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5.2.6 Fibrations over the $n$-Simplex

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a cocartesian fibration $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$. For $0 \leq i \leq n$, let $\operatorname{\mathcal{C}}(i)$ denote the fiber $\{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}$. For each integer $1 \leq i \leq n$, Proposition 5.2.2.4 supplies a functor $F(i): \operatorname{\mathcal{C}}(i-1) \rightarrow \operatorname{\mathcal{C}}(i)$, given by covariant transport along the edge $\operatorname{N}_{\bullet }( \{ i-1 < i \} ) \subseteq \Delta ^ n$. Our goal in this section is to show that $\operatorname{\mathcal{C}}$ can be reconstructed (up to equivalence) from the sequence of covariant transport functors

\[ \operatorname{\mathcal{C}}(0) \xrightarrow {F(1)} \operatorname{\mathcal{C}}(1) \xrightarrow {F(2)} \operatorname{\mathcal{C}}(2) \cdots \xrightarrow {F(n)} \operatorname{\mathcal{C}}(n); \]

see Proposition 5.2.6.19 and Remark 5.2.6.20. We begin by using the functors $\{ F(i) \} _{1 \leq i \leq n}$ to construct an approximation to the $\infty $-category $\operatorname{\mathcal{C}}$ which is well-defined up to isomorphism.

Notation 5.2.6.1 (The Universal Mapping Simplex). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets and let $\operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }})$ denote its nerve. For each $n \geq 0$, we can identify $\operatorname{N}_{n}(\operatorname{Set_{\Delta }})$ with the set of functors from the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ to the category $\operatorname{Set_{\Delta }}$. In what follows, we will typically denote such a functor by $\overrightarrow {X}$ and will denote its value on an integer $0 \leq i \leq n$ by $X(i)$. Let $\operatorname{N}^{+}_{n}(\operatorname{Set_{\Delta }})$ denote the set of pairs $( \overrightarrow {X}, \sigma )$, where $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$ is a functor and $\sigma $ is an $n$-simplex of the simplicial set $X(0)$. Every nondecreasing function $\alpha : [m] \rightarrow [n]$ determines a map $\alpha ^{\ast }: \operatorname{N}^{+}_{n}(\operatorname{Set_{\Delta }}) \rightarrow \operatorname{N}^{+}_{m}(\operatorname{Set_{\Delta }})$, which carries $( \overrightarrow {X}, \sigma )$ to the pair $( \overrightarrow {X} \circ \alpha , \sigma ' )$ where $\sigma '$ denotes the composite map

\[ \Delta ^{m} \xrightarrow {\alpha } \Delta ^{n} \xrightarrow {\sigma } X(0) \rightarrow X( \alpha (0) ). \]

The construction $[n] \mapsto \operatorname{N}^{+}_{n}( \operatorname{Set_{\Delta }})$ determines a simplicial set $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$, which we will refer to as the universal mapping simplex.

Remark 5.2.6.2. The universal mapping simplex $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$ is equipped with a morphism of simplicial sets $\pi : \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})$, given on $n$-simplices by the formula $\pi ( \overrightarrow {X}, \sigma ) = \overrightarrow {X}$. We will refer to $\pi $ as the projection map. Unwinding the definitions, we see that for every simplicial set $K$, there is a canonical isomorphism of simplicial sets

\[ K \simeq \{ K\} \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}). \]

Construction 5.2.6.3 (The Mapping Simplex). Suppose we are given a diagram of simplicial sets

\[ X(0) \xrightarrow {F(1)} X(1) \xrightarrow { F(2) } X(2) \xrightarrow {F(3)} \cdots \xrightarrow {F(n)} X(n), \]

which we identify with an $n$-simplex $\overrightarrow {X}$ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }})$. We let $M( \overrightarrow {X} )$ denote the fiber product $\Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$, where $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$ is the universal mapping simplex of Notation 5.2.6.1. We will refer to $M( \overrightarrow {X} )$ as the mapping simplex of the diagram $\overrightarrow {X}$.

Remark 5.2.6.4. Let $\overrightarrow {X}$ be a diagram of simplicial sets

\[ X(0) \xrightarrow {F(1)} X(1) \xrightarrow { F(2) } X(2) \xrightarrow {F(3)} \cdots \xrightarrow {F(n)} X(n). \]

By construction, the mapping simplex $M(\overrightarrow {X})$ is equipped with a projection map $M(\overrightarrow {X}) \rightarrow \Delta ^ n$. Moreover, for each integer $0 \leq i \leq n$, the fiber $\{ i\} \times _{\Delta ^ n} M( \overrightarrow {X} )$ is canonically isomorphic to the simplicial set $X(i)$ (Remark 5.2.6.2).

Remark 5.2.6.5 (Functoriality in $\Delta ^ n$). Let $\alpha : [m] \rightarrow [n]$ be a nondecreasing function of linearly ordered sets. Let $\overrightarrow {X}$ be a diagram of simplicial sets

\[ X(0) \xrightarrow {F(1)} X(1) \xrightarrow { F(2) } X(2) \xrightarrow {F(3)} \cdots \xrightarrow {F(n)} X(n), \]

and let $\overrightarrow {X}'$ denote the diagram

\[ X( \alpha (0) ) \rightarrow X( \alpha (1) ) \rightarrow \cdots \rightarrow X( \alpha (m) ) \]

obtained by precomposition with $\alpha $. Then the mapping simplices of $\overrightarrow {X}$ and $\overrightarrow {X}'$ fit into a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ M( \overrightarrow {X}' ) \ar [r] \ar [d] & M( \overrightarrow {X} ) \ar [d] \\ \Delta ^{m} \ar [r]^-{ \alpha } & \Delta ^ n. } \]

Example 5.2.6.6. Let $X$ be a simplicial set, let $n$ be a nonnegative integer, and let $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$ denote the constant diagram

\[ X \xrightarrow { \operatorname{id}_ X } X \xrightarrow { \operatorname{id}_ X } X \xrightarrow { \operatorname{id}_ X } \cdots \xrightarrow { \operatorname{id}_ X} X. \]

Then the mapping simplex $M( \overrightarrow {X} )$ can be identified with the cartesian product $\Delta ^ n \times X$.

Example 5.2.6.7. Let $\overrightarrow {X}$ be a diagram of simplicial sets

\[ X(0) \xrightarrow {F(1)} X(1) \xrightarrow { F(2) } X(2) \xrightarrow {F(3)} \cdots \xrightarrow {F(n)} X(n), \]

and suppose that the simplicial set $X(0)$ is empty. It follows from Remark 5.2.6.4 that the fiber $\{ 0\} \times _{ \Delta ^ n } M( \overrightarrow {X} )$ is also empty, so the inclusion map

\[ \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} ) \times _{ \Delta ^ n } M( \overrightarrow {X} ) \hookrightarrow M( \overrightarrow {X} ) \]

is bijective. If $n > 0$, then we can use Remark 5.2.6.5 to identify $M( \overrightarrow {X} )$ with the mapping simplex of the truncated diagram $X(1) \rightarrow X(2) \rightarrow X(3) \rightarrow \cdots \rightarrow X(n)$.

Remark 5.2.6.8 (Functoriality in $\overrightarrow {X}$). Let $n \geq 0$ be an integer. Then the mapping simplex construction $\overrightarrow {X} \mapsto M( \overrightarrow {X} )$ determines a functor from the category $\operatorname{Fun}( [n], \operatorname{Set_{\Delta }})$ to the category of simplicial sets. This functor carries colimits of diagrams to colimits in the category of simplicial sets, and carries limits of diagrams to limits in the slice category $(\operatorname{Set_{\Delta }})_{ / \Delta ^ n }$.

Remark 5.2.6.9. Let $n \geq 1$ and let $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets which we denote by

\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n). \]

Let $\overrightarrow {X}': [n] \rightarrow \operatorname{Set_{\Delta }}$ denote the constant diagram taking the value $X(0)$. Let $\overrightarrow {X}_0 \subseteq \overrightarrow {X}$ be the subfunctor given by the diagram

\[ \emptyset \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n), \]

and define $\overrightarrow {X}'_0 \subseteq \overrightarrow {X}'$ similarly, so that we have a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \overrightarrow {X}'_0 \ar [r] \ar [d] & \overrightarrow {X}_0 \ar [d] \\ \overrightarrow {X}' \ar [r] & \overrightarrow {X} } \]

in the category $\operatorname{Fun}( [n], \operatorname{Set_{\Delta }})$. Combining Remark 5.2.6.8, Example 5.2.6.7, and Example 5.2.6.6, we obtain a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} ) \times X(0) \ar [r] \ar [d] & M( X(1) \rightarrow \cdots \rightarrow X(n) ) \ar [d] \\ \Delta ^ n \times X(0) \ar [r] & M( X(0) \rightarrow \cdots \rightarrow X(n) ). } \]

Example 5.2.6.10. Let $F: X(0) \rightarrow X(1)$ be a morphism of simplicial sets, which we identify with a diagram $\overrightarrow {X}: [1] \rightarrow \operatorname{Set_{\Delta }}$. Then the mapping simplex $M( \overrightarrow {X} )$ can be identified with the pushout $( \Delta ^1 \times X(0) ) \coprod _{ (\{ 1\} \times X(0))} X(1)$.

Example 5.2.6.11. Let $n \geq 0$ and let $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets which we denote by

\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n). \]

For each integer $0 \leq i \leq n$, let $\Delta ^ n_{\geq i}$ denote the nerve of the linearly ordered set $\{ i < i+1 < \cdots < n \} $, which we regard as a simplicial subset of $\Delta ^ n$. Applying Remark 5.2.6.9 repeatedly, we can identify the mapping simplex $M( \overrightarrow {X} )$ with the iterated pushout

\[ (\Delta ^ n \times X(0)) \coprod _{ ( \Delta ^{n}_{\geq 1} \times X(0)) } (\Delta ^{n}_{\geq 1} \times X(1) ) \coprod _{ ( \Delta ^{n}_{\geq 2} \times X(1))} \cdots \coprod _{ (\{ n\} \times X(n-1))} (\{ n\} \times X(n)). \]

Definition 5.2.6.12. Let $n \geq 0$ and let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a functor $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$. For $0 \leq i \leq n$, let $\operatorname{\mathcal{C}}(i)$ denote the fiber $\{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}$, which we regard as a full subcategory of $\operatorname{\mathcal{C}}$. A scaffold of $\pi $ is a sequence of functors

\[ \operatorname{\mathcal{C}}(0) \xrightarrow { F(1) } \operatorname{\mathcal{C}}(1) \xrightarrow { F(2) } \operatorname{\mathcal{C}}(2) \xrightarrow { F(3) } \cdots \xrightarrow {F(n)} \operatorname{\mathcal{C}}(n) \]

together with a morphism of simplicial sets

\[ U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}} \]

with the following properties:

$(1)$

For each integer $0 \leq i \leq n$, the composite map

\[ \operatorname{\mathcal{C}}(i) \simeq \{ i \} \times _{\Delta ^ n} M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

coincides with the inclusion of $\operatorname{\mathcal{C}}(i)$ into $\operatorname{\mathcal{C}}$.

$(2)$

For every pair of integers $0 \leq i \leq j \leq n$ and every object $C \in \operatorname{\mathcal{C}}(i)$, the composition

\[ \Delta ^1 \times \{ C\} \rightarrow \operatorname{N}_{\bullet }( \{ i \leq j \} ) \times \operatorname{\mathcal{C}}(i) \rightarrow M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

is a $\pi $-cocartesian edge of $\operatorname{\mathcal{C}}$.

Example 5.2.6.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a functor $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^1$ having fibers $\operatorname{\mathcal{C}}(0) = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}(1) = \{ 1\} \times _{ \Delta ^1} \operatorname{\mathcal{C}}$. By virtue of Example 5.2.6.10, choosing a scaffold of $\pi $ is equivalent to choosing a functor $F: \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1)$ together with a map $h: \Delta ^1 \times \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}$ which witnesses $F$ as given by covariant transport along the nondegenerate edge of $\Delta ^1$, in the sense of Definition 5.2.2.1.

Remark 5.2.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a functor $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$. If there exists a scaffold of $\pi $, then $\pi $ is a cocartesian fibration (note that $\pi $ is automatically an inner fibration, by virtue of Proposition 4.1.1.10).

Remark 5.2.6.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a functor $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ having fibers $\{ \operatorname{\mathcal{C}}(i) = \{ i \} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\} _{0 \leq i \leq n}$. Suppose that we are given a sequence of functors

\[ \operatorname{\mathcal{C}}(0) \xrightarrow { F(1) } \operatorname{\mathcal{C}}(1) \xrightarrow { F(2)} \operatorname{\mathcal{C}}(2) \xrightarrow { F(3)} \cdots \xrightarrow {F(n)} \operatorname{\mathcal{C}}(n) \]

and a morphism of simplicial sets $U: M( \operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}}$ satisfying condition $(1)$ of Definition 5.2.6.12. Since the collection of $\pi $-cocartesian morphisms of $\operatorname{\mathcal{C}}$ is closed under composition (Corollary 5.1.2.4), we can replace $(2)$ with the following a priori weaker condition:

$(2')$

For every integer $1 \leq i \leq n$ and every object $C \in \operatorname{\mathcal{C}}(i-1)$, the composition

\[ \Delta ^1 \times \{ C\} \rightarrow \operatorname{N}_{\bullet }( \{ i - 1 < i \} ) \times \operatorname{\mathcal{C}}(i-1) \rightarrow M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

is a $\pi $-cocartesian edge of $\operatorname{\mathcal{C}}$.

If $\pi $ is a cocartesian fibration, then a morphism of $\operatorname{\mathcal{C}}$ is $\pi $-cocartesian if and only if it is locally $\pi $-cocartesian (Remark 5.1.4.5). In this case, we can restate $(2')$ as follows:

$(2'')$

For every integer $1 \leq i \leq n$, the composition

\[ \operatorname{N}_{\bullet }( \{ i-1 < i \} ) \times \operatorname{\mathcal{C}}(i-1) \rightarrow M(\operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

witnesses the functor $F(i): \operatorname{\mathcal{C}}(i-1) \rightarrow \operatorname{\mathcal{C}}(i)$ as given by covariant transport along the edge $\operatorname{N}_{\bullet }( \{ i-1 < i \} ) \subseteq \Delta ^ n$ (in the sense of Definition 5.2.2.1).

Remark 5.2.6.16 (Compatibility with Pullback). Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a functor of $\infty $-categories having fibers $\{ \operatorname{\mathcal{C}}(i) = \{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\} _{0 \leq i \leq n}$ and let

\[ U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}} \]

be a scaffold of $\pi $. For every morphism of simplices $\alpha : \Delta ^{m} \rightarrow \Delta ^{n}$, the pullback

\[ \Delta ^{m} \times _{ \Delta ^{n} } M( \operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {\operatorname{id}\times U} \Delta ^{m} \times _{\Delta ^ n} \operatorname{\mathcal{C}} \]

is a scaffold of the projection map $\Delta ^{m} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\rightarrow \Delta ^ m$; here we implicitly invoke Remark 5.2.6.5 to identify $\Delta ^{m} \times _{\Delta ^ n} M( \operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) )$ with the mapping simplex of the diagram

\[ \operatorname{\mathcal{C}}( \alpha (0) ) \rightarrow \operatorname{\mathcal{C}}( \alpha (1) ) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}( \alpha (m) ). \]

Our next goal is to show that every cocartesian fibration $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ admits a scaffold (Proposition 5.2.6.18). The proof will make use of the following observation concerning the universal mapping simplex $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$ of Notation 5.2.6.1:

Lemma 5.2.6.17. Let $f: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets, and let $g: B \rightarrow \operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }})$ be any morphism of simplicial sets. Then the induced map

\[ \theta _{g}: A \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}) \rightarrow B \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}) \]

is inner anodyne.

Proof. Let $S$ be the collection of all morphisms of simplicial sets $f: A \rightarrow B$ having the property that, for every morphism $g: B \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})$, the map $\theta _{g}$ is inner anodyne. It follows immediately from the definitions that $S$ is weakly saturated (in the sense of Definition 1.4.4.15). Consequently, to show that every inner anodyne morphism belongs to $S$, it will suffice to prove that $S$ contained every inner horn inclusion $f: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i < n$. In this case, we can identify a morphism $g: B \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})$ with a diagram of simplicial sets $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$, which we denote by

\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n). \]

To complete the proof, we must show that the inclusion map $\theta : \Lambda ^{n}_{i} \times _{\Delta ^ n} M( \overrightarrow {X} ) \hookrightarrow M( \overrightarrow {X} )$ is inner anodyne. Using Example 5.2.6.11, we see that $\theta $ is a pushout of the inclusion $\Lambda ^{n}_{i} \times X(0) \hookrightarrow \Delta ^ n \times X(0)$, which is inner anodyne by virtue of Lemma 1.4.7.5. $\square$

Proposition 5.2.6.18. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a cocartesian fibration of $\infty $-categories having fibers $\{ \operatorname{\mathcal{C}}(i) = \{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\} _{0 \leq i \leq n}$. Then there exists a scaffold

\[ U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}}. \]

Proof. For $1 \leq i \leq n$, Proposition 5.2.2.4 guarantees that there exists a functor $F(i): \operatorname{\mathcal{C}}(i-1) \rightarrow \operatorname{\mathcal{C}}(i)$ and a natural transformation $h(i): \Delta ^1 \times \operatorname{\mathcal{C}}(i-1) \rightarrow \operatorname{\mathcal{C}}$ which witnesses the functor $F(i)$ as given by covariant transport along the edge $\operatorname{N}_{\bullet }( \{ i-1 < i \} ) \subseteq \Delta ^ n$. Let $M$ denote the mapping simplex for the diagram $\operatorname{\mathcal{C}}(0) \xrightarrow {F(1)} \operatorname{\mathcal{C}}(1) \xrightarrow {F(2)} \operatorname{\mathcal{C}}(2) \xrightarrow {F(3)} \cdots \xrightarrow { F(n)} \operatorname{\mathcal{C}}(n)$. Unwinding the definitions, we see that the inclusion maps $\operatorname{\mathcal{C}}(i) \hookrightarrow \operatorname{\mathcal{C}}$ and the natural transformations $\{ h(i) \} _{1 \leq i \leq n}$ determine a morphism of simplicial sets $U_0: \operatorname{Spine}[n] \times _{\Delta ^ n} M \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{Spine}[n] \subseteq \Delta ^ n$ denotes the spine of the $n$-simplex (see Example 1.4.7.7). Since the inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.4.7.7), Lemma 5.2.6.17 guarantees that the inclusion $\operatorname{Spine}[n] \times _{\Delta ^ n} M \hookrightarrow M$ is also inner anodyne. Because $\operatorname{\mathcal{C}}$ is an $\infty $-category, we can extend $U_0$ to a morphism of simplicial sets $U: M \rightarrow \operatorname{\mathcal{C}}$ (Proposition 1.4.6.7). Using the criterion of Remark 5.2.6.15, we conclude that $U$ is a scaffold of $\pi $. $\square$

Proposition 5.2.6.19. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a cocartesian fibration of $\infty $-categories having fibers $\{ \operatorname{\mathcal{C}}(i) = \{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\} _{0 \leq i \leq n}$, and let

\[ U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}}. \]

be a scaffold of $\pi $. Then $U$ is a categorical equivalence of simplicial sets.

Remark 5.2.6.20. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a cocartesian fibration of $\infty $-categories. It follows from Proposition 5.2.6.19 that $\operatorname{\mathcal{C}}$ is determined, up to equivalence, by the diagram of covariant transport functors

\[ \operatorname{\mathcal{C}}(0) \xrightarrow {F(1)} \operatorname{\mathcal{C}}(1) \xrightarrow {F(2)} \operatorname{\mathcal{C}}(2) \xrightarrow {F(3)} \cdots \xrightarrow {F(n)} \operatorname{\mathcal{C}}(n). \]

Proof of Proposition 5.2.6.19. We proceed by induction on $n$. If $n=0$, then $U$ is an isomorphism and there is nothing to prove. Assume that $n \geq 1$, let $\operatorname{\mathcal{C}}(\geq 1)$ denote the fiber product $\operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} ) \times _{\Delta ^ n} \operatorname{\mathcal{C}}$, and let $A \subseteq \Delta ^ n$ be the simplicial subset given by the pushout

\[ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \coprod _{ \{ 1\} } \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} ). \]

Since the inclusion map $\{ 1\} \hookrightarrow \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} )$ is left anodyne (Lemma 4.3.7.8), it follows from Example 4.3.6.5 that the inclusion map $A \hookrightarrow \Delta ^ n$ is inner anodyne. Applying Lemma 5.2.6.17, we deduce that the inclusion

\[ A \times _{\Delta ^ n} M(\operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n)) \hookrightarrow M(\operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \]

is also inner anodyne. It will therefore suffice to show that the restriction of $U$ to $A \times _{\Delta ^ n} M(\operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n))$ is a categorical equivalence. Equivalently, we wish to show that the outer rectangle of the diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}(0) \ar [r] \ar@ {=}[d] & M( \operatorname{\mathcal{C}}(1) \rightarrow \operatorname{\mathcal{C}}(2) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \ar [d] \\ \{ 1\} \times \operatorname{\mathcal{C}}(0) \ar [r] \ar [d] & \operatorname{\mathcal{C}}(\geq 1) \ar [d] \\ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times \operatorname{\mathcal{C}}(0) \ar [r]^-{h} & \operatorname{\mathcal{C}}} \]

is a categorical pushout square. Our inductive hypothesis guarantees that the upper vertical maps in this diagram are categorical equivalences, so that the upper square is a categorical pushout (Proposition 4.5.3.7). By virtue of Proposition 4.5.3.5, it will suffice to show that the lower square is a categorical pushout diagram. Let $\rho : \Delta ^{n} \rightarrow \Delta ^1$ be the morphism given on vertices by the formula $\rho (i) = \begin{cases} 0 & \textnormal{ if } i = 0 \\ 1 & \textnormal{ if } i > 0.\end{cases}$ Then $\rho $ is a cocartesian fibration of simplicial sets, and the edge $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Delta ^ n$ is $\rho $-cocartesian. Note that $\operatorname{\mathcal{C}}(\geq 1)$ can be identified with the inverse image $(\rho \circ \pi )^{-1} \{ 1\} $. For every object $C \in \operatorname{\mathcal{C}}(0)$, it follows from Remark 5.1.1.6 that the composition

\[ \Delta ^1 \times \{ C\} \hookrightarrow \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times \operatorname{\mathcal{C}}(0) \rightarrow M( \operatorname{\mathcal{C}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \xrightarrow {U} \operatorname{\mathcal{C}} \]

is a $(\rho \circ \pi )$-cocartesian edge of $\operatorname{\mathcal{C}}$. The desired result now follows by applying the criterion of Theorem 5.2.5.1. $\square$

Corollary 5.2.6.21. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a cocartesian fibration of $\infty $-categories having fibers $\{ \operatorname{\mathcal{C}}(i) = \{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\} _{0 \leq i \leq n}$, and let

\[ U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}}. \]

be a scaffold of $\pi $. Then, for every morphism of simplicial sets $X \rightarrow \Delta ^ n$, the induced map

\[ U': X \times _{\Delta ^ n} M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow X \times _{\Delta ^ n} \operatorname{\mathcal{C}} \]

is a categorical equivalence of simplicial sets.

Proof. By virtue of Corollary 4.5.4.4, we may assume without loss of generality that $X = \Delta ^ m$ is a standard simplex. In this case, we can identify $U'$ with a scaffold of the projection map $\Delta ^ m \times _{\Delta ^ n} \operatorname{\mathcal{C}}\rightarrow \Delta ^ m$ (Remark 5.2.6.16), so the desired result follows from Proposition 5.2.6.19. $\square$

Corollary 5.2.6.22. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a cocartesian fibration of $\infty $-categories and let $\operatorname{\mathcal{C}}(0)$ denote the fiber $\{ 0\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}$. Then there exists a functor $V: \Delta ^ n \times \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}$ with the following properties:

$(1)$

The composition $\Delta ^ n \times \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}\xrightarrow {\pi } \Delta ^ n$ is given by projection onto the first factor (that is, $V$ is a morphism in the category $(\operatorname{Set_{\Delta }})_{/\Delta ^ n}$).

$(2)$

The restriction $V|_{ \{ 0\} \times \operatorname{\mathcal{C}}(0) }$ is equal to the identity map $\operatorname{id}_{\operatorname{\mathcal{C}}(0)}$.

$(3)$

For each object $C \in \operatorname{\mathcal{C}}(0)$, the restriction $V|_{ \Delta ^ n \times \{ C\} }: \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ carries each edge of $\Delta ^ n$ to a $\pi $-cocartesian morphism of $\operatorname{\mathcal{C}}$.

$(4)$

The diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \times \operatorname{\mathcal{C}}(0) \ar [r] \ar [d] & \operatorname{\partial \Delta }^ n \times _{\Delta ^ n} \operatorname{\mathcal{C}}\ar [d] \\ \Delta ^ n \times \operatorname{\mathcal{C}}(0) \ar [r]^-{V} & \operatorname{\mathcal{C}}} \]

is a categorical pushout square of simplicial sets.

Proof. For $0 \leq i \leq n$, let $\operatorname{\mathcal{C}}(i)$ denote the fiber $\{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{C}}$. By virtue of Proposition 5.2.6.18, there exists a sequence of functors

\[ \operatorname{\mathcal{C}}(0) \xrightarrow {F(1)} \operatorname{\mathcal{C}}(1) \xrightarrow {F(2)} \operatorname{\mathcal{C}}(2) \rightarrow \cdots \xrightarrow {F(n)} \operatorname{\mathcal{C}}(n) \]

and a morphism $U: M \rightarrow \operatorname{\mathcal{C}}$ which is a scaffold of $\pi $, where $M = M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) )$ is the mapping simplex of Construction 5.2.6.3. Let $V$ denote the composite map

\[ \Delta ^ n \times \operatorname{\mathcal{C}}(0) \rightarrow M \xrightarrow {U} \operatorname{\mathcal{C}}. \]

It follows immediately from the definitions that $V$ satisfies conditions $(1)$, $(2)$, and $(3)$. To prove $(4)$, we observe that there is a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \times \operatorname{\mathcal{C}}(0) \ar [r] \ar [d] & \operatorname{\partial \Delta }^ n \times _{\Delta ^ n} M \ar [r] \ar [d] & \operatorname{\partial \Delta }^ n \times _{\Delta ^ n} \operatorname{\mathcal{C}}\ar [d] \\ \Delta ^ n \times \operatorname{\mathcal{C}}(0) \ar [r] & M \ar [r]^-{U} & \operatorname{\mathcal{C}}. } \]

Note that the square on the left is a pushout diagram in which the vertical maps are monomorphisms, hence a categorical pushout diagram (Example 4.5.3.9). Corollary 5.2.6.21 implies that both of the horizontal maps on the right are categorical equivalences, so that the right square is also a categorical pushout diagram (Proposition 4.5.3.7). Applying Proposition 4.5.3.5, we deduce that the outer rectangle is also a categorical pushout square. $\square$

Exercise 5.2.6.23. In the situation of Corollary 5.2.6.22, show that any functor $V: \Delta ^ n \times \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}$ satisfying conditions $(1)$, $(2)$, and $(3)$ also satisfies condition $(4)$.

We conclude with a sample application of Proposition 5.2.6.19.

Proposition 5.2.6.24. Suppose we are given a pullback diagram of simplicial sets

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

where $\operatorname{\mathcal{D}}$ is an $\infty $-category and $\overline{F}$ is a categorical equivalence of simplicial sets. If $q$ is either a cartesian fibration or a cocartesian fibration, then $F$ is also a categorical equivalence of simplicial sets.

Remark 5.2.6.25. In the statement of Proposition 5.2.6.24, the hypothesis that $\operatorname{\mathcal{D}}$ is an $\infty $-category is not necessary: see Corollary 5.6.4.6.

The proof of Proposition 5.2.6.24 will make use of the following:

Lemma 5.2.6.26. Suppose we are given a pullback diagram of simplicial sets

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

where $\overline{F}$ is inner anodyne. If $q$ is either a cartesian fibration or a cocartesian fibration, then $F$ is a categorical equivalence of simplicial sets.

Proof. We will give the proof under the assumption that $q$ is a cocartesian fibration; the proof when $q$ is a cartesian fibration is similar. Let $S$ be the collection of all monomorphisms of simplicial sets $f: A \hookrightarrow B$ with the following property: for every morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{D}}$, the induced map $A \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\hookrightarrow B \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is a categorical equivalence. To complete the proof, it will suffice to show that the morphism $\overline{F}': \operatorname{\mathcal{D}}' \hookrightarrow \operatorname{\mathcal{D}}''$ belongs to $S$. In fact, we claim that every inner anodyne morphism of simplicial sets belongs to $S$. Using Remark 4.5.2.6, Remark 4.5.2.5, Corollary 4.5.4.2, and Remark 4.5.3.10, we see that $S$ is weakly saturated (see Definition 1.4.4.15). It will therefore suffice to show that $S$ contains every inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i < n$. To prove this, fix a morphism $f: \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$, and let $\operatorname{\mathcal{E}}$ denote the fiber product $\Delta ^ n \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. Then the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$ is a projection map having fibers $\{ \operatorname{\mathcal{E}}(j) = \{ j\} \times _{\Delta ^ n} \operatorname{\mathcal{E}}\} _{0 \leq j \leq n}$. By virtue of Proposition 5.2.6.18, the cocartesian fibration $\pi $ admits a scaffold $U: M( \operatorname{\mathcal{E}}(0) \rightarrow \operatorname{\mathcal{E}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{E}}(n) ) \rightarrow \operatorname{\mathcal{E}}$. We then have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \times _{\Delta ^ n} M(\operatorname{\mathcal{E}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{E}}(n) ) \ar [r] \ar [d] & \Lambda ^{n}_{i} \times _{\Delta ^ n} \operatorname{\mathcal{E}}\ar [d] \\ M( \operatorname{\mathcal{E}}(0) \rightarrow \cdots \rightarrow \operatorname{\mathcal{E}}(n) ) \ar [r]^-{U} & \operatorname{\mathcal{E}}, } \]

where the horizontal maps are categorical equivalences (Corollary 5.2.6.21). Consequently, to show that the right vertical map is a categorical equivalence, it will suffice to show that the left vertical map is a categorical equivalence, which follows from Lemma 5.2.6.17. $\square$

Proof of Proposition 5.2.6.24. Suppose we are given a pullback square

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

where $q$ is a cocartesian fibration of $\infty $-categories and $\overline{F}$ is a categorical equivalence; we wish to show that $F$ is also a categorical equivalence (the analogous assertion for cartesian fibrations follows by a similar argument). By virtue of Proposition 4.1.3.2, the morphism $\overline{F}$ factors as a composition

\[ \operatorname{\mathcal{D}}' \xrightarrow { \overline{F}' } \operatorname{\mathcal{D}}'' \xrightarrow { \overline{F}'' } \operatorname{\mathcal{D}}, \]

where $\overline{F}'$ is inner anodyne and $\overline{F}''$ is an inner fibration. Since $\operatorname{\mathcal{D}}$ is an $\infty $-category, it follows that $\operatorname{\mathcal{D}}''$ is also an $\infty $-category (Remark 4.1.1.9). Because $\overline{F}'$ is a categorical equivalence (Corollary 4.5.2.13), our assumption that $\overline{F}$ is a categorical equivalence guarantees that $\overline{F}''$ is an equivalence of $\infty $-categories (Remark 4.5.2.5). Proposition 5.1.4.8 guarantees that $q$ is an isofibration of $\infty $-categories, so that the projection map $\operatorname{\mathcal{D}}'' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is also an equivalence of $\infty $-categories (Corollary 4.5.4.6). It will therefore suffice to show that $\overline{F}'$ induces a categorical equivalence $\operatorname{\mathcal{C}}' \simeq \operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}'' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$, which is a special case of Lemma 5.2.6.26. $\square$