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4.5.9 Direct Image Fibrations

Suppose we are given morphisms of simplicial sets $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, where $V$ is an inner fibration. For every vertex $C \in \operatorname{\mathcal{C}}$, set $\operatorname{\mathcal{D}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, so that $V$ restricts to an inner fibration $V_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{D}}_{C}$. Sections of the inner fibration $V_{C}$ are parametrized by the simplicial set

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{C} }( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ) = \operatorname{Fun}( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{D}}_{C} ) } \{ \operatorname{id}_{ \operatorname{\mathcal{D}}_{C} } \} , \]

which is an $\infty $-category (Proposition 4.1.4.6). Our goal in this section is to study the dependence of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{C} }( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} )$ on the vertex $C \in \operatorname{\mathcal{C}}$. We begin by observing that $\operatorname{Fun}_{/ \operatorname{\mathcal{D}}_{C} }( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} )$ can be identified with the the fiber over $C$ of a morphism $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$.

Construction 4.5.9.1 (Direct Images). Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets. For every integer $n \geq 0$, we let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n}$ denote the collection of pairs $(\sigma , f)$, where $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$ and $f: \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism for which the composition

\[ \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}\xrightarrow {V} \operatorname{\mathcal{D}} \]

coincides with projection onto the second factor. Note that every nondecreasing function $\alpha : [m] \rightarrow [n]$ induces a map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n} \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{m} \quad \quad (\sigma ,f) \mapsto (\alpha ^{\ast }(\sigma ), f' ), \]

where $f'$ denotes the composite map

\[ \Delta ^{m} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow { \alpha \times \operatorname{id}} \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}. \]

This construction is compatible with composition, and therefore endows $\{ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})_{n} \} _{n \geq 0}$ with the structure of a simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) = \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{\bullet }$ which we will refer to as the direct image of $\operatorname{\mathcal{E}}$ along $U$.

Note that the construction $(\sigma ,f) \mapsto \sigma $ determines a morphism of simplicial sets $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$. Moreover, there is a tautological evaluation map $\operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$, which carries an $n$-simplex $( \widetilde{\sigma }, (\sigma ,f) )$ of the fiber product $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ to the $n$-simplex of $\operatorname{\mathcal{E}}$ given by the composite map $\Delta ^{n} \xrightarrow { \operatorname{id}\times \widetilde{\sigma } } \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}$.

The direct image $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ of Construction 4.5.9.1 is characterized by a universal property:

Proposition 4.5.9.2. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets. For every morphism of simplicial sets $\sigma : \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, postcomposition with the evaluation map $\operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$ of Construction 4.5.9.1 induces a bijection

\[ \operatorname{Hom}_{ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} }( \operatorname{\mathcal{C}}', \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Hom}_{ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} }( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}). \]

Proof. Writing $\operatorname{\mathcal{C}}'$ as a colimit of simplices, we may reduce to the case where $\operatorname{\mathcal{C}}' = \Delta ^{n}$, so that $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$. In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$. $\square$

Remark 4.5.9.3. In the situation of Proposition 4.5.9.2, composition with the evaluation map $\operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$ induces an isomorphism of simplicial sets

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) ) \xrightarrow {\sim } \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}). \]

The bijectivity of this map on $n$-simplices follows by applying Proposition 4.5.9.2 after replacing $\operatorname{\mathcal{C}}'$ by the product $\Delta ^ n \times \operatorname{\mathcal{C}}'$.

Corollary 4.5.9.4. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the pullback functor

\[ U^{\ast }: (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}} \]

has a right adjoint, given on objects by the construction $\operatorname{\mathcal{E}}\mapsto \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$.

Example 4.5.9.5. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ be the identity map. Then the projection map $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ is an isomorphism.

Example 4.5.9.6. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, and let $U: \operatorname{\mathcal{D}}\rightarrow \Delta ^{0}$ denote the projection map. Then the direct image $\operatorname{Res}_{ \operatorname{\mathcal{D}}/ \Delta ^{0} }( \operatorname{\mathcal{E}})$ can be identified with the simplicial set

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) = \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}) } \{ \operatorname{id}_{ \operatorname{\mathcal{D}}} \} , \]

which parametrizes sections of $V$.

Remark 4.5.9.7. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r] \ar [d] & \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}' \ar [d] \\ \operatorname{\mathcal{E}}\ar [r] & \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{C}}, } \]

where both squares are pullbacks. Then there is a canonical isomorphism of simplicial sets

\[ \operatorname{Res}_{ \operatorname{\mathcal{D}}' / \operatorname{\mathcal{C}}' }( \operatorname{\mathcal{E}}' ) \simeq \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}). \]

Remark 4.5.9.8. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets, and let $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection map of Construction 4.5.9.1. For every vertex $C \in \operatorname{\mathcal{C}}$, Remark 4.5.9.7 and Example 4.5.9.6 furnish an isomorphism of simplicial sets

\[ \pi ^{-1} \{ C\} = \{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{C} }(\operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ). \]

Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. It follows from Remark 4.5.9.8 and Proposition 4.1.4.6 that, for any morphism of simplicial sets $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, the fibers of the induced map $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ are $\infty $-categories.

Exercise 4.5.9.9. Let $\operatorname{\mathcal{C}}= \Delta ^2$ be the standard $2$-simplex, let $\operatorname{\mathcal{D}}= \operatorname{N}_{\bullet }( \{ 0 < 2 \} )$ be the long edge of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}= \{ 0\} \coprod \{ 2\} $ be its boundary. Let $V: \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion maps. Show that $U$ are $V$ are isofibrations of $\infty $-categories but that the projection map $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ can be identified with the horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$, which is not an inner fibration.

To avoid the behavior described in Exercise 4.5.9.9, we need to impose an additional condition on the morphism $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

Definition 4.5.9.10. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will say that $U$ is exponentiable if it satisfies the following condition:

$(\ast )$

For every diagram of simplicial sets

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

in which both squares are pullbacks, if $\overline{F}$ is a categorical equivalence, then $F$ is also a categorical equivalence.

Remark 4.5.9.11. We will be primarily interested in the special case of Definition 4.5.9.10 where $U$ is an inner fibration of simplicial sets. In this case, Definition 4.5.9.10 can be considerably simplified: to show that an inner fibration of simplicial sets $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is exponentiable, it suffices to verify condition $(\ast )$ in the special case where $\overline{F}: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$ is the inner horn $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$ (see Proposition ).

Remark 4.5.9.12. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be exponentiable morphisms of simplicial sets. Then the composition $(U \circ V): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is also exponentiable.

Remark 4.5.9.13. Suppose we are given a pullback diagram of simplicial sets

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

If $U$ is exponentiable, then $U'$ is also exponentiable.

Remark 4.5.9.14. The collection of exponentiable morphisms of simplicial sets is closed under retracts. That is, if we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =40pt@R=40pt{ \operatorname{\mathcal{D}}\ar [r] \ar [d]^{U} & \operatorname{\mathcal{D}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}} \]

where $U'$ is exponentiable and both horizontal compositions are the identity, then $U$ is also exponentiable.

Example 4.5.9.15. Let $\operatorname{\mathcal{D}}$ be any simplicial set. Then the projection map $\operatorname{\mathcal{D}}\rightarrow \Delta ^{0}$ is exponentiable (this is a reformulation of Remark 4.5.3.7).

Example 4.5.9.16. The inclusion map $\operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \hookrightarrow \Delta ^2$ is an isofibration of $\infty $-categories which is not exponentiable. Note that there is a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \{ 0 \} \coprod \{ 2\} \ar [r] \ar [d] & \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \ar [d] \\ \Lambda ^{2}_{1} \ar [r] & \Delta ^2 } \]

where the lower horizontal map is a categorical equivalence, but the upper horizontal map is not.

The terminology of Definition 4.5.9.10 is motivated by the following:

Proposition 4.5.9.17. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ is exponentiable (Definition 4.5.9.10).

$(2)$

Let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a simplicial subset for which the restriction $U|_{\operatorname{\mathcal{D}}_0}$ is exponentiable, let $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be an isofibration in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'_0 = \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}'$. Then the induced map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Res}_{ \operatorname{\mathcal{D}}_0/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}_0 ) \times _{ \operatorname{Res}_{\operatorname{\mathcal{D}}_0/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}'_0) } \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}' ) \]

is also an isofibration.

$(3)$

For every isofibration $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, the induced map

\[ \operatorname{Res}_{ \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}') \]

is also an isofibration.

$(4)$

For every isofibration of $\infty $-categories $T_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$, the induced map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}_0 ) \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}'_0 ) \]

is also an isofibration.

Proof. We first show that $(1)$ implies $(2)$. Assume that $U$ is exponentiable, let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a simplicial subset for which $U|_{\operatorname{\mathcal{D}}_0}$ is also exponentiable, let $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be an isofibration in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets which is a categorical equivalence; we wish to show that every lifting problem

4.45
\begin{equation} \begin{gathered}\label{equation:exponentiable-equivalent-condition} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \ar [d] \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{Res}_{ \operatorname{\mathcal{D}}_0/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}_0 ) \times _{ \operatorname{Res}_{\operatorname{\mathcal{D}}_0/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}'_0) } \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}' )} \end{gathered} \end{equation}

admits a solution. Note that the bottom horizontal map determines a morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{C}}$. Invoking the universal property of direct images (Proposition 4.5.9.2), we can rewrite (4.45) as a lifting problem

4.46
\begin{equation} \begin{gathered}\label{equation:silly-diagram-for-fun2} \xymatrix@R =50pt@C=50pt{ (A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}) \coprod _{ (A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0)} (B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 ) \ar [d]^{j} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{T} \\ B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{E}}'. } \end{gathered} \end{equation}

Since $U$ and $U|_{\operatorname{\mathcal{D}}_0}$ are exponentiable, the horizontal maps in the diagram

4.47
\begin{equation} \begin{gathered}\label{equation:silly-diagram-for-fun} \xymatrix@R =50pt@C=50pt{ A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 \ar [r] \ar [d] & B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 \ar [d] \\ A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\ar [r] & B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

are categorical equivalences. In particular, the diagram (4.47) is a categorical pushout square (Proposition 4.5.4.10). It follows that the morphism $j$ appearing in (4.46) is also a categorical equivalence (Proposition 4.5.4.11). Since $T$ is an isofibration of simplicial sets, it follows that the lifting problem (4.46) admits a solution.

The implication $(2) \Rightarrow (3)$ and $(3) \Rightarrow (4)$ are immediate. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that condition $(4)$ is satisfied and suppose that we are given a commutative diagram of simplicial sets

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

where both squares are pullbacks and $\overline{F}$ is a categorical equivalence; we wish to show that $F$ is also a categorical equivalence. By virtue of Exercise 3.1.7.11, there exists a monomorphism of simplicial sets $\iota : \operatorname{\mathcal{C}}'' \hookrightarrow Q$, where $Q$ is a contractible Kan complex. Replacing $\overline{F}$ by the morphism $(\iota , \overline{F}): \operatorname{\mathcal{C}}'' \hookrightarrow Q \times \operatorname{\mathcal{C}}'$ (and $F$ by the morphism $(\iota ,F): \operatorname{\mathcal{D}}'' \hookrightarrow Q \times \operatorname{\mathcal{D}}'$), we can reduce to the case where $\overline{F}$ is a monomorphism of simplicial sets, so that $F$ is also a monomorphism of simplicial sets. To show that $F$ is a categorical equivalence, it will suffice to show that if $T_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$ is an isofibration of $\infty $-categories, then every lifting problem

4.48
\begin{equation} \begin{gathered}\label{equation:exponentiable-equivalent-condition2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}'' \ar [d]^{F} \ar [r] & \operatorname{\mathcal{E}}_0 \ar [d]^{T_0} \\ \operatorname{\mathcal{D}}' \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}'_0 } \end{gathered} \end{equation}

admits a solution (Proposition 4.5.5.4). Invoking the universal property of direct images (Proposition 4.5.9.2), we can rewrite (4.48) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'' \ar [d]^{ \overline{F} } \ar [r] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}_0 ) \ar [d] \\ \operatorname{\mathcal{C}}' \ar [r] \ar@ {-->}[ur] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}'_0 ). } \]

Condition $(4)$ guarantees that the right vertical map is an isofibration, so that the solution exists by virtue of our assumption that $\overline{F}$ is a categorical equivalence. $\square$

Corollary 4.5.9.18. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be an exponentiable morphism of simplicial sets. For every isofibration of simplicial sets $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$, the projection map $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ is also an isofibration of simplicial sets.

Proof. Applying Proposition 4.5.9.17 in the special case $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{D}}$. $\square$