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Theorem 5.7.8.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Suppose that, for every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small. Then the simplicial set $\operatorname{TW}(\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is a contractible Kan complex.

Proof of Theorem 5.7.8.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small. We wish to show that the simplicial set $\operatorname{TW}(\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is a contractible Kan complex.

For every simplicial set $\operatorname{\mathcal{C}}_0$ equipped with a morphism $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, let $X( \operatorname{\mathcal{C}}_0 )$ denote the simplicial set $\operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$, where $\operatorname{\mathcal{E}}_0$ is the fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Note that the simplicial set $X(\operatorname{\mathcal{C}}) = \operatorname{TW}(\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ can be realized as the inverse limit of the tower

\[ \cdots \rightarrow X( \operatorname{sk}_{2}(\operatorname{\mathcal{C}}) ) \rightarrow X( \operatorname{sk}_{1}(\operatorname{\mathcal{C}}) ) \rightarrow X(\operatorname{sk}_{0}(\operatorname{\mathcal{C}}) ), \]

where each of the transition maps is a Kan fibration (Lemma 5.7.8.10). Consequently, to show that $X(\operatorname{\mathcal{C}})$ is a contractible Kan complex, it will suffice to show that each of the simplicial sets $X( \operatorname{sk}_{k}(\operatorname{\mathcal{C}}) )$ is a contractible Kan complex. Replacing $\operatorname{\mathcal{C}}$ by $\operatorname{sk}_{k}(\operatorname{\mathcal{C}})$, we can assume that the simplicial set $\operatorname{\mathcal{C}}$ has dimension $\leq k$, for some integer $k \geq -1$.

We now proceed by induction on $k$. In the case $k=-1$, the simplicial set $\operatorname{\mathcal{C}}$ is empty and $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is isomorphic to $\Delta ^0$. We may therefore assume without loss of generality that $k \geq 0$. Let $S$ be the collection of nondegenerate $k$-simplices of $\operatorname{\mathcal{C}}$, so that Proposition 1.1.3.13 supplies a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in S }{\coprod } \operatorname{\partial \Delta }^{k} \ar [r] \ar [d] & \underset { \sigma \in S }{\coprod } \Delta ^{k} \ar [d] \\ \operatorname{\mathcal{C}}_0 \ar [r] & \operatorname{\mathcal{C}}, } \]

where $\operatorname{\mathcal{C}}_0 = \operatorname{sk}_{k-1}(\operatorname{\mathcal{C}})$ is the $(k-1)$-skeleton of $\operatorname{\mathcal{C}}$. It follows from our inductive hypothesis that the simplicial set $X(\operatorname{\mathcal{C}}_0)$ is a contractible Kan complex. Consequently, to show that $X(\operatorname{\mathcal{C}})$ is a contractible Kan complex, it will suffice to show that the restriction map $\theta : X(\operatorname{\mathcal{C}}) \rightarrow X(\operatorname{\mathcal{C}}_0)$ is a trivial Kan fibration. Note that $\theta $ is a pullback of the restriction map

\[ \theta _0: X( \underset { \sigma \in S }{\coprod } \Delta ^{k} ) \rightarrow X( \underset { \sigma \in S }{\coprod } \operatorname{\partial \Delta }^{k} ). \]

We will complete the proof by showing that $\theta _0$ is a trivial Kan fibration. Since $\theta _0$ is a Kan fibration (Lemma 5.7.8.10), this is equivalent to the assertion that $\theta _0$ is a homotopy equivalence (Corollary 3.2.7.4). Our inductive hypothesis guarantees that the Kan complex $X( \underset { \sigma \in S }{\coprod } \operatorname{\partial \Delta }^{k} )$ is contractible. We are therefore reduced to showing that the Kan complex $X( \underset { \sigma \in S }{\coprod } \Delta ^{k} )$ is also contractible. Since the collection of contractible Kan complexes is closed under products, we are reduced to verifying the contractibility of the simplicial set $X( \operatorname{\mathcal{C}}_0 )$ in the special case where $\operatorname{\mathcal{C}}_0 = \Delta ^{k}$ is a standard simplex of dimension $k$. We now consider several cases:

  • In the case $k=0$, the desired result follows from Lemma 5.7.9.2.

  • In the case $k=1$, Lemma 5.7.9.1 supplies a trivial Kan fibration $X( \Delta ^1 ) \rightarrow X( \operatorname{\partial \Delta }^1 )$. Our inductive hypothesis guarantees that the Kan complex $X( \operatorname{\partial \Delta }^1 )$ is contractible, so that $X( \Delta ^1)$ is also contractible.

  • In the case $k \geq 2$, we can choose an integer $0 < i < k$. In this case, the inclusion $\Lambda ^{k}_{i} \hookrightarrow \Delta ^{k}$ is inner anodyne, so the restriction map $X( \Delta ^{k} ) \rightarrow X( \Lambda ^{k}_{i} )$ is a trivial Kan fibration (Lemma 5.7.8.10). Our inductive hypothesis guarantees that the Kan complex $X( \Lambda ^{k}_{i} )$ is contractible, so that $X( \Delta ^{k} )$ is also contractible.

$\square$