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5.3.1 Exponentiation for Cartesian Fibrations

In this section, we study the behavior of cartesian fibrations with respect to the formation of functor $\infty $-categories. Our main result can be stated as follows:

Theorem 5.3.1.1. Let $q: X \rightarrow S$ be a cartesian fibration of simplicial sets, let $B$ be a simplicial set, and let $q': \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(B,S)$ be the morphism given by postcomposition with $q$. Then:

$(1)$

The morphism $q'$ is a cartesian fibration of simplicial sets.

$(2)$

Let $e$ be an edge of the simplicial set $\operatorname{Fun}(B,X)$. Then $e$ is $q'$-cartesian if and only if, for every vertex $b \in B$, the evaluation map $\operatorname{ev}_{b}: \operatorname{Fun}(B, X) \rightarrow \operatorname{Fun}( \{ b\} , X) \simeq X$ carries $e$ to a $q$-cartesian edge of $X$.

Remark 5.3.1.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, so that the projection map $q: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is a cartesian fibration (Example 5.2.4.3). In this case, part $(1)$ of Theorem 5.3.1.1 is equivalent to the assertion that for every simplicial set $B$, the simplicial set $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$ is also an $\infty $-category (Theorem 1.4.3.7). By virtue of Proposition 5.2.4.9, part $(2)$ is equivalent to the assertion that a morphism of $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is an isomorphism if and only if, for every vertex $b \in B$, its image under the evaluation functor $\operatorname{ev}_{b}: \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is an isomorphism in $\operatorname{\mathcal{C}}$ (Theorem 4.4.4.4).

Corollary 5.3.1.3. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets, let $K$ be a simplicial set, and let $q': S \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X) \rightarrow S$ be the projection map onto the first factor. Then:

$(1)$

The morphism $q'$ is a locally cartesian fibration of simplicial sets.

$(2)$

Let $e$ be an edge of the simplicial set $S \times _{ \operatorname{Fun}(B,S)} \operatorname{Fun}(B,X)$. Then $e$ is locally $q'$-cartesian if and only if, for every vertex $b \in B$, the image of $e$ under the evaluation functor $S \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}( \{ b\} , X) \simeq X$ is locally $q$-cartesian.

Proof. By virtue of Remark 5.2.6.6, we may assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration. The desired result now follows by combining Theorem 5.3.1.1 with Remark 5.2.4.6. $\square$

The proof of Theorem 5.3.1.1 will require some preliminaries. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets. By definition, $q$ is a cartesian fibration if and only if for every vertex $z \in X$ and every edge $\overline{e}: s \rightarrow q(z)$ of $S$, there exists a $q$-cartesian edge $e: y \rightarrow z$ in $X$ satisfying $q(e) = \overline{e}$. To prove Theorem 5.3.1.1, we need to show that the edge $e$ can be chosen to depend functorially on $z$.

Proposition 5.3.1.4. Let $q: X \rightarrow S$ be a cartesian fibration of simplicial sets, and let $Y \subseteq \operatorname{Fun}( \Delta ^1, X)$ be the full simplicial subset of $\operatorname{Fun}( \Delta ^1, X)$ spanned by those edges $e: \Delta ^1 \rightarrow X$ which are $q$-cartesian (see Definition 4.1.2.17). Then the restriction map

\[ \theta : Y \rightarrow \operatorname{Fun}( \Delta ^1, S) \times _{ \operatorname{Fun}( \{ 1\} , S) } \operatorname{Fun}( \{ 1\} , X) \]

is a trivial Kan fibration of simplicial sets.

Proof. Let $n \geq 0$ be an integer; we wish to show that every lifting problem of the form

5.11
\begin{equation} \begin{gathered}\label{equation:lifting-problem-for-cartesian-lift-uniqueness} \[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & Y \ar [d]^{\theta } \\ \Delta ^ n \ar [r] \ar@ {-->}[r] & \operatorname{Fun}( \Delta ^1, S) \times _{ \operatorname{Fun}( \{ 1\} , S) } \operatorname{Fun}( \{ 1\} , X) } \] \end{gathered} \end{equation}

admits a solution. In the case $n = 0$, this follows immediately from our assumption that $q$ is a cartesian fibration. Let us therefore assume that $n > 0$. Unwinding the definitions, we can rephrase (5.11) as a lifting problem

\[ \xymatrix@R =50pt@C=75pt{ (\Delta ^1 \times \operatorname{\partial \Delta }^ n) \coprod _{ (\{ 1\} \times \operatorname{\partial \Delta }^ n)} (\{ 1\} \times \Delta ^ n) \ar [r]^-{h_0} \ar [d] & X \ar [d]^{q} \\ \Delta ^1 \times \Delta ^ n \ar [r]^-{\overline{h}} \ar@ {-->}[ur]^{h} & S, } \]

where the morphism $h_0$ has the property that $h_0|_{ \Delta ^1 \times \{ i\} }$ is a $q$-cartesian edge of $X$ for $0 \leq i \leq n$. Let

\[ (\Delta ^1 \times \operatorname{\partial \Delta }^ n) \cup (\{ 1\} \times \Delta ^ n) = Y(0) \subset Y(1) \subset X(2) \subset \cdots \subset Y(n+1) = \Delta ^{1} \times \Delta ^ n \]

be the sequence of simplicial subsets appearing in the proof of Lemma 3.1.2.9, so that $h_0$ can be identified with morphism of simplicial sets from $Y(0)$ to $X$. We will show that, for $0 \leq j \leq n+1$, there exists a morphism of simplicial sets $h_ j: Y(j) \rightarrow X$ satisfying $h_ j|_{Y(0)} = h_0$ and $q \circ h_ j = \overline{h}|_{ Y(j)}$ (taking $j = n+1$, this will complete the proof of Proposition 5.3.1.4). We proceed by induction on $j$, the case $j=0$ being vacuous. Assume that $j > 0$ and that we have already constructed a morphism $h_{j-1}: Y(j-1) \rightarrow X$ satisfying $h_{j-1}|_{Y(0)} = h_0$ and $q \circ h_{j-1} = \overline{h}|_{ Y(j-1)}$. By virtue of Lemma 3.1.2.9, we have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{j} \ar [r]^-{\sigma _0} \ar [d] & Y(j-1) \ar [d] \\ \Delta ^{n+1} \ar [r]^-{\sigma } & Y(j). } \]

Consequently, to prove the existence of $h_{i}$, it suffices to solve the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{j} \ar [r]^-{ h_{j-1} \circ \sigma _0} \ar [d] & X \ar [d]^{q} \\ \Delta ^{n+1} \ar [r]^-{ \overline{h} \circ \sigma } \ar@ {-->}[ur] & S. } \]

For $0 < j < n+1$, the existence of the desired solution follows from our assumption that $q$ is an inner fibration. In the case $j = n+1$, the existence follows from the fact that the composite map

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n < n+1 \} ) \hookrightarrow \Lambda ^{n+1}_{j} \xrightarrow { \sigma _0} Y(n) \xrightarrow { h_ n} X \]

is the edge of $X$ the restriction $h_0|_{ \Delta ^1 \times \{ n\} }$, and is therefore $q$-cartesian. $\square$

Lemma 5.3.1.5. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, let $B$ be a simplicial set, and let $q': \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(B,S)$ be the map given by postcomposition with $q$ (so that $q'$ is also an inner fibration; see Corollary 4.1.4.3). Let $e$ be an edge of the simplicial set $\operatorname{Fun}(B,X)$.

$(1)$

Suppose that, for every vertex $b \in B$, the evaluation map

\[ \operatorname{ev}_{b}: \operatorname{Fun}(B, X) \rightarrow \operatorname{Fun}( \{ b\} , X) \simeq X \]

carries $e$ to a $q$-cartesian edge of $X$. Then $e$ is $q'$-cartesian.

$(2)$

Suppose that, for every vertex $b \in B$, the evaluation map

\[ \operatorname{ev}_{b}: \operatorname{Fun}(B, X) \rightarrow \operatorname{Fun}( \{ b\} , X) \simeq X \]

carries $e$ to a $q$-cocartesian edge of $X$. Then $e$ is $q'$-cocartesian.

Proof. We will give a proof of $(2)$; assertion $(1)$ follows by a similar argument. We proceed as in the proof of Lemma 4.4.4.8. Suppose we are given an integer $n \geq 2$; we wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{Fun}(B,X) \ar [d]^{q'} \\ \Delta ^ n \ar@ {-->}[ur]^{\sigma } \ar [r]^-{\overline{\sigma } } & \operatorname{Fun}(B,S) } \]

admits a solution, provided that the composite map

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^ n_0 \xrightarrow {\sigma _0} \operatorname{Fun}(B,X) \]

is the edge $e$. Unwinding the definitions, we can rewrite this as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ B \times \Lambda ^{n}_0 \ar [r]^-{F_0} \ar [d] & X \ar [d]^{q} \\ B \times \Delta ^ n \ar [r]^-{ \overline{F} } \ar@ {-->}[ur]^{F} & S. } \]

Let $P$ denote the collection of all pairs $(A, F_ A)$, where $A \subseteq B$ is a simplicial subset and $F_ A: A \times \Delta ^ n \rightarrow X$ is a morphism of simplicial sets satisfying

\[ F_{A} |_{ A \times \Lambda ^{n}_0} = F_0 |_{ A \times \Lambda ^{n}_{0}} \quad \quad q \circ F_ A = \overline{F}|_{ A \times \Delta ^ n} \]

We regard $P$ as partially ordered set, where $(A, F_ A) \leq (A', F_{A'} )$ if $A \subseteq A'$ and $F_ A = F_{A'} |_{K \times \Delta ^ n}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma, and therefore has a maximal element $(A_{\mathrm{max}}, F_{A_{\mathrm{max}}})$. We will complete the proof by showing that $A_{\mathrm{max}} = B$. Assume otherwise. Then there exists some nondegenerate $m$-simplex $\tau : \Delta ^{m} \rightarrow B$ whose image is not contained in $A_{ \mathrm{max} }$. Choosing $m$ as small as possible, we can assume that $\tau $ carries the boundary $\operatorname{\partial \Delta }^{m}$ into $B$. Let $A' \subseteq B$ be the union of $A_{\mathrm{max}}$ with the image of $\tau $, so that we have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [d] \ar [r] & A_{\mathrm{max}} \ar [d] \\ \Delta ^{m} \ar [r] & A'. } \]

We will complete the proof by showing that the lifting problem

\[ \xymatrix@R =50pt@C=100pt{ (A_{\mathrm{max}} \times \Delta ^ n) \coprod _{ (A_{\mathrm{max}} \times \Lambda ^ n_0 )} (A' \times \Lambda ^ n_0) \ar [r]^-{(F_{A_{\mathrm{max}}}, F_0|_{A' \times \Lambda ^{n}_{0}})} \ar [d] & X \ar [d]^{q} \\ A' \times \Delta ^ n \ar [r] \ar@ {-->}[ur] & S } \]

admit a solution (contradicting the maximality of the pair $(A_{\mathrm{max}}, F_{A_{\mathrm{max}}} )$).

Choose a sequence of simplicial subsets

\[ Y(0) \subset Y(1) \subset Y(2) \subset \cdots \subset Y(t) = \Delta ^{m} \times \Delta ^{n} \]

satisfying the requirements of Lemma 4.4.4.7, so that $F_{A_{\mathrm{max}}}$ determines a map of simplicial sets $G_0: Y(0) \rightarrow X$. We will show that, for $0 \leq s \leq t$, there exists a morphism of simplicial sets $G_{s}: Y(s) \rightarrow X$ satisfying $G_ s|_{Y(0)} = G_0$ and $q \circ G_ s = \overline{F}|_{ Y(s) }$ (in the case $s = t$, this will complete the proof of Lemma 5.3.1.5). We proceed by induction on $s$, the case $s=0$ being vacuous. Assume that $s > 0$ and that we have already constructed a morphism $G_{s-1}: Y(s-1) \rightarrow X$ satisfying $G_{s-1}|_{Y(0)} = F_0$ and $q \circ G_{s-1} = \overline{F}|_{ Y(s-1)}$. By construction, there exist integers $\ell \geq 2$, $0 \leq k< \ell $ and a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{\ell }_{k} \ar [r]^-{\tau _0} \ar [d] & Y(s-1) \ar [d] \\ \Delta ^{\ell } \ar [r]^-{\tau } & Y(s). } \]

Moreover, in the special case $p=0$, we can assume that $\tau (0) = (0,0)$ and $\tau (1) = (0,1)$, so that the composite map

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^{\ell }_{k} \xrightarrow {\sigma _0} Y(s-1) \xrightarrow { G_{s-1} } X \]

corresponds to $q$-cocartesian edge $e'$ of $X$. To construct the desired extension $F_ s$, it suffices to solve the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{\ell }_{k} \ar [r]^-{ G_{s-1} \circ \tau _0} \ar [d] & X \ar [d]^{q} \\ \Delta ^{\ell } \ar [r]^-{ \overline{F} \circ \tau } \ar@ {-->}[ur] & S. } \]

For $0 < k < \ell $, the existence of the desired solution follows from our assumption that $q$ is an inner fibration; when $k = 0$, it follows from the fact that $e'$ is $q$-cocartesian. $\square$

Proof of Theorem 5.3.1.1. Let $q: X \rightarrow S$ be a cartesian fibration of simplicial sets, let $B$ be any simplicial set, and let $q': \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(B,S)$ be the map given by postcomposition with $q$. Then $q'$ is an inner fibration (Corollary 4.1.4.3). Let us say that an edge $e$ of the simplicial set $\operatorname{Fun}(B,X)$ is special if, for every vertex $b \in B$, the evaluation map $\operatorname{ev}_{b}: \operatorname{Fun}(B, X) \rightarrow \operatorname{Fun}( \{ b\} , X) \simeq X$ carries $e$ to a $q$-cocartesian edge of $X$. By virtue of Lemma 5.3.1.5, every special edge of $\operatorname{Fun}(B,X)$ is $q'$-cartesian. Moreover, Proposition 5.3.1.4 guarantees that for every vertex $z \in \operatorname{Fun}(B,X)$ and every edge $\overline{e}: \overline{y} \rightarrow q'(z)$ of $\operatorname{Fun}(B,S)$, there exists a special edge $e: y \rightarrow z$ of $\operatorname{Fun}(B,X)$ satisfying $q'(e) = \overline{e}$. It follows that $q'$ is a cartesian fibration.

To complete the proof, it will suffice to show that every $q'$-cartesian edge $e: x \rightarrow z$ of the simplicial set $\operatorname{Fun}(B,X)$ is special. By virtue of the preceding argument, there exists a special edge $e': y \rightarrow z$ of $\operatorname{Fun}(B,X)$ satisfying $q'(e') = q'(e)$, which is also $q'$-cartesian. Applying Remark 5.2.3.9, we can choose a $2$-simplex $\sigma $ of $\operatorname{Fun}(B,X)$ as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{ e' } & \\ x \ar [ur]^{e''} \ar [rr]^{e} & & z, } \]

where $e''$ is an isomorphism in the $\infty $-category $\{ q'(x) \} \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X)$. For each vertex $b \in B$, the evaluation functor $\operatorname{ev}_{b}$ carries $\sigma $ to a $2$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{ev}_ b(y) \ar [dr]^{ \operatorname{ev}_ b(e') } & \\ \operatorname{ev}_ b(x) \ar [ur]^{\operatorname{ev}_ b(x'')} \ar [rr]^{\operatorname{ev}_ b(e)} & & \operatorname{ev}_ b(z) } \]

in the simplicial set $X$. Since $e'$ is special, the edge $\operatorname{ev}_ b(e')$ is $q$-cartesian. The edge $\operatorname{ev}_ b(e'')$ is an isomorphism in a fiber of $q$, and is therfefore also $q$-cartesian (Proposition 5.2.4.9). Applying Proposition 5.2.4.10, we deduce that $\operatorname{ev}_ b(x)$ is $q$-cartesian. Allowing the vertex $b$ to vary, we conclude that $e$ is a special edge of $\operatorname{Fun}(B,X)$, as desired. $\square$