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5.2.5 Fibrations over the $1$-Simplex

Let $\operatorname{\mathcal{M}}$ be an $\infty $-category equipped with a cocartesian fibration $\pi : \operatorname{\mathcal{M}}\rightarrow \Delta ^1$. Our goal in this section is to show that $\operatorname{\mathcal{M}}$ is determined (up to equivalence) by the $\infty $-categories $\operatorname{\mathcal{C}}= \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$, $\operatorname{\mathcal{D}}= \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$, and the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ given by covariant transport along the nondegenerate edge of $\Delta ^1$. This is a consequence of the following:

Theorem 5.2.5.1. Let $\pi : \operatorname{\mathcal{M}}\rightarrow \Delta ^1$ be a functor of $\infty $-categories, and suppose we are given a commutative diagram $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r] \ar [d] & \{ 1\} \times \operatorname{\mathcal{D}}\ar [d]^-{u} \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r]^-{h} & \operatorname{\mathcal{M}}} \]

in the category $(\operatorname{Set_{\Delta }})_{/\Delta ^1}$. Then $\sigma $ is a categorical pushout diagram of simplicial sets (Definition 4.5.3.1) if and only if the following conditions are satisfied:

$(1)$

The restriction $h|_{ \{ 0\} \times \operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ is a categorical equivalence of simplicial sets.

$(2)$

The morphism $u: \operatorname{\mathcal{D}}\rightarrow \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ is a categorical equivalence of simplicial sets.

$(3)$

For every vertex $C \in \operatorname{\mathcal{C}}$, the restriction $h|_{ \Delta ^1 \times \{ C\} }$ is a $\pi $-cocartesian morphism of $\operatorname{\mathcal{M}}$.

Moreover, if these conditions are satisfied, then $\pi $ is a cocartesian fibration.

Corollary 5.2.5.2. Let $\pi : \operatorname{\mathcal{M}}\rightarrow \Delta ^1$ be a cocartesian fibration of $\infty $-categories with fibers $\operatorname{\mathcal{C}}= \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ and $\operatorname{\mathcal{D}}= \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$. Let $h: \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{M}}$ be a functor which witnesses the functor $F = h|_{ \{ 1\} \times \operatorname{\mathcal{C}}}$ as given by covariant transport along the nondegenerate edge of $\Delta ^1$ (Definition 5.2.2.1). Then $h$ induces a categorical equivalence of simplicial sets

\[ (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ (\{ 1\} \times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{M}}. \]

Remark 5.2.5.3. Let $\pi : \operatorname{\mathcal{M}}\rightarrow \Delta ^1$ be a cocartesian fibration of $\infty $-categories. It follows from Corollary 5.2.5.2 that the $\infty $-category $\operatorname{\mathcal{M}}$ can be recovered (up to equivalence) from the $\infty $-categories $\operatorname{\mathcal{C}}= \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$, $\operatorname{\mathcal{D}}= \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$, and the covariant transport functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Similarly, if $\pi : \operatorname{\mathcal{M}}\rightarrow \Delta ^1$ is a cartesian fibration, then the $\infty $-category $\operatorname{\mathcal{M}}$ can be recovered from $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and the contravariant transport functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

We will deduce Theorem 5.2.5.1 from the following:

Proposition 5.2.5.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then the diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d] & \operatorname{\mathcal{D}}\ar [d] \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r]^-{\rho } & \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}} \]

is a categorical pushout square of simplicial sets. Here $\rho $ denotes the composite map

\[ \Delta ^1 \times \operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow { \operatorname{id}\star _{F} F} \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}. \]

Example 5.2.5.5. In the special case $\operatorname{\mathcal{D}}= \Delta ^0$, Proposition 5.2.5.4 asserts that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r] \ar [d] & \Delta ^{0} \ar [d] \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright } } \]

is a categorical pushout square: that is, that the comparison map $\operatorname{\mathcal{C}}\diamond \Delta ^{0} \rightarrow \operatorname{\mathcal{C}}\star \Delta ^{0}$ of Notation 4.5.5.3 is a categorical equivalence. This is the content of Proposition 4.5.5.12 (which is a special case of Theorem 4.5.5.8).

Proof of Theorem 5.2.5.1 from Proposition 5.2.5.4. Let $\pi : \operatorname{\mathcal{M}}\rightarrow \Delta ^1$ be a functor of $\infty $-categories and suppose we are given a commutative diagram $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d] & \{ 1\} \times \operatorname{\mathcal{D}}\ar [d]^-{u} \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r]^-{h} & \operatorname{\mathcal{M}}} \]

in the category $(\operatorname{Set_{\Delta }})_{/\Delta ^1}$. We wish to show that $\sigma $ is a categorical pushout square if and only if conditions $(1)$ through $(4)$ of Theorem 5.2.5.1 are satisfied.

We first reduce to the case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories. Choose inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{D}}'$, where $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{D}}'$ are $\infty $-categories (Corollary 4.1.3.3). Since the fiber $\{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ is an $\infty $-category, we can extend $u$ to a functor $u': \operatorname{\mathcal{D}}' \rightarrow \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$. Similarly, the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{D}}'$ extends to a functor of $\infty $-categories $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}'$. Using Exercise 3.1.7.11, we can factor $F'$ as a composition $\operatorname{\mathcal{C}}' \xrightarrow {F''} \operatorname{\mathcal{D}}'' \xrightarrow {v} \operatorname{\mathcal{D}}'$, where $F''$ is a monomorphism and $v$ is a trivial Kan fibration. It follows from Lemma 1.4.7.5 that the inclusion map

\[ ( \Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ ( \{ 1\} \times \operatorname{\mathcal{C}}) } (\{ 1\} \times \operatorname{\mathcal{C}}' ) \hookrightarrow \Delta ^1 \times \operatorname{\mathcal{C}}' \]

is inner anodyne, so that we can extend $h$ to a functor $h': \Delta ^1 \times \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{M}}$ satisfying $h'|_{ \{ 1\} \times \operatorname{\mathcal{C}}'} = u' \circ F'$. By virtue of Proposition 4.5.3.5, $\sigma $ is a categorical pushout square if and only if the diagram $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}' \ar [r]^-{F''} \ar [d] & \{ 1\} \times \operatorname{\mathcal{D}}'' \ar [d]^-{u' \circ v} \\ \Delta ^1 \times \operatorname{\mathcal{C}}' \ar [r]^-{h'} & \operatorname{\mathcal{M}}} \]

is a categorical pushout square. We may therefore replace $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ by $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{D}}''$, and thereby reduce to the case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories and $F$ is a monomorphism.

The assumption that $F$ is a monomorphism guarantees that the natural map

\[ \iota : (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ ( \{ 1\} \times \operatorname{\mathcal{C}})} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}} \]

is also a monomorphism, and Proposition 5.2.5.4 guarantees that $\iota $ is a categorical equivalence of simplicial sets. Since $\operatorname{\mathcal{M}}$ is an $\infty $-category, Lemma 4.5.6.2 guarantees the existence of a functor $U: \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{M}}$ satisfying $U|_{ \Delta ^1 \times \operatorname{\mathcal{C}}} = h$ and $U|_{\operatorname{\mathcal{D}}} = u$. By virtue of Proposition 4.5.3.5, the diagram $\sigma $ is a categorical pushout square if and only if the functor $U$ is an equivalence of $\infty $-categories.

Note that the functor $U$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [rr]^{U} \ar [dr]^{\pi '} & & \operatorname{\mathcal{M}}\ar [dl]^{\pi } \\ & \Delta ^1, & } \]

where $\pi '$ is the cocartesian fibration of Proposition 5.2.4.16, and the functor $\pi $ is automatically an isofibration (Example 4.4.1.6). The desired result now follows by applying the criterion of Theorem 5.1.5.1 (and invoking Remark 5.1.5.8). $\square$

Proof of Proposition 5.2.5.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. We wish to show that the natural map

\[ \theta _{\operatorname{\mathcal{C}}}: (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ ( \{ 1\} \times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}} \]

is a categorical equivalence of simplicial sets. By virtue of Corollary 4.5.4.3, it will suffice to prove that for every map $\Delta ^{n} \rightarrow \operatorname{\mathcal{D}}$, the induced map

\[ \Delta ^{n} \times _{\operatorname{\mathcal{D}}} ((\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ ( \{ 1\} \times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{D}}) \rightarrow \Delta ^{n} \times _{\operatorname{\mathcal{D}}} ( \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}) \]

is a categorical equivalence. Using Remark 5.2.4.9, we can replace $\operatorname{\mathcal{D}}$ by $\Delta ^ n$ and $\operatorname{\mathcal{C}}$ by the fiber product $\Delta ^ n \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$, and thereby reduce the proof of Proposition 5.2.5.4 to the special case where $\operatorname{\mathcal{D}}= \Delta ^ n$ is a standard simplex.

Since the collection of categorical equivalences is closed under the formation of filtered colimits (Corollary 4.5.4.2), we may assume without loss of generality that the simplicial set $\operatorname{\mathcal{C}}$ is finite (see Remark 3.5.1.8). In particular, $\operatorname{\mathcal{C}}$ has dimension $\leq m$ for some integer $m \geq -1$. We proceed by induction on $m$. If $m = -1$, then $\operatorname{\mathcal{C}}$ is empty and the morphism $\theta _{\operatorname{\mathcal{C}}}$ is an isomorphism (see Example 5.2.4.4). Assume that $m \geq 0$; we now proceed by induction on the number of nondegenerate $m$-simplices of $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ does not have dimension $\leq m-1$, then a choice of nondegenerate $m$-simplex of $\operatorname{\mathcal{C}}$ determines a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ m \ar [r] \ar [d] & \Delta ^ m \ar [d] \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}, } \]

where the horizontal maps are monomorphisms (Proposition 1.1.3.13). We then obtain a cubical diagram

\[ \xymatrix@R =50pt@C=10pt{ (\Delta ^1 \times \operatorname{\partial \Delta }^ m) \coprod _{ ( \{ 1\} \times \operatorname{\partial \Delta }^ m ) } \operatorname{\mathcal{D}}\ar [dr]_{ \theta _{ \operatorname{\partial \Delta }^{m} }} \ar [rr] \ar [dd] & & (\Delta ^1 \times \Delta ^ m) \coprod _{ ( \{ 1\} \times \Delta ^ m ) } \operatorname{\mathcal{D}}\ar [dr]_{ \theta _{\Delta ^{m}} } \ar [dd] & \\ & \operatorname{\partial \Delta }^ m \star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [dd] \ar [rr] & & \Delta ^{m} \star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [dd] \\ (\Delta ^1 \times \operatorname{\mathcal{C}}' ) \coprod _{ (\{ 1\} \times \operatorname{\mathcal{C}}')} \operatorname{\mathcal{D}}\ar [dr]_{\theta _{\operatorname{\mathcal{C}}'} } \ar [rr] & & (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ (\{ 1\} \times \operatorname{\mathcal{C}})} \operatorname{\mathcal{D}}\ar [dr]_{\theta _{\operatorname{\mathcal{C}}} } & \\ & \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [rr] & & \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, } \]

where the front and back faces are categorical pushout squares (Proposition 4.5.3.7). Our inductive hypothesis guarantees that the morphisms $\theta _{\operatorname{\mathcal{C}}'}$ and $\theta _{\operatorname{\partial \Delta }^ m}$ are categorical equivalences. Consequently, to show that $\theta _{\operatorname{\mathcal{C}}}$ is a categorical equivalence, it will suffice to show that $\theta _{\Delta ^ m}$ is a categorical equivalence. We can therefore replace $\operatorname{\mathcal{C}}$ by $\Delta ^ m$, and thereby reduce the proof of Proposition 5.2.5.4 to the special case where $F: \Delta ^{m} \rightarrow \Delta ^{n}$ is a morphism between standard simplices.

Suppose that $F(m) < n$. In this case, we can identify $F$ with a morphism from $\operatorname{\mathcal{C}}= \Delta ^{m}$ to the simplex $\Delta ^{n-1}$ (regarded as a simplicial subset of $\Delta ^{n}$), and we can identify $\operatorname{\mathcal{C}}\star _{\Delta ^{n}} \Delta ^{n}$ with the right cone $( \operatorname{\mathcal{C}}\star _{\Delta ^{n-1} } \Delta ^{n-1} )^{\triangleright }$. Under this identification, $\theta _{\operatorname{\mathcal{C}}}$ corresponds to the composition the composition

\begin{eqnarray*} (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ (\{ 1\} \times \operatorname{\mathcal{C}})} (\Delta ^{n-1})^{\triangleright } & \xrightarrow {\theta '} & (\Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleright } \coprod _{ (\{ 1\} \times \operatorname{\mathcal{C}})^{\triangleright }} (\Delta ^{n-1})^{\triangleright } \\ & \simeq & (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ (\{ 1\} \times \operatorname{\mathcal{C}})} \Delta ^{n-1} )^{\triangleright } \\ & \xrightarrow {\theta ''} & ( \operatorname{\mathcal{C}}\star _{\Delta ^{n-1} } \Delta ^{n-1} )^{\triangleright }, \end{eqnarray*}

where $\theta '$ is a pushout of the map

\[ ( \Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ ( \{ 1\} \times \operatorname{\mathcal{C}})} ( \{ 1\} \times \operatorname{\mathcal{C}})^{\triangleright } \rightarrow (\Delta ^1 \times \operatorname{\mathcal{C}})^{\triangleright } \]

and is therefore inner anodyne by virtue of Example 4.3.6.5 (since the inclusion $\{ 1\} \times \operatorname{\mathcal{C}}\hookrightarrow \Delta ^1 \times \operatorname{\mathcal{C}}$ is right anodyne; see Proposition 4.2.5.3). Consequently, to show that $\theta _{\operatorname{\mathcal{C}}}$ is a categorical equivalence, it will suffice to show that $\theta ''$ is a categorical equivalence. By virtue of Corollary 4.5.5.9, we are reduced to proving Proposition 5.2.5.4 for the map $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^{n-1}$. Applying this argument repeatedly, we can reduce to the case where $F(m) = n$.

Let $X(0)$ denote the simplicial subset of $\Delta ^1 \times \Delta ^ m$ given by the union of $\Delta ^1 \times \operatorname{\partial \Delta }^ m$ with $\{ 1\} \times \Delta ^ m$, and let

\[ X(0) \subset X(1) \subset X(2) \subset \cdots \subset X(m) \subset X(m+1) = \Delta ^1 \times \Delta ^ m \]

be the sequence of simplicial subsets appearing in Lemma 3.1.2.10. Note that $\theta _{\Delta ^ m}$ carries $X(m)$ into the simplicial subset $\operatorname{\partial \Delta }^ m \star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\subseteq \Delta ^{m} \star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. We therefore obtain a cubical diagram of simplicial sets

\[ \xymatrix@R =50pt@C=40pt{ X(0) \ar [dr] \ar [rr] \ar [dd] & & (\Delta ^1 \times \operatorname{\partial \Delta }^ m) \coprod _{ (\{ 1\} \times \operatorname{\partial \Delta }^ m) } \operatorname{\mathcal{D}}\ar [dr]_{ \theta _{\operatorname{\partial \Delta }^ m} } \ar [dd] & \\ & X(m) \ar [rr] \ar [dd] & & \operatorname{\partial \Delta }^{m} \star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [dd] \\ \Delta ^{1} \times \Delta ^{m} \ar [dr]_{\operatorname{id}} \ar [rr] & & (\Delta ^{1} \times \Delta ^ m) \coprod _{ (\{ 1\} \times \Delta ^ m) } \operatorname{\mathcal{D}}\ar [dr]_{ \theta _{\Delta ^{m}} } & \\ & \Delta ^{1} \times \Delta ^{m} \ar [rr] & & \Delta ^{m} \star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}} \]

where the front and back faces are pushout squares and the vertical maps are monomorphisms. It follows that the front and back faces are categorical pushout squares (Example 4.5.3.8). Our inductive hypothesis guarantees that $\theta _{\operatorname{\partial \Delta }^{m}}$ is a categorical equivalence, and the inclusion $X(0) \hookrightarrow X(m)$ is inner anodyne by construction (see Lemma 3.1.2.10). Applying Proposition 4.5.3.5, we conclude that $\theta _{\Delta ^ m}$ is also a categorical equivalence. $\square$