11.9.6 Classification of Fibrations
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 5.2.2.4).
Proposition 11.9.6.5. 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
\[ U: M( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) ) \rightarrow \operatorname{\mathcal{C}}. \]
Corollary 11.9.6.6. 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.
Corollary 11.9.6.7. 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 11.9.6.5, 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 $M = \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{C}}(n) )$ is the mapping simplex of Notation 5.3.2.11. 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 4.5.4.12). Proposition 11.5.0.61 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 4.5.4.10). Applying Proposition 4.5.4.8, we deduce that the outer rectangle is also a categorical pushout square.
$\square$
Exercise 11.9.6.8. In the situation of Corollary 11.9.6.7, 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 $(4)$.