# Kerodon

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### 9.3.6 Application: Flat Inner Fibrations

Recall that an inner fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is exponentiable if the pullback functor

$(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{ / \operatorname{\mathcal{E}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

preserves categorical equivalences of simplicial sets (Definition 4.5.9.10). In this section, we study a weaker version of this condition.

Definition 9.3.6.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. We say that $U$ is flat if, for every $2$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$, the inclusion map

$\Lambda ^{2}_{1} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \Delta ^2 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

is a categorical equivalence of simplicial sets.

Example 9.3.6.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. If $U$ is exponentiable, then it is flat.

Remark 9.3.6.3 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{F} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}, & }$

where the vertical maps are inner fibrations and $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$ (see Definition 5.1.7.1). Then $U$ is flat if and only if $U'$ is flat.

Remark 9.3.6.4. Suppose we are given a pullback diagram of simplicial sets

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

If $U$ is a flat inner fibration, then $U'$ is a flat inner fibration. The converse holds if $F$ is surjective on $2$-simplices.

Remark 9.3.6.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibraiton of simplicial sets. Then $U$ is flat if and only if the opposite inner fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is flat.

Our first goal in this section is to establish a weak converse to Example 9.3.6.2.

Theorem 9.3.6.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets. Then, for every Morita equivalence of simplicial sets $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ is also a Morita equivalence.

Corollary 9.3.6.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. The following conditions are equivalent:

$(1)$

The pullback functor

$(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{ / \operatorname{\mathcal{E}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

preserves Morita equivalences of simplicial sets. That is, if $F: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$ is a Morita equivalence in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$, then the induced map $F_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}'' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is also a Morita equivalence.

$(2)$

If $F: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$ is a categorical equivalence in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$, then $F_{\operatorname{\mathcal{E}}}$ is a Morita equivalence.

$(3)$

If $F: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$ is a categorical equivalence in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ which is surjective on vertices, then $F_{\operatorname{\mathcal{E}}}$ is a categorical equivalence.

$(4)$

The inner fibration $U$ is flat.

Proof. The implications $(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (4)$ are immediate from the definitions (see Corollary 9.3.5.9), and the implication $(4) \Rightarrow (1)$ is a reformulation of Theorem 9.3.6.6. $\square$

Corollary 9.3.6.8. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be flat inner fibrations of simplicial sets. Then the composite map $(U \circ V): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is also a flat inner fibration.

Our proof of Theorem 9.3.6.6 will make use of the following flatness criterion:

Proposition 9.3.6.9. Let $\kappa$ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is essentially $\kappa$-small, and suppose we are given a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$. Then $U$ is flat if and only if $\widehat{U}$ is a cartesian fibration.

Example 9.3.6.10. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{U} \ar [rr]^-{F} & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}, & }$

where $F$ is fully faithful and induces a Morita equivalence $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$, for each object $C \in \operatorname{\mathcal{C}}$. If $\kappa$ is an uncountable regular cardinal such that $U'$ is essentially $\kappa$-small, then any fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}'$ can also be regarded as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$ (see Proposition 9.3.5.10). Applying the criterion of Proposition 9.3.6.9, we conclude that $U$ is flat if and only if $U'$ is flat. In particular, an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is flat if and only if its fiberwise idempotent completion is flat.

The proof of Proposition 9.3.6.9 will require some preliminaries.

Lemma 9.3.6.11. Let $K$ be a simplicial set and let $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty$-category $\widehat{\operatorname{\mathcal{E}}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ admits $K$-indexed colimits. Let $\operatorname{\mathcal{D}}$ be an $\infty$-category which admits $K$-indexed colimits, let $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a diagram, and let $S$ be the collection of vertices $C \in \operatorname{\mathcal{C}}$ for which the functor $F_{C} = F|_{ \operatorname{\mathcal{E}}_{C} }$ preserves $K$-indexed colimits. Then $S$ is closed under retracts in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Proof. Using Corollary 5.6.7.3, we can reduce to the case where $\operatorname{\mathcal{C}}$ is an $\infty$-category. Fix objects $C, C' \in \operatorname{\mathcal{C}}$ such that $C$ is a retract of $C'$, and assume that $F_{C'}$ preserves $K$-indexed colimits; we wish to show that $F_{C}$ also preserves $K$-indexed colimits. Choose a colimit diagram $\overline{u}: K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$; we will show that $F_{C} \circ \overline{u}$ is a colimit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$. Set $u = \overline{u}|_{K}$. It follows from Theorem 5.2.1.1 that $\widehat{U}$ induces a cartesian fibration $\operatorname{Fun}( K, \widehat{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Applying Remark 8.5.1.23, we deduce that there is a diagram $u': K \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C'}$ having the property that $u$ is a retract of $u'$ in the $\infty$-category $\operatorname{Fun}(K, \widehat{\operatorname{\mathcal{E}}} )$. Since $\widehat{\operatorname{\mathcal{E}}}_{C'}$ admits $K$-indexed colimits, we can extend $u'$ to a colimit diagram $\overline{u}': K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C'}$. Since $\widehat{U}$ is a cartesian fibration, $\overline{u}$ and $\overline{u}'$ are $\widehat{U}$-colimit diagrams in the $\infty$-category $\widehat{\operatorname{\mathcal{E}}}$ (Corollary 7.3.3.23). Using Theorem 7.3.6.14, we see that any diagram which exhibits $u$ as a retract of $u'$ can be extended to a diagram which exhibits $\overline{u}$ as a retract of $\overline{u}'$. It follows that $F_{C} \circ \overline{u}$ is a retract of $F_{C'} \circ \overline{u}'$ (in the $\infty$-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{D}})$). Consequently, to show that $F_{C} \circ \overline{u}$ is a colimit diagram in $\operatorname{\mathcal{D}}$, it will suffice to show that $F_{C'} \circ \overline{u}'$ is a colimit diagram in $\operatorname{\mathcal{D}}$ (Corollary 8.5.1.12). This follows from our assumption that $F_{C'}$ preserves $K$-indexed colimits. $\square$

Lemma 9.3.6.12. Let $\kappa$ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is essentially $\kappa$-small, and suppose we are given a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$. Let $\overline{F}: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a Morita equivalence of simplicial sets. If $\widehat{U}$ is a cartesian fibration, then the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ is also a Morita equivalence of simplicial sets.

Proof. Set $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\widehat{\operatorname{\mathcal{E}}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$, so that $\overline{F}$ induces projection maps $F: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$ and $\widehat{F}: \widehat{\operatorname{\mathcal{E}}}' \rightarrow \widehat{\operatorname{\mathcal{E}}}$. We wish to show that, for every idempotent complete $\infty$-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces an equivalence of $\infty$-categories $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{D}})$. By virtue of Proposition 9.3.5.10, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is $\kappa$-cocomplete (in fact, we can assume that $\operatorname{\mathcal{D}}= \operatorname{\mathcal{S}}^{< \lambda }$ for some uncountable regular cardinal $\lambda \geq \kappa$). Let $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ spanned by those diagrams $G: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{D}}$ having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the functor $G_{C} = G|_{ \widehat{\operatorname{\mathcal{E}}}_{C} }$ preserves $\kappa$-small colimits, and define $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ similarly. We then have a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \ar [r]^-{ \circ \widehat{F} } \ar [d] & \operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \ar [r]^-{\circ F} & \operatorname{Fun}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{D}}), }$

where the vertical maps are equivalences by virtue of the universal property of fiberwise $\kappa$-cocompletions (Theorem 9.3.1.20). It will therefore suffice to show that precomposition with $\widehat{F}$ induces an equivalence of $\infty$-categories $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}})$. Since $\overline{F}$ is a Morita equivalence and $\widehat{U}$ is a cartesian fibration, the morphism $\widehat{F}$ is also a Morita equivalence (Proposition 9.3.5.11). In particular, precomposition with $\widehat{F}$ induces an equivalence of $\infty$-categories $\operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}})$. To complete the proof, it will suffice to show that an object $G \in \operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ belongs to the subcategory $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ if and only if $G \circ \widehat{F}$ belongs to $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}})$. This follows from Lemma 9.3.6.11. $\square$

Lemma 9.3.6.13. Let $\kappa$ be an uncountable regular cardinal, let $\operatorname{\mathcal{E}}$ be an $\infty$-category which is essentially $\kappa$-small, and suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \Delta ^2 & }$

which exhibits $\widehat{E}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$. The following conditions are equivalent:

$(1)$

The inclusion map $\Lambda ^{2}_{1} \times _{\Delta ^2} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets.

$(2)$

The functor $\widehat{U}$ is a cartesian fibration.

Proof. The implication $(2) \Rightarrow (1)$ follows from Lemma 9.3.6.12 and Corollary 9.3.5.9. Conversely, suppose that condition $(1)$ is satisfied. Set $\operatorname{\mathcal{E}}_0 = \Lambda ^{2}_{1} \times _{\Delta ^2} \operatorname{\mathcal{E}}$ and $\widehat{\operatorname{\mathcal{E}}}_0 = \Lambda ^{2}_{1} \times _{\Delta ^2} \widehat{\operatorname{\mathcal{E}}}$. Using Proposition 5.6.7.2, we can choose a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \widehat{\operatorname{\mathcal{E}}}_0 \ar [d]^{ \widehat{U}_0 } \ar [r] & \widehat{\operatorname{\mathcal{E}}}' \ar [d]^{ \widehat{U}' } \\ \Lambda ^{2}_{1} \ar [r] & \Delta ^2, }$

where $\widehat{U}'$ is a cartesian fibration. Let us abuse notation by identifying $\widehat{\operatorname{\mathcal{E}}}_0$ with its image in $\widehat{\operatorname{\mathcal{E}}}'$. It follows from Proposition 5.3.6.1 that the inclusion map $\widehat{\operatorname{\mathcal{E}}}_0 \hookrightarrow \widehat{\operatorname{\mathcal{E}}}'$ is a categorical equivalence of simplicial sets, so there exists a functor $F: \widehat{\operatorname{\mathcal{E}}}' \rightarrow \widehat{\operatorname{\mathcal{E}}}$ which is the identity on $\widehat{\operatorname{\mathcal{E}}}_0$. We will complete the proof by showing that $F$ is an equivalence of inner fibrations over $\Delta ^2$, so that $\widehat{U}$ is also a cartesian fibration (Proposition 5.1.7.14).

Note that the inner $\widehat{U}$ and $\widehat{U}'$ are $\kappa$-cocomplete. It will therefore suffice to show that, for every $\kappa$-cocomplete inner fibration $V: \operatorname{\mathcal{D}}\rightarrow \Delta ^2$, precomposition with $F$ induces an equivalence of $\infty$-categories

$\operatorname{Fun}^{\kappa }_{ / \Delta ^2 }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\kappa }_{ / \Delta ^2 }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}}).$

Since the inclusion $\widehat{\operatorname{\mathcal{E}}}_0 \hookrightarrow \widehat{\operatorname{\mathcal{E}}}'$ is a categorical equivalence, this is equivalent to the statement that the restriction map

$\operatorname{Fun}^{\kappa }_{ / \Delta ^2 }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\kappa }_{ / \Lambda ^2_1}( \widehat{\operatorname{\mathcal{E}}}_0, \operatorname{\mathcal{D}}_0 )$

is an equivalence, where $\operatorname{\mathcal{D}}_0 = \Lambda ^{2}_{1} \times _{\Delta ^2} \operatorname{\mathcal{D}}$. Invoking the universal property of fiberwise cocompletion (Theorem 9.3.1.20), we are reduced to showing that the restriction map $\operatorname{Fun}_{ / \Delta ^2 }( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{ / \Delta ^2 }( \Lambda ^{2}_{1} \times _{\Delta ^2}, \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}})$ is an equivalence of $\infty$-categories, which follows immediately from assumption $(1)$. $\square$

Proof of Proposition 9.3.6.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially $\kappa$-small inner fibration and let

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& }$

be a diagram which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$. It follows from Corollary 5.1.5.11 that $\widehat{U}$ is a cartesian fibration if and only if, for every $2$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$, the projection map $\widehat{U}_{\sigma }: \Delta ^2 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}} \rightarrow \Delta ^2$ is a cartesian fibration. By virtue of Lemma 9.3.6.13, this is equivalent to the requirement that $U$ is flat. $\square$

Proof of Theorem 9.3.6.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets and let $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a Morita equivalence; we wish to show that the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ is also a Morita equivalence. Choose an uncountable regular cardinal $\kappa$ such that $U$ is essentially $\kappa$-small. Using Theorem 9.3.4.1, we can choose a diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \Delta ^1 & }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$. By virtue of Lemma 9.3.6.12, it will suffice to show that $\widehat{U}$ is a cartesian fibration, which is a reformulation of our assumption that $U$ is flat (Proposition 9.3.6.9). $\square$

We record another consequence of Proposition 9.3.6.9.

Corollary 9.3.6.14. Every functor of $\infty$-categories $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is a flat inner fibration.

Proof. Choose an uncountable regular cardinal $\kappa$ for which $\operatorname{\mathcal{E}}$ is essentially $\kappa$-small. Using Theorem 9.3.4.1, we can choose a diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \Delta ^1 & }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$. It follows from Proposition 9.3.1.11 that $\widehat{U}$ is a cartesian fibration, so that $U$ is flat by virtue of Proposition 9.3.6.9. $\square$

Remark 9.3.6.15. By definition, an inner fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is flat if, for every $2$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$, the inclusion map

$\Lambda ^{2}_{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \Delta ^2 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

is a categorical equivalence of simplicial sets. It follows from Corollary 9.3.6.14 that it suffices to check this condition in the case where $\sigma$ is nondegenerate.

Corollary 9.3.6.16. A functor of $\infty$-categories $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ is a flat inner fibration if and only if the inclusion map $\Lambda ^{2}_{1} \times _{\Delta ^2} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets.

Proposition 9.3.6.17. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty$-categories. Then $U$ is exponentiable if and only if it is a flat isofibration.

Proof. Assume that $U$ is a flat isofibration; we will show that $U$ is exponentiable (the converse follows from Remark 4.5.9.12 and Example 9.3.6.2). Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}'' \ar [r]^-{ F} \ar [d] & \operatorname{\mathcal{E}}_0 \ar [r]^-{G} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}_0 \ar [r]^-{ \overline{G} } & \operatorname{\mathcal{C}}}$

in which both squares are pullbacks, where $\overline{F}$ is a categorical equivalence; we wish to show that $F$ is also a categorical equivalence. Using Proposition 4.1.3.2, we can factor $\overline{G}$ as a composition $\operatorname{\mathcal{C}}_0 \xrightarrow {\iota _0} \operatorname{\mathcal{C}}_{0}^{+} \xrightarrow {V_0} \operatorname{\mathcal{C}}$, where $\iota _0$ is inner anodyne and $V_0$ is an inner fibration. In particular, $\operatorname{\mathcal{C}}^{+}_{0}$ is an $\infty$-category. Replacing $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}^{+}_{0}$ (and $U$ by the projection map $\operatorname{\mathcal{C}}_0^{+} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}^{+}_{0}$), we can assume that $\overline{G}$ is inner anodyne. In this case, Corollary 9.3.6.7 guarantees that $G$ is a categorical equivalence of simplicial sets. It will therefore suffice to show that the composition $(G \circ F): \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence. Applying Proposition 4.1.3.2 again, we can factor $\overline{G} \circ \overline{F}$ as a composition $\operatorname{\mathcal{C}}_1 \xrightarrow {\iota _1} \operatorname{\mathcal{C}}_{1}^{+} \xrightarrow {V_{1}} \operatorname{\mathcal{C}}$, where $\iota _1$ is inner anodyne and $V_1$ is an inner fibration. Applying Corollary 9.3.6.7 again, we conclude that the inclusion map $\operatorname{\mathcal{E}}_1 \hookrightarrow \operatorname{\mathcal{C}}_{1}^{+} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets. We are therefore reduced to showing that the projection map $\operatorname{\mathcal{C}}_{1}^{+} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ is an equivalence of $\infty$-categories. Note that $\overline{F}$, $\overline{G}$, and $\iota _{1}$ are categorical equivalences of simplicial sets, so the inner fibration $V_1$ is an equivalence of $\infty$-categories. The desired result now follows from Corollary 4.5.2.29 (since $U$ is an isofibration). $\square$

Warning 9.3.6.18. In the formulation of Proposition 9.3.6.17, the assumption that $U$ is an isofibration cannot be omitted. For example, let $\operatorname{\mathcal{C}}$ be a contractible Kan complex containing two vertices $C_0$ and $C_1$, let $\operatorname{\mathcal{D}}_1$ be an $\infty$-category which is the idempotent completion of a full subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}_1$, and let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}_1$ be the full subcategory spanned by those objects $(C_ i, D)$ where $D$ is contained in $\operatorname{\mathcal{D}}_ i$. It follows from Example 9.3.6.10 that the projection map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a flat inner fibration. However, if $\operatorname{\mathcal{D}}_0$ is not idempotent complete, then $U$ is not an isofibration.

Corollary 9.3.6.19. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. Then $U$ is flat if and only if, for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the projection map $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$ is exponentiable. Moreover, it suffices to verify this condition in the case $n = 2$.

We now show that the counterexample of Warning 9.3.6.18 is essentially the only way that a flat inner fibration can fail to be exponentiable.

Lemma 9.3.6.20. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{F} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{ U' } \\ & \operatorname{\mathcal{C}}, & }$

where $U$ is a flat inner fibration, $U'$ is an inner fibration, and $F$ is an equivalence of $\infty$-categories. Then:

$(1)$

For every object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ is a Morita equivalence.

$(2)$

The inner fibration $U'$ is flat.

Note that, in the statement of Lemma 9.3.6.20, we do not assume that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$ (otherwise, there would be nothing to prove).

Proof of Lemma 9.3.6.20. We will prove $(1)$; assertion $(2)$ then follows from Example 9.3.6.10. Using Corollary 4.5.2.23, we can factor the inclusion map $\{ C\} \hookrightarrow \operatorname{\mathcal{C}}$ as a composition $\{ C\} \hookrightarrow \widetilde{\operatorname{\mathcal{C}}} \xrightarrow {V} \operatorname{\mathcal{C}}$, where $V$ is an isofibration and $\widetilde{\operatorname{\mathcal{C}}}$ is a contractible Kan complex. Set $\widetilde{\operatorname{\mathcal{E}}} = \widetilde{\operatorname{\mathcal{C}}} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\widetilde{\operatorname{\mathcal{E}}}' = \widetilde{\operatorname{\mathcal{C}}} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$, so that we have a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{C} \ar [r]^-{F_{C}} \ar [d] & \operatorname{\mathcal{E}}'_{C} \ar [d] \\ \widetilde{\operatorname{\mathcal{E}}} \ar [r]^-{\widetilde{F}} & \widetilde{\operatorname{\mathcal{E}}}'. }$

It follows from Theorem 9.3.6.6 that the vertical maps are Morita equivalences, and from Corollary 4.5.2.29 that $\widetilde{F}$ is an equivalence of $\infty$-categories. Applying Remark 9.3.5.5, we conclude that $F_{C}$ is a Morita equivalence. $\square$

Warning 9.3.6.21. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{F} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{ U' } \\ & \operatorname{\mathcal{C}}, & }$

where $U$ and $U'$ are inner fibrations and $F$ is an equivalence of $\infty$-categories (but not necessarily an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$). Lemma 9.3.6.20 asserts that if $U$ is flat, then $U'$ is also flat. Beware that the converse generally does not hold: if $U'$ is flat, then $U$ need not be flat. For example, suppose that $U'$ is an isomorphism and that $F: \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}'$ is an isomorphism from $\operatorname{\mathcal{E}}$ to a full subcategory of $\operatorname{\mathcal{E}}'$. If $F$ is essentially surjective, then it is an equivalence of $\infty$-categories. In this case, the inner fibration $U = U' \circ F$ is flat if and only if $F$ is an isomorphism.

Proposition 9.3.6.22. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories. Then $U$ factors as a composition $\operatorname{\mathcal{E}}\xrightarrow {F} \operatorname{\mathcal{E}}' \xrightarrow {U'} \operatorname{\mathcal{C}}$, where $F$ is an equivalence of $\infty$-categories and $U'$ is a flat isofibration. Moreover, for each object $C \in \operatorname{\mathcal{C}}$, the induced map $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ is a Morita equivalence.

Corollary 9.3.6.23. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories. Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty$-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is idempotent complete. Then $U$ is an isofibration. In particular, $U$ is exponentiable.

Proof. Using Proposition 9.3.6.22, we can factor $U$ as a composition $\operatorname{\mathcal{E}}\xrightarrow {F} \operatorname{\mathcal{E}}' \xrightarrow {U'} \operatorname{\mathcal{C}}$, where $F$ is an equivalence of $\infty$-categories and $U'$ is an isofibration. For each object $C \in \operatorname{\mathcal{C}}$, Lemma 9.3.6.20 guarantees that the induced map $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ is a Morita equivalence; in particular, every object of $\operatorname{\mathcal{E}}'_{C}$ is a retract of $F(X)$, for some object $X \in \operatorname{\mathcal{E}}_{C}$ (see Proposition 9.3.5.8). Since $\operatorname{\mathcal{E}}_{C}$ is idempotent complete, it follows that $F_{C}$ is an equivalence of $\infty$-categories. Applying Corollary 5.1.7.10, we conclude that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$, so that $U$ is also an isofibration (Proposition 5.1.7.14). $\square$

We now extend some of the preceding results to the case of inner fibrations between simplicial sets.

Proposition 9.3.6.24. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset which contains every vertex of $\operatorname{\mathcal{C}}$, and let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be a flat inner fibration. If the inclusion map $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is inner anodyne, then there exists a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [d]^{U_0} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}_0 \ar [r] & \operatorname{\mathcal{C}}, }$

where $U$ is a flat inner fibration.

Proof. Choose an uncountable regular cardinal $\kappa$ for which the inner fibration $U_0$ is essentially $\kappa$-small. Using Theorem 9.3.4.1, we can choose a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [rr]^-{H_0} \ar [dr]_{U_0} & & \widehat{\operatorname{\mathcal{E}}}_0 \ar [dl]^{ \widehat{U}_0 } \\ & \operatorname{\mathcal{C}}& }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}_0$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}_0$. Let $\operatorname{\mathcal{E}}'_0$ be the full simplicial subset of $\widehat{\operatorname{\mathcal{E}}}_0$ spanned by those vertices which belong to the image of $H_0$. It follows from Remark 9.3.1.15 that $H_0$ induces an equivalence $\operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$ of inner fibrations over $\operatorname{\mathcal{C}}_0$. Using Lemma 5.6.7.1 (and Remark 9.3.6.3), we can replace $\operatorname{\mathcal{E}}_0$ by $\operatorname{\mathcal{E}}'_0$ and thereby reduce to the case where $H_0$ is an isomorphism from $\operatorname{\mathcal{E}}_0$ onto a full simplicial subset of $\widehat{\operatorname{\mathcal{E}}}_0$.

Since $U_0$ is flat, the morphism $\widehat{U}_0$ is a cartesian fibration (Proposition 9.3.6.9). Using Proposition 5.6.7.2, we can choose a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \widehat{\operatorname{\mathcal{E}}}_0 \ar [r] \ar [d]^{ \widehat{U}_0 } & \widehat{\operatorname{\mathcal{E}}} \ar [d]^{ \widehat{U} } \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{ \iota } & \operatorname{\mathcal{C}}}$

where $\widehat{U}$ is a cartesian fibration. Let $\operatorname{Tr}_{\widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}$ be the homotopy transport representation of $\widehat{U}$. Since $\iota$ induces an isomorphism of homotopy categories $\mathrm{h} \mathit{ \operatorname{\mathcal{C}}_0 } \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, Proposition 9.3.1.16 guarantees that the functor $\operatorname{Tr}_{\widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}}$ carries each object of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to a $\kappa$-cocomplete $\infty$-category and each morphism of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to a functor which preserves $\kappa$-small colimits.

Let $\operatorname{\mathcal{E}}\subseteq \widehat{\operatorname{\mathcal{E}}}$ be the full simplicial subset spanned by those vertices which belong to the image of $\operatorname{\mathcal{E}}_0$, and set $U = \widehat{U}|_{\operatorname{\mathcal{E}}}$. Applying the criterion of Proposition 9.3.1.16, we see that the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr] \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& }$

exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}$. Since $\widehat{U}$ is a cartesian fibration, Proposition 9.3.6.9 guarantees that the inner fibration $U$ is flat. By construction, the isomorphism $\widehat{\operatorname{\mathcal{E}}}_0 \xrightarrow {\sim } \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ restricts to an isomorphism of $\operatorname{\mathcal{E}}_0$ with the fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. $\square$

Exercise 9.3.6.25. In the special case $\operatorname{\mathcal{C}}= \Delta ^2$ and $\operatorname{\mathcal{C}}_0 = \Lambda ^{2}_{1}$, Proposition 9.3.6.24 reduces to the assertion that every inner fibration $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \Lambda ^{2}_{1}$ fits into a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r]^-{F} \ar [d]^{U_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Lambda ^{2}_{1} \ar [r] & \Delta ^2, }$

where $\operatorname{\mathcal{E}}$ is an $\infty$-category and $F$ is a categorical equivalence of simplicial sets (see Corollary 9.3.6.16). Use the small object argument to give a more direct proof of this statement.

Corollary 9.3.6.26. Let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be a flat inner fibration of simplicial sets. Then there exists a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{U_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{\iota } & \operatorname{\mathcal{C}}, }$

where $U$ is a flat inner fibration of $\infty$-categories. Moreover, we may assume that $\iota$ is inner anodyne.

Corollary 9.3.6.27. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets. Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty$-category $\operatorname{\mathcal{E}}_{C}$ is idempotent complete. Then $U$ is an isofibration.

Corollary 9.3.6.28. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets and suppose we are given a digram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{\widehat{U}} \\ & \operatorname{\mathcal{C}}& }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise idempotent completion of $\operatorname{\mathcal{E}}$. Then $\widehat{U}$ is a flat isofibration.

Proof. It follows from Example 9.3.6.10 that $\widehat{U}$ is a flat inner fibration. Since the fibers of $\widehat{U}$ are idempotent complete, it is an isofibration (Corollary 9.3.6.27). $\square$

Proposition 9.3.6.29. Let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be a flat isofibration of simplicial sets. Then there exists a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{U_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{\iota } & \operatorname{\mathcal{C}}, }$

where $U$ is a flat isofibration of $\infty$-categories. Moreover, we may assume that $\iota$ is inner anodyne.

Proof. Using Corollary 9.3.6.26, we can choose an inner anodyne map $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ and a pullback diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{U_0} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{\iota } & \operatorname{\mathcal{C}}, }$

where $U'$ is a flat isofibration. Using Proposition 9.3.6.22, we can factor $U'$ as a composition $\operatorname{\mathcal{E}}' \xrightarrow {F} \operatorname{\mathcal{E}}'' \xrightarrow {U''} \operatorname{\mathcal{C}}$, where $F$ is an equivalence of $\infty$-categories and $U''$ is a flat isofibration. Set $\operatorname{\mathcal{E}}''_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}''$, so that we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{F_0} & \operatorname{\mathcal{E}}' \ar [d]^{F} \\ \operatorname{\mathcal{E}}''_0 \ar [r] & \operatorname{\mathcal{E}}''. }$

Since $U'$ and $U''$ are flat, Proposition 9.3.6.6 guarantees that the horizontal maps are categorical equivalences. Since $F$ is an equivalence of $\infty$-categories, it follows that $F_0$ is a categorical equivalence of simplicial sets (Remark 4.5.3.5), and therefore an equivalence of isofibrations over $\operatorname{\mathcal{C}}_0$ (Proposition 5.1.7.5). Applying Lemma 5.6.7.1, we conclude that there exists a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [d] \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{G} & \operatorname{\mathcal{E}}'' \ar [d]^{U''} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{\iota } & \operatorname{\mathcal{C}}\ar [r]^-{\operatorname{id}} & \operatorname{\mathcal{C}}, }$

where the left square is a pullback, the upper horizontal composition is $F_0$, and $G$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$. Since $U''$ is a flat isofibration, it follows that $U$ is also a flat isofibration (Remark 9.3.6.3 and Proposition 5.1.7.14). $\square$

Corollary 9.3.6.30. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat isofibration of simplicial sets. Then $U$ is exponentiable.

Proof. By virtue of Proposition 9.3.6.29, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty$-category. In this case, the result follows from Proposition 9.3.6.17. $\square$