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4.4.5 Exponentiation for Isofibrations

We now show that the formation of $\infty$-categories of functors behaves well with respect to isofibrations.

Proposition 4.4.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty$-categories, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset. Then the restriction map

$F': \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}})$

is an isofibration of $\infty$-categories.

We will give the proof of Proposition 4.4.5.1 at the end of this section.

Corollary 4.4.5.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset. Then the restriction map $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is an isofibration of $\infty$-categories.

Proof. Apply Proposition 4.4.5.1 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$. $\square$

Corollary 4.4.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset. Then the restriction functor $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})$ induces a Kan fibration of simplicial sets $\operatorname{Fun}(B, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})^{\simeq }$.

Proof. Combine Corollary 4.4.5.3 with Proposition 4.4.3.7. $\square$

Corollary 4.4.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, and let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms. Then the restriction map

$\operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\quad \quad (f: X \rightarrow Y) \mapsto (X,Y)$

is an isofibration of $\infty$-categories.

Corollary 4.4.5.6. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty$-categories. For every simplicial set $B$, the induced map $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ is also an isofibration of $\infty$-categories.

Proof. Apply Proposition 4.4.5.1 in the special case $A = \emptyset$. $\square$

Corollary 4.4.5.7. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty$-categories. For every simplicial set $B$, the induced map $\operatorname{Fun}(B, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})^{\simeq }$ is a Kan fibration of Kan complexes.

Proof. Combine Corollary 4.4.5.6 with Proposition 4.4.3.7. $\square$

The main ingredient needed in our proof of Proposition 4.4.5.1 is the following isomorphism extension result:

Proposition 4.4.5.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset which contains every vertex of $B$, and suppose we are given a lifting problem

$\xymatrix@R =50pt@C=50pt{ (\Delta ^1 \times A) \coprod _{ (\{ 1\} \times A) } (\{ 1 \} \times B) \ar [r]^-{h_0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^1 \times B \ar [r]^-{ \overline{h} } \ar@ {-->}[ur]^{h} & \operatorname{\mathcal{D}}}$

with the following property:

$(\ast )$

For every simplex $\tau : \Delta ^{n} \rightarrow B$ which is not contained in $A$ having final vertex $b = \tau (n)$, the edge

$\Delta ^1 \simeq \Delta ^1 \times \{ b\} \xrightarrow {h_0} \operatorname{\mathcal{C}}$

is an isomorphism in $\operatorname{\mathcal{C}}$.

Then $h_0$ can be extended to a diagram $h: \Delta ^1 \times B \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{h} = F \circ h$.

Proof. We proceed as in the proof of Lemma 4.4.4.8, with some minor modifications. Let $P$ denote the collection of all pairs $(K, h_ K)$, where $K \subseteq B$ is a simplicial subset containing $A$ and $h_ K: \Delta ^1 \times K \rightarrow \operatorname{\mathcal{C}}$ is a morphism of simplicial sets satisfying

$h_ K |_{\Delta ^1 \times A } = h_0 |_{ \Delta ^1 \times A } \quad \quad h_{K} |_{ \{ 1\} \times K} = h_0 |_{ \{ 1\} \times K }.$

We regard $P$ as partially ordered set, where $(K, h_ K) \leq (K', h_{K'} )$ if $K \subseteq K'$ and $h_ K = h_{K'} |_{\Delta ^1 \times K}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma, and therefore has a maximal element $(K_{\mathrm{max}}, h_{K_{\mathrm{max}}})$. We will complete the proof by showing that $K_{\mathrm{max}} = B$. Assume otherwise. Then there exists some nondegenerate $n$-simplex $\tau : \Delta ^{n} \rightarrow B$ whose image is not contained in $K_{ \mathrm{max} }$. Choosing $n$ as small as possible, we can assume that $\tau$ carries the boundary $\operatorname{\partial \Delta }^{n}$ into $K_{\mathrm{max}}$. Note that, since $A$ contains every vertex of $B$, we must have $n > 0$. Let $K' \subseteq B$ be the union of $K_{\mathrm{max}}$ with the image of $\tau$, so that we have a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [d] \ar [r] & K_{\mathrm{max}} \ar [d] \\ \Delta ^{n} \ar [r] & K'. }$

We will complete the proof by showing that the lifting problem

$\xymatrix@R =50pt@C=75pt{ (\Delta ^1 \times K_{\mathrm{max}}) \coprod _{ (\{ 1\} \times K_{\mathrm{max}})} (\{ 1\} \times K') \ar [r]^-{(h_ K, h_0|_{\{ 1\} \times K'})} \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \Delta ^1 \times K' \ar [r] \ar@ {-->}[ur] & \Delta ^0 }$

admits a solution, where the dotted arrow carries each edge $\Delta ^1 \times \{ x\}$ to an isomorphism in $\operatorname{\mathcal{C}}$ (contradicting the maximality of the pair $(K_{\mathrm{max}}, h_{K_{\mathrm{max}}} )$). To prove this, we can replace the inclusion $K_{\mathrm{max}} \hookrightarrow K'$ by $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$. We are therefore reduced to proving Lemma 4.4.4.8 in the special case where $B = \Delta ^ n$ is a simplex and $A = \operatorname{\partial \Delta }^ n$ is its boundary.

Let

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

be the sequence of simplicial subsets appearing in the proof of Lemma 3.1.2.11, so that $h_0$ can be identified with a morphism of simplicial sets from $X(0)$ to $\operatorname{\mathcal{C}}$. We will show that, for $0 \leq i \leq n+1$, there exists a morphism of simplicial sets $h_ i: X(i) \rightarrow \operatorname{\mathcal{C}}$ satisfying $h_ i|_{X(0)} = h_0$ and $F \circ h_ i = \overline{h}|_{ X(i)}$ (taking $i = n+1$, this will complete the proof of Proposition 4.4.5.8). We proceed by induction on $i$, the case $i=0$ being vacuous. Assume that $i > 0$ and that we have already constructed a morphism $h_{i-1}: X(i-1) \rightarrow \operatorname{\mathcal{C}}$ satisfying $h_{i-1}|_{X(0)} = h_0$ and $F \circ h_{i-1} = \overline{h}|_{ X(i-1)}$. By virtue of Lemma 3.1.2.11, we have a pushout diagram of simplicial sets

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

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

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i} \ar [r]^-{ h_{i-1} \circ \sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n+1} \ar [r]^-{ \overline{h} \circ \sigma } \ar@ {-->}[ur] & \operatorname{\mathcal{D}}. }$

For $0 < i < n+1$, the existence of the desired solution follows from our assumption that $F$ is an inner fibration. In the case $i = n+1$, the existence follows from Proposition 4.4.2.13, since the map $\sigma : \Delta ^{n+1} \rightarrow \Delta ^1 \times \Delta ^ n$ carries the final edge $\operatorname{N}_{ \bullet }( \{ n < n+1 \} ) \subseteq \Delta ^{n+1}$ to the edge $\Delta ^1 \times \{ n\} \subseteq \Delta ^1 \times \Delta ^ n$, which $h_0$ carries to an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$ by virtue of assumption $(\ast )$. $\square$

Corollary 4.4.5.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty$-categories, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset, and suppose we are given a lifting problem

$\xymatrix@R =50pt@C=50pt{ (\Delta ^1 \times A) \coprod _{ (\{ 1\} \times A) } (\{ 1 \} \times B) \ar [r]^-{h_0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^1 \times B \ar [r]^-{ \overline{h} } \ar@ {-->}[ur]^{h} & \operatorname{\mathcal{D}}}$

with the following properties:

• For every vertex $a \in A$, the edge

$\Delta ^1 \simeq \Delta ^1 \times \{ a\} \xrightarrow {h_0} \operatorname{\mathcal{C}}$

is an isomorphism in $\operatorname{\mathcal{C}}$.

• For every vertex $b \in B$, the edge

$\Delta ^1 \simeq \Delta ^1 \times \{ b\} \xrightarrow { \overline{h} } \operatorname{\mathcal{D}}$

is an isomorphism in $\operatorname{\mathcal{D}}$.

Then $h_0$ can be extended to a diagram $h: \Delta ^1 \times B \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{h} = F \circ h$. Moreover, we can arrange that for every vertex $b \in B$, the edge $\Delta ^1 \simeq \Delta ^1 \times \{ b\} \xrightarrow {h} \operatorname{\mathcal{C}}$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$ (so that $h$ can be regarded as an isomorphism in the diagram $\infty$-category $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$, by virtue of Theorem 4.4.4.4).

Proof. Let $A'$ be the union of $A$ with the $0$-skeleton $\operatorname{sk}_0(B)$, regarded as a simplicial subset of $B$. For each vertex $b \in B$ which does not belong to $A$, our assumption that $F$ is an isofibration allows us to choose an edge $e_{b}: \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ which is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$ satisfying $e_ b(1) = h_0( 1, b)$ and $F \circ e_{b} = \overline{h}|_{ \Delta ^1 \times \{ b\} }$. The morphism $h_0$ and the edges $e_{b}$ can then be amalgamated to a map $h'_0: (\Delta ^1 \times A') \coprod _{ (\{ 1\} \times A') } (\{ 1 \} \times B) \rightarrow \operatorname{\mathcal{C}}$. The desired result now follows by applying Proposition 4.4.5.8 to the commutative diagram

$\xymatrix@R =50pt@C=50pt{ (\Delta ^1 \times A') \coprod _{ (\{ 1\} \times A') } (\{ 1 \} \times B) \ar [r]^-{h'_0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^1 \times B \ar [r]^-{ \overline{h} } & \operatorname{\mathcal{D}}. }$
$\square$

Specializing Corollary 4.4.5.9 to the case $\operatorname{\mathcal{D}}= \Delta ^0$, we obtain the following:

Corollary 4.4.5.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ be the full subcategory spanned by the isomorphisms, and let $\operatorname{ev}_{0},\operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the functors given by evaluation at the vertices $0,1 \in \Delta ^1$. Then the functors $\operatorname{ev}_{0}$ and $\operatorname{ev}_{1}$ are trivial Kan fibrations.

Proof of Proposition 4.4.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty$-categories, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset. We wish to show that the restriction map

$F': \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}})$

is an isofibration of $\infty$-categories. We first note that the projection map

$\operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})$

is a pullback of the inner fibration $\operatorname{Fun}(B, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{D}})$ (see Corollary 4.1.4.2). Since $\operatorname{Fun}(A, \operatorname{\mathcal{C}})$ is an $\infty$-category (Theorem 1.4.3.7), it follows that $\operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ is also an $\infty$-category (Remark 4.1.1.9). It follows from Proposition 4.1.4.1 that $F'$ is an inner fibration. It will therefore suffice to show that, for every object $Y \in \operatorname{Fun}(B,\operatorname{\mathcal{C}})$, every isomorphism $u: X \rightarrow F'(Y)$ in the $\infty$-category $\operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ can be lifted to an isomorphism $\overline{u}: \overline{X} \rightarrow Y$ in the $\infty$-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$. This follows immediately from Corollary 4.4.5.9. $\square$

Replacing Corollary 4.4.5.9 by Proposition 4.4.5.8 in the preceding argument, we obtain the following:

Variant 4.4.5.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset which contains every vertex of $B$. Then the induced map

$F': \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}})$

is an isofibration of $\infty$-categories.