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5.6.5 Transport Witnesses

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, and assume that for every object $X \in \operatorname{\mathcal{C}}$ the fiber $\operatorname{\mathcal{E}}_{X} = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small. Theorem 5.6.0.2 asserts that there exists a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^{ \widetilde{\mathscr {F}} } \ar [d]^{U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^{\mathscr {F} } & \operatorname{\mathcal{QC}}} \]

which witnesses $\mathscr {F}$ as a covariant transport representation for $U$; here $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ is the cocartesian fibration of Proposition 5.4.6.11. In this section, we formulate a stronger statement, which asserts that the collection of all such diagrams is parametrized by a contractible Kan complex (Theorem 5.6.5.3).

Notation 5.6.5.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. We let $\operatorname{TW}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ denote the simplicial subset of the fiber product

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \]

whose $n$-simplices are diagrams

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^ n \times \operatorname{\mathcal{E}}\ar [r]^{ \widetilde{\mathscr {F}} } \ar [d]^{\operatorname{id}_{\Delta ^{n}} \times U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \Delta ^{n} \times \operatorname{\mathcal{C}}\ar [r]^{\mathscr {F} } & \operatorname{\mathcal{QC}}} \]

which witness $\mathscr {F}$ as a covariant transport representation for the cocartesian fibration $(\operatorname{id}_{ \Delta ^{n} } \times U): \Delta ^{n} \times \operatorname{\mathcal{E}}\rightarrow \Delta ^{n} \times \operatorname{\mathcal{C}}$.

Example 5.6.5.2. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^0$ denote the projection map. Note that projection onto the first factor determines a morphism of simplicial sets

\[ \operatorname{TW}( \operatorname{\mathcal{E}}/ \Delta ^{0} ) \rightarrow \operatorname{Fun}( \Delta ^0, \operatorname{\mathcal{QC}}) = \operatorname{\mathcal{QC}}. \]

Unwinding the definitions, we see that the fiber of this morphism over a small $\infty $-category $\operatorname{\mathcal{E}}'$ can be identified with the full subcategory

\[ \operatorname{Equiv}( \operatorname{\mathcal{E}}, \{ \operatorname{\mathcal{E}}' \} \times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{E}}, \{ \operatorname{\mathcal{E}}' \} \times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}})^{\simeq } \]

spanned by the equivalences of $\infty $-categories $\operatorname{\mathcal{E}}\rightarrow \{ \operatorname{\mathcal{E}}' \} \times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$.

We will prove the following result in §5.6.6:

Theorem 5.6.5.3. 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.

Remark 5.6.5.4. Theorem 5.6.5.3 is an immediate consequence of Theorem 5.6.2.8. We will see at the end of this section that the converse is also true.

The remainder of this section is devoted to establishing some formal properties of the simplicial sets $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ which will be useful for the proof of Theorem 5.6.5.3.

Lemma 5.6.5.5. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{\mathscr {F} }} \ar [d]^{U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F} } & \operatorname{\mathcal{QC}}, } \]

where $U$ is a cocartesian fibration. Let $j: \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ be an inner anodyne morphism of simplicial sets, let $\operatorname{\mathcal{E}}_0$ denote the fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, and let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ denote the projection map. If $\widetilde{\mathscr {F}}|_{ \operatorname{\mathcal{E}}_0}$ witnesses $\mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ as a covariant transport representation for $U_0$, then $\widehat{\mathscr {F}}$ witnesses $\mathscr {F}$ as a covariant transport representation for $U$.

Proof. Let $S$ denote the collection of all morphisms of simplicial sets $i: A \rightarrow B$ with the following property: for every morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{C}}$, if the restriction $\widetilde{\mathscr {F}}|_{ A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}}$ witnesses $\mathscr {F}|_{ A }$ as a covariant transport representation for the projection map $A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow A$, then $\widetilde{\mathscr {F}}|_{ B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}}$ witnesses $\mathscr {F}|_{ B }$ as a covariant transport representation for the projection map $B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow B$. To prove Lemma 5.6.5.5, it will suffice to show that every inner anodyne morphism of simplicial sets belongs to $S$. It is not difficult to see that the collection of morphisms $S$ is weakly saturated, in the sense of Definition 1.4.4.15. It will therefore suffice to show that, for every pair of integers $0 < i < n$, the inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $S$. We may therefore assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ and $\operatorname{\mathcal{C}}_0 = \Lambda ^{n}_{i}$ is an inner horn.

Since every vertex of $\Delta ^ n$ is contained in $\Lambda ^{n}_{i}$, it follows immediately that the pair $(\mathscr {F}, \widetilde{\mathscr {F}})$ satisfies condition $(a)$ of Remark 5.6.2.3. To verify $(b)$, let $e: X \rightarrow Z$ be an $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ having image $\overline{e} = U(e)$ in $\Delta ^ n$; we wish to show that $\widetilde{\mathscr {F}}(e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}'$. If $\overline{e}$ belongs to the horn $\Lambda ^ n_{i}$, then this follows from our assumption on $\widetilde{F}|_{ \operatorname{\mathcal{E}}_0 }$. We may therefore assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^{2}$ and that $\overline{e}: 0 \rightarrow 2$ is the “long” edge of the simplex $\Delta ^{2}$. Since $U$ is a cocartesian fibration, there exists a $U$-cocartesian edge $e': X \rightarrow Y$ of $\operatorname{\mathcal{E}}$, where $U(Y) = 1$. Our assumption that $e'$ is $U$-cocartesian guarantees the existence of a $2$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{e''} & \\ X \ar [ur]^{ e'} \ar [rr]^{e} & & Z } \]

of $\operatorname{\mathcal{E}}$, and Proposition 5.1.4.12 implies that $e''$ is also $U$-cocartesian. Since $\widetilde{\mathscr {F}}|_{ \operatorname{\mathcal{C}}_0}$ carries $U_0$-cocartesian morphisms of $\operatorname{\mathcal{E}}_0$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$, it follows that $\widetilde{\mathscr {F}}(e')$ and $\widetilde{\mathscr {F}}(e'')$ are $V$-cocartesian edges of $\operatorname{\mathcal{E}}'$. Applying Proposition 5.1.4.12 again, we deduce that $\widetilde{\mathscr {F}}(e)$ is also $V$-cocartesian. $\square$

Lemma 5.6.5.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then:

$(1)$

The fiber product $\operatorname{\mathcal{M}}= \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} )$ is an $\infty $-category.

$(2)$

The simplicial set $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is a replete subcategory of $\operatorname{\mathcal{M}}$ (see Example 4.4.1.11).

In particular, the simplicial set $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is an $\infty $-category.

Proof. Since $V$ is an inner fibration, the induced map $V': \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}})$ is also an inner fibration (Corollary 4.1.4.3). The projection map $\operatorname{\mathcal{M}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ is a pullback of $V'$, and is therefore also an inner fibration. Since $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ is an $\infty $-category (Theorem 1.4.3.7), assertion $(1)$ follows from Remark 4.1.1.9.

We now prove $(2)$. We first show that $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is a subcategory of $\operatorname{\mathcal{M}}$: that is, that the inclusion map $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{\mathcal{M}}$ is an inner fibration. Fix integers $0 < i < n$ and let $\sigma $ be an $n$-simplex of $\operatorname{\mathcal{M}}$ for which the restriction $\sigma |_{ \Lambda ^{n}_{i} }$ belongs to $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$; we wish to show that $\sigma $ is an $n$-simplex of $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$. Unwinding the definitions, we can identify $\sigma $ with a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^ n \times \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{\mathscr {F}} } \ar [d]^{ \operatorname{id}_{\Delta ^ n} \times U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \Delta ^ n \times \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{\mathcal{QC}}; } \]

we wish to show that $\widetilde{\mathscr {F}}$ witnesses $\mathscr {F}$ as a covariant transport representation for the cocartesian fibration $\operatorname{id}_{\Delta ^{n}} \times U$. This follows from Lemma 5.6.5.5, since the inclusion $\Lambda ^{n}_{i} \times \operatorname{\mathcal{C}}\hookrightarrow \Delta ^{n} \times \operatorname{\mathcal{C}}$ is inner anodyne (Lemma 1.4.7.5).

We now complete the proof by showing that the subcategory $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \subseteq \operatorname{\mathcal{M}}$ is replete. Let $u$ be an isomorphism in the $\infty $-category $\operatorname{\mathcal{M}}$, which we identify with a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{\mathscr {F}} } \ar [d]^{ \operatorname{id}_{\Delta ^1} \times U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{\mathcal{QC}}. } \]

Set $\mathscr {F}_0 = \mathscr {F}|_{ \{ 0\} \times \operatorname{\mathcal{C}}}$ and $\widetilde{\mathscr {F}}_{0} = \widetilde{\mathscr {F}}|_{ \{ 0\} \times \operatorname{\mathcal{E}}}$, and suppose that the pair $(\mathscr {F}_0, \widetilde{\mathscr {F}}_0)$ is an object of the $\infty $-category $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ (that is, $\widetilde{\mathscr {F}}_0$ witnesses $\mathscr {F}_0$ as a covariant transport representation for $U$. We wish to show that $\widetilde{\mathscr {F}}$ witnesses $\mathscr {F}$ as a covariant transport representation for $( \operatorname{id}_{\Delta ^1} \times U): \Delta ^1 \times \operatorname{\mathcal{E}}\rightarrow \Delta ^1 \times \operatorname{\mathcal{C}}$.

We first verify condition $(b)$ of Remark 5.6.2.3. Let $e$ be an $(\operatorname{id}_{\Delta ^1} \times U)$-cocartesian edge of the simplicial set $\Delta ^1 \times \operatorname{\mathcal{E}}$; we wish to show that $\widetilde{\mathscr {F}}( e )$ is a $V$-cocartesian morphism of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$. Write $e = ( \varphi _{ij}, \overline{e})$, where $\varphi _{ij}: i \rightarrow j$ is an edge of $\Delta ^1$ and $\overline{e}: X \rightarrow Y$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$. We consider three cases:

$(1)$

Suppose that $i=j=0$. Then $\widetilde{\mathscr {F}}(e) = \widetilde{\mathscr {F}}_0( \overline{e} )$ is $V$-cocartesian by virtue of our assumption that $\widetilde{\mathscr {F}}_{0}$ witnesses $\mathscr {F}_0$ as a covariant transport representation for $U$.

$(2)$

Suppose that $i=0$ and $j=1$. In this case, there exists a $2$-simplex of $\Delta ^1 \times \operatorname{\mathcal{E}}$ whose boundary is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & (0,Y) \ar [dr]^{ ( \varphi _{01}, \operatorname{id}_ Y) } & \\ (0,X) \ar [ur]^{ (\varphi _{00}, \overline{e} )} \ar [rr]^{ ( \varphi _{01}, \overline{e} ) } & & (1,Y). } \]

Our assumption that $u$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{M}}$ guarantees that $\widetilde{\mathscr {F}}( \varphi _{01}, \operatorname{id}_ Y)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$, and is therefore $V$-cocartesian (Proposition 5.1.1.8). It follows from case $(1)$ that $\widetilde{\mathscr {F}}( \varphi _{00}, \overline{e} )$ is also a $V$-cocartesian morphism of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$. Since the collection of $V$-cocartesian morphisms of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ is closed under composition (Corollary 5.1.2.4), we conclude that $\widetilde{\mathscr {F}}( \varphi _{01}, \overline{e})$ is also $V$-cocartesian.

$(3)$

Suppose that $i=j=1$. In this case, there exists a $2$-simplex of $\Delta ^1 \times \operatorname{\mathcal{E}}$ whose boundary is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & (1,X) \ar [dr]^{ ( \varphi _{11}, \overline{e}) } & \\ (0,X) \ar [ur]^{ (\varphi _{01}, \operatorname{id}_{X} )} \ar [rr]^{ ( \varphi _{01}, \overline{e} ) } & & (1,Y). } \]

Our assumption that $u$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{M}}$ guarantees that $\widetilde{\mathscr {F}}( \varphi _{01}, \operatorname{id}_ X)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$, and is therefore $V$-cocartesian (Proposition 5.1.1.8). It follows from case $(2)$ that $\widetilde{\mathscr {F}}( \varphi _{01}, \overline{e} )$ is also a $V$-cocartesian morphism of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$, so that $\widetilde{\mathscr {F}}( \varphi _{11}, \overline{e} )$ is $V$-cocartesian by virtue of Corollary 5.1.2.4.

We now complete the proof by showing that the pair $( \widetilde{\mathscr {F}}, \mathscr {F})$ satisfies condition $(a)$ of Remark 5.6.2.3. Let $(i,C)$ be a vertex of the product $\Delta ^1 \times \operatorname{\mathcal{C}}$, so that $\widetilde{\mathscr {F}}$ restricts to a functor of $\infty $-categories

\[ \widetilde{\mathscr {F}}_{(i,C)}: \{ i \} \times \operatorname{\mathcal{E}}_{C} \rightarrow \{ \mathscr {F}(i,C) \} \times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}. \]

We wish to show that the functor $\widetilde{\mathscr {F}}_{(i,C)}$ is an equivalence of $\infty $-categories. If $i=0$, this follows from our assumption that $\widetilde{\mathscr {F}}_{0}$ witnesses $\mathscr {F}_0$ as a covariant transport representation for $U$. We may therefore assume without loss of generality that $i=1$. Set $v = \mathscr {F}( \varphi _{01}, \operatorname{id}_{C})$ and let

\[ v_{!}: \{ \mathscr {F}( 0, C) \} \times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \{ \mathscr {F}(1,C) \} \times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \]

be the functor given by covariant transport along $v$. Since $u$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{M}}$, $v$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{QC}}$ so that $v_{!}$ is an equivalence of $\infty $-categories (Remark 5.2.3.4). Combining the first part of the proof with Remark 5.2.7.5, we deduce that the diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ \widetilde{\mathscr {F}}_{(0,C)} } \ar [d]^{\sim } & \{ \mathscr {F}( 0, C) \} \times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{ v_{!} } \\ \{ 1\} \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ \widetilde{\mathscr {F}}_{(1,C)}} & \{ \mathscr {F}( 1, C) \} \times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} } \]

commutes up to isomorphism (that is, it determines a commutative diagram in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$). Since $v_{!}$ and $\widetilde{\mathscr {F}}_{(0,C)}$ are equivalences of $\infty $-categories, it follows that $\widetilde{\mathscr {F}}_{(1,C)}$ is also an equivalence of $\infty $-categories. $\square$

Lemma 5.6.5.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then the simplicial set $\operatorname{TW}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ is a Kan complex.

Proof. Since $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is an $\infty $-category (Lemma 5.6.5.6), it will suffice to show that every morphism $u$ of $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is an isomorphism (Proposition 4.4.2.1). Let us identify $u$ with a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{\mathscr {F}} } \ar [d]^{ \operatorname{id}_{\Delta ^1} \times U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{\mathcal{QC}}} \]

satisfying conditions $(a)$ and $(b)$ of Remark 5.6.2.3.

Passing to homotopy categories, we see that $\mathscr {F}$ induces a functor $\mathrm{h} \mathit{\mathscr {F}}: [1] \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \simeq \mathrm{h} \mathit{\operatorname{QCat}}$. Applying Remark 5.6.2.6, we see that $\mathrm{h} \mathit{\mathscr {F}}$ is isomorphic to the composite functor $[1] \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \twoheadrightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \xrightarrow { \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } \mathrm{h} \mathit{\operatorname{QCat}}$, where $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ denotes the homotopy transport representation of Construction 5.2.3.2. It follows that, for every vertex $C \in \operatorname{\mathcal{C}}$, the morphism $\mathscr {F}$ carries the edge $\Delta ^1 \times \{ C\} $ to an isomorphism $\overline{e}$ in $\operatorname{\mathcal{QC}}$. If $X$ is an object of $\operatorname{\mathcal{E}}$ satisfying $U(X) = C$, then $\widetilde{\mathscr {F}}$ carries $\Delta ^1 \times \{ X\} $ to a $V$-cocartesian morphism $e$ of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ satisfying $V(e) = \overline{e}$, which is then also an isomorphism by virtue of Corollary 5.1.1.10. Allowing $C$ and $X$ to vary and applying Theorem 4.4.4.4, we deduce that $\mathscr {F}$ and $\widetilde{\mathscr {F}}$ are isomorphisms when regarded as morphisms in the $\infty $-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ and $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} )$, respectively.

Set $\operatorname{\mathcal{M}}= \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}})$. Applying Corollary 4.4.3.19 to the pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{M}}\ar [r] \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \ar [d] \\ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}), } \]

we deduce that $u$ is an isomorphism when regarded as a morphism of the $\infty $-category $\operatorname{\mathcal{M}}$. Since $\operatorname{TW}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ is replete subcategory of $\operatorname{\mathcal{M}}$ (Lemma 5.6.5.6), it follows that $u$ is also an isomorphism when regarded as a morphism of $\operatorname{TW}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ (Example 4.4.2.9). $\square$

Remark 5.6.5.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. It follows from Lemmas 5.6.5.6 and 5.6.5.7 that $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ can be identified with the full subcategory of the Kan complex

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})^{\simeq } \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}})^{\simeq } } \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} )^{\simeq } \]

spanned by those pairs $(\mathscr {F}, \widetilde{\mathscr {F}} )$ which witness $\mathscr {F}$ as a covariant transport representation for $U$.

Notation 5.6.5.9 (Functoriality). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Suppose we are given an arbitrary morphism of simplicial sets $f: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Precomposition with $f$ and with the projection map $\operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}$ determines a morphism of simplicial sets

\[ f^{\ast }: \operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \rightarrow \operatorname{TW}( \operatorname{\mathcal{E}}_0 /\operatorname{\mathcal{C}}_0 ), \]

which we will refer to as the restriction map. Note that the construction $\operatorname{\mathcal{C}}_0 \mapsto \operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$ carries colimits in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ to limits in the category of simplicial sets.

Lemma 5.6.5.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Let $\operatorname{\mathcal{C}}_0$ be a simplicial subset of $\operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Then:

$(1)$

The restriction map $\theta : \operatorname{TW}(\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \rightarrow \operatorname{TW}( \operatorname{\mathcal{E}}_0/ \operatorname{\mathcal{C}}_0 )$ of Notation 5.6.5.9 is a Kan fibration between Kan complexes.

$(2)$

If the inclusion $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is inner anodyne, then $\theta $ is a trivial Kan fibration.

Proof. We first prove $(1)$. Since the simplicial set $\operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$ is a Kan complex (Lemma 5.6.5.7), it will suffice to show that $\theta $ is an isofibration. Define fiber products

\[ \operatorname{\mathcal{M}}= \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \]

\[ \operatorname{\mathcal{M}}_0 = \operatorname{Fun}(\operatorname{\mathcal{C}}_0, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{QC}}_{\operatorname{Obj}}), \]

so that we have a commutative diagram

5.54
\begin{equation} \begin{gathered}\label{equation:classifier-infinity5} \xymatrix@R =50pt@C=50pt{ \operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \ar [r] \ar [d]^{\theta } & \operatorname{\mathcal{M}}\ar [d]^{\overline{\theta }} \\ \operatorname{TW}(\operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0) \ar [r] & \operatorname{\mathcal{M}}_0. } \end{gathered} \end{equation}

It follows from Lemma 5.6.5.6 that $\operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ is a replete subcategory of $\operatorname{\mathcal{M}}$, and therefore also a replete subcategory of the fiber product $\operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 ) \times _{ \operatorname{\mathcal{M}}_0 } \operatorname{\mathcal{M}}$. It will therefore suffice to show that the projection map $\operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 ) \times _{ \operatorname{\mathcal{M}}_0 } \operatorname{\mathcal{M}}\rightarrow \operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$ is an isofibration of $\infty $-categories. Since the collection of isofibrations is stable under pullback, we are reduced to showing that the map $\overline{\theta }: \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{M}}_0$ is an isofibration. We now observe that $\overline{\theta }$ factors as a composition

\begin{eqnarray*} \operatorname{\mathcal{M}}& = & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}})} \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \\ & \xrightarrow {\overline{\theta }'} & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \\ & \xrightarrow {\overline{\theta }''} & \operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{QC}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \\ & = & \operatorname{\mathcal{M}}_0, \end{eqnarray*}

where $\overline{\theta }''$ is a pullback of the restriction map

\[ \psi '': \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{QC}}_{\operatorname{Obj}} ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{QC}}) } \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{QC}}). \]

Since the forgetful functor $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ is an isofibration, $\psi ''$ is also an isofibration (Propositions 4.4.5.1). Similarly, $\overline{\theta }'$ is a pullback of the restriction map $\psi ': \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}_0, \operatorname{\mathcal{QC}})$, which is an isofibration by virtue of Corollary 4.4.5.3. It follows that $\overline{\theta } = \overline{\theta }'' \circ \overline{\theta }'$ is also an isofibration. This completes the proof of $(1)$.

We now prove $(2)$. Suppose that the inclusion map $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is inner anodyne; we wish to show that $\theta $ is a trivial Kan fibration. Applying Proposition 1.4.7.6, we deduce that $\psi '$ is a trivial Kan fibration of simplicial sets. Since $U$ is a cocartesian fibration, the inclusion map $\operatorname{\mathcal{E}}_0 \hookrightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence (Lemma 5.2.6.26). Applying Proposition 4.5.7.14, we deduce that $\psi ''$ is a trivial Kan fibration. It follows that the morphisms $\overline{\theta }'$ and $\overline{\theta }''$ are also trivial Kan fibrations, so that $\overline{\theta } = \overline{\theta }'' \circ \overline{\theta }'$ is a trivial Kan fibration. Applying Lemma 5.6.5.5, we see that the diagram (5.54) is a pullback square, so that $\theta $ is also a trivial Kan fibration. $\square$

Proof of Theorem 5.6.2.8 from Theorem 5.6.5.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having the property 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. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Applying Theorem 5.6.5.3, we see that the simplicial sets $\operatorname{TW}(\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ and $\operatorname{TW}( \operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$ are contractible Kan complexes. It follows that the restriction map $\theta : \operatorname{TW}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \rightarrow \operatorname{TW}(\operatorname{\mathcal{E}}_0 / \operatorname{\mathcal{C}}_0 )$ is a homotopy equivalence. Since $\theta $ is also Kan fibration (Lemma 5.6.5.10), it is a trivial Kan fibration (Corollary 3.2.7.4). In particular, $\theta $ is surjective on vertices, which is a restatement of Theorem 5.6.2.8. $\square$