4.5.9 Relative Exponentiation
Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets. For every vertex $B \in \operatorname{\mathcal{B}}$, let $\operatorname{\mathcal{C}}_{B} = \{ B\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$ denote the corresponding fiber of $U$. If $\operatorname{\mathcal{D}}$ is an $\infty $-category, then Theorem 1.5.3.7 guarantees that the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}_{B}, \operatorname{\mathcal{D}})$ is also an $\infty $-category. Our goal in this section is to study the dependence of this construction on the vertex $B \in \operatorname{\mathcal{B}}$. We begin by introducing a relative version of Construction 1.5.3.1.
Construction 4.5.9.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets and let $\operatorname{\mathcal{D}}$ be another simplicial set. For every integer $n \geq 0$, we let $\operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})_{n}$ denote the collection of pairs $(\sigma , f)$, where $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{B}}$ and $f: \Delta ^ n \times _{ \operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a morphism of simplicial sets. Note that every nondecreasing function $\alpha : [m] \rightarrow [n]$ induces a map
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})_{n} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})_{m} \quad \quad (\sigma , f) \mapsto (\alpha ^{\ast }(\sigma ), f'), \]
where $f'$ denotes the composite map
\[ \Delta ^{m} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\xrightarrow { \alpha \times \operatorname{id}} \Delta ^{n} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\xrightarrow {f} \operatorname{\mathcal{D}}. \]
This construction is compatible with composition, and therefore endows $\{ \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}), \operatorname{\mathcal{D}})_{n} \} _{n \geq 0}$ with the structure of a simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$.
Note that the assignment $(\sigma ,f) \mapsto \sigma $ determines a morphism of simplicial sets $\pi : \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$. Moreover, there is an evaluation map $\operatorname{ev}: \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{B}}} \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{D}}$, which carries an $n$-simplex $( \widetilde{\sigma }, (\sigma ,f) )$ of the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{B}}} \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ to the $n$-simplex of $\operatorname{\mathcal{D}}$ given by the composite map $\Delta ^{n} \xrightarrow { \operatorname{id}\times \widetilde{\sigma } } \Delta ^{n} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\xrightarrow {f} \operatorname{\mathcal{D}}$.
Example 4.5.9.2. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets and let $U: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ denote the projection map. Then the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/ \Delta ^0, \operatorname{\mathcal{D}})$ of Construction 4.5.9.1 can be identified with the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of Construction 1.5.3.1.
Example 4.5.9.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets and let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the identity morphjism. Then the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of Construction 4.5.9.1 can be identified with the product $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$.
Example 4.5.9.4. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets. Then the projection map $\pi : \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \Delta ^0 ) \rightarrow \operatorname{\mathcal{B}}$ is an isomorphism.
The direct image $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ of Construction 4.5.9.1 can be characterized by a universal mapping property:
Proposition 4.5.9.5. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets and let $\operatorname{\mathcal{D}}$ be a simplicial set. For every morphism of simplicial sets $\operatorname{\mathcal{B}}' \rightarrow \operatorname{\mathcal{B}}$, postcomposition with the evaluation map $\operatorname{ev}: \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{B}}} \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{D}}$ induces a bijection
\[ \operatorname{Hom}_{ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{B}}} }( \operatorname{\mathcal{B}}', \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}', \operatorname{\mathcal{D}}). \]
Proof.
Writing $\operatorname{\mathcal{B}}'$ as a colimit of simplices, we may reduce to the case where $\operatorname{\mathcal{B}}' = \Delta ^{n}$, so that $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{B}}$. In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$.
$\square$
Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. It follows from Remark 4.5.9.8 Theorem 1.5.3.7 that every fiber of the projection map $\pi : \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ is an $\infty $-category. Beware that $\pi $ is generally not an inner fibration.
Exercise 4.5.9.9. Let $\operatorname{\mathcal{B}}= \Delta ^2$ be the standard $2$-simplex and let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \{ 0 < 2 \} )$ be the long edge of $\operatorname{\mathcal{C}}$. Show that $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \Delta ^1 )$ is not an $\infty $-category.
To avoid the behavior described in Exercise 4.5.9.9, we need to impose an additional condition on the morphism $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$.
Definition 4.5.9.10. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ 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
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'' \ar [r]^-{F} \ar [d] & \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{B}}'' \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{B}}' \ar [r] & \operatorname{\mathcal{B}}} \]
in which both squares are pullbacks, if $\overline{F}$ is a categorical equivalence, then $F$ is also a categorical equivalence.
Example 4.5.9.15. Let $\operatorname{\mathcal{C}}$ be any simplicial set. Then the projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^{0}$ 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 $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $U$ is exponentiable (Definition 4.5.9.10).
- $(2)$
For every isofibration of simplicial sets $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, the induced map
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \xrightarrow {V \circ } \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) \]
is also an isofibration of simplicial sets.
- $(3)$
For every isofibration of $\infty $-categories $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, the induced map
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \xrightarrow {V \circ } \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) \]
is also an isofibration.
Proof.
We first show that $(1)$ implies $(2)$. Assume that $U$ is exponentiable, let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of simplicial sets, 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
4.46
\begin{equation} \begin{gathered}\label{equation:exponentiable-equivalent-condition} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \ar [d]^{V \circ } \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) } \end{gathered} \end{equation}
admits a solution. Note that the lower horizontal map determines a morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{B}}$. Invoking the universal property of Proposition 4.5.9.5, we can rewrite (4.46) as a lifting problem
4.47
\begin{equation} \begin{gathered}\label{equation:silly-diagram-for-fun2} \xymatrix@R =50pt@C=50pt{ A \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\ar [d]^{j} \ar [r] & \operatorname{\mathcal{D}}\ar [d]^{V} \\ B \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{E}}. } \end{gathered} \end{equation}
Because $U$ is exponentiable, the left vertical map is a categorical equivalence of simplicial sets. Our assumption that $V$ is an isofibration then guarantees the existence of a solution.
The implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3)$ implies $(1)$. Assume that condition $(3)$ is satisfied and suppose that we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'' \ar [r]^-{F} \ar [d] & \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{B}}'' \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{B}}' \ar [r] & \operatorname{\mathcal{B}}} \]
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 $\iota : \operatorname{\mathcal{B}}'' \hookrightarrow Q$, where $Q$ is a contractible Kan complex. Replacing $\overline{F}$ by the morphism $(\iota , \overline{F}): \operatorname{\mathcal{B}}'' \hookrightarrow Q \times \operatorname{\mathcal{B}}'$ (and $F$ by the morphism $(\iota ,F): \operatorname{\mathcal{C}}'' \hookrightarrow Q \times \operatorname{\mathcal{C}}'$), 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 $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ 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{C}}'' \ar [d]^{F} \ar [r] & \operatorname{\mathcal{D}}\ar [d]^{V} \\ \operatorname{\mathcal{C}}' \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}
admits a solution (Proposition 4.5.5.4). Invoking the universal property of direct images (Proposition 4.5.9.5), we can rewrite (4.48) as a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{B}}'' \ar [d]^{ \overline{F} } \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \ar [d]^{V \circ } \\ \operatorname{\mathcal{B}}' \ar [r] \ar@ {-->}[ur] & \operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}). } \]
Condition $(3)$ 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 $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be an exponentiable morphism of simplicial sets. For every $\infty $-category $\operatorname{\mathcal{D}}$, the projection map $\pi : \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ is an isofibration of simplicial sets.
Proof.
Apply Proposition 4.5.9.17 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$ (see Example 4.5.9.4).
$\square$
