operatorname{\mathcal{D}}\ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}', } \]

where $q$ and $q'$ are right fibrations and the functor $\overline{F}$ 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 $F_{D}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}'_{D'}$ is a homotopy equivalence of Kan complexes.