Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

9.9.2 Classification of Fibrations

Remark 9.9.2.1. Let $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ be a cocartesian fibration of $\infty $-categories. It follows from Proposition 9.5.0.8 that $\operatorname{\mathcal{C}}$ is determined, up to equivalence, by the diagram of covariant transport functors

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

Remark 9.9.2.2. 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 9.9.2.3. 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 5.3.4.2. 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 5.2.2.4).

Remark 9.9.2.4 (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 9.6.0.8 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

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

Proposition 9.9.2.5. 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 9.9.2.6. 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 9.9.2.7. 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 9.9.2.5, 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 9.5.0.8 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 9.9.2.8. In the situation of Corollary 9.9.2.7, 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 $(4)$.