5.6.2 Elements of $\operatorname{\mathcal{QC}}$-Valued Functors
Let $\operatorname{QCat}$ denote the ordinary category whose objects are $\infty $-categories and whose morphisms are functors (Construction 5.5.4.1). To every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, the weighted nerve construction of Definition 5.3.3.1 supplies a cocartesian fibration of $\infty $-categories $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Corollary 5.3.3.16), whose fiber over an object $C \in \operatorname{\mathcal{C}}$ is isomorphic to the $\infty $-category $\mathscr {F}(C)$ (Example 5.3.3.8). The utility of this construction is limited by the fact that it applies only to strictly commutative diagrams in $\operatorname{QCat}$: that is, Definition 5.3.3.1 requires $\operatorname{\mathcal{C}}$ to be an ordinary category and $\mathscr {F}$ to be a functor of ordinary categories. Our goal in this section is to introduce a homotopy coherent variant of the weighted nerve which is associated to any functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$; here $\operatorname{\mathcal{QC}}$ denotes the $\infty $-category of $\infty $-categories introduced in Construction 5.5.4.1.
Definition 5.6.2.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$ be a morphism of simplicial sets. We let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{ \Delta ^{0} / }$, so that we have a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d]^-{U} & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{\Delta ^{0}/} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }}). } \]
We will refer to $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ as the projection map.
The simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ of Definition 5.6.2.1 is defined for an arbitrary morphism $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$. However, we will be primarily interested in the case where $\mathscr {F}$ takes values in the simplicial subset $\operatorname{\mathcal{QC}}\subseteq \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$ introduced in Construction 5.5.4.1.
Proposition 5.6.2.2. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets. Then the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets.
Proof.
By construction, the morphism $U$ fits into a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d]^-{U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{\mathcal{QC}}, } \]
where
\[ \operatorname{\mathcal{QC}}_{\operatorname{Obj}} = \operatorname{\mathcal{QC}}\times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{ \Delta ^{0} / } \]
is the $\infty $-category introduced in Construction 5.5.6.10. It will therefore suffice to show that the projection map $\operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ is a cocartesian fibration of simplicial sets, which follows from Proposition 5.5.6.11.
$\square$
Corollary 5.6.2.3. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor of $\infty $-categories. Then the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ of Definition 5.6.2.1 is an $\infty $-category.
Definition 5.6.2.4. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor of $\infty $-categories. We will refer to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as the $\infty $-category of elements of $\mathscr {F}$.
Warning 5.6.2.6. In §5.6.1, we introduced a variant of the category of elements construction for contravariant $\mathbf{Cat}$-valued functors $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ (see Definition 5.6.1.4), which is characterized by the formula
\[ \int ^{\operatorname{\mathcal{C}}} \mathscr {F} = ( \int _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F}^{\operatorname{op}} )^{\operatorname{op}}. \]
In the $\infty $-categorical setting, the situation is more subtle: the involution $\operatorname{\mathcal{E}}\mapsto \operatorname{\mathcal{E}}^{\operatorname{op}}$ does not preserve the simplicial structure on the category $\operatorname{QCat}$ and therefore does not induce an involution on the simplicial set $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$. We will return to this point in §
.
Warning 5.6.2.7. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor of ordinary categories. Passing to the homotopy coherent nerve, we obtain a functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$. Beware that the simplicial set $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ is usually not isomorphic to the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Definition 5.3.3.1, even in the special case $\operatorname{\mathcal{C}}= \Delta ^{0}$. However, in §5.6.4 we will construct a comparison map
\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]
which is an equivalence of $\infty $-categories (Proposition 5.6.4.8).
Example 5.6.2.8 ($\operatorname{Set}$-Valued Functors). Let $\operatorname{Set}$ denote the category of sets, and let us regard the nerve $\operatorname{N}_{\bullet }(\operatorname{Set})$ as a simplicial subset of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$ be a morphism of simplicial sets, which we can identify with a functor of categories $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$. Using Example 5.5.3.12 and Remark 5.2.6.6, we obtain a canonical isomorphism of simplicial sets
\[ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \simeq \operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) } \operatorname{N}_{\bullet }( \int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \mathrm{h} \mathit{\mathscr {F}}), \]
where $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is the simplicial set of Definition 5.6.2.1 and $\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathrm{h} \mathit{\mathscr {F}}$ is the category of elements introduced in Construction 5.2.6.1. In particular, the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering map.
Example 5.6.2.9 ($\operatorname{\mathcal{S}}$-Valued Functors). Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1), which we view as a full simplicial subset of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a morphism of simplicial sets. Then the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ fits into pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d]^-{\pi } & \operatorname{\mathcal{S}}_{\ast } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{\mathcal{S}}, } \]
where $\operatorname{\mathcal{S}}_{\ast }$ is the $\infty $-category of pointed spaces (Construction 5.5.3.1). In this case, Proposition 5.5.3.2 guarantees that the projection map $\pi : \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets.
Example 5.6.2.10 ($\operatorname{ \pmb {\mathcal{QC}} }$-Valued Functors). Let $\operatorname{ \pmb {\mathcal{QC}} }$ denote the $(\infty ,2)$-category of $\infty $-categories (Construction 5.5.5.1), which we view as a full simplicial subset of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ be a morphism of simplicial sets. We then have a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d]^-{\pi } & \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{ \pmb {\mathcal{QC}} }, } \]
where $\operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}}$ is the $(\infty ,2)$-category of Construction 5.5.6.10 (Construction 5.5.6.8). If $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ is a functor of $(\infty ,2)$-categories, then Proposition 5.5.6.9 and Remark 5.4.2.4 guarantee that $\pi : \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration; in particular, $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is also an $(\infty ,2)$-category.
Warning 5.6.2.11. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ be a morphism of simplicial sets. If $\mathscr {F}$ is not a functor, then $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ need not be an $(\infty ,2)$-category (this phenomenon arises already in the case $\operatorname{\mathcal{C}}= \Delta ^2$).
Example 5.6.2.12 (Objects of the $\infty $-Category of Elements). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$ be a morphism of simplicial sets. Then vertices of the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ can be identified with pairs $(C, X)$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and $X$ is a vertex of the simplicial set $\mathscr {F}(C)$ (see Example 5.5.6.12). Moreover, the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is given on vertices by the construction $U(C,X) = C$.
Example 5.6.2.13 (Morphisms of the $\infty $-Category of Elements). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$ be a morphism of simplicial sets. Let $(C,X)$ and $(D,Y)$ be vertices of the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Edges of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ from $(C,X)$ to $(D,Y)$ can be identified with pairs $(f,u)$, where $f: C \rightarrow D$ is an edge of the simplicial set $\operatorname{\mathcal{C}}$ and $u: \mathscr {F}(f)(X) \rightarrow Y$ is an edge of the simplicial set $\mathscr {F}(D)$ (see Example 5.5.6.12). Moreover, the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is given on edges by the construction $U(f,u) = f$.
Example 5.6.2.16 ($2$-Simplices of the $\infty $-Category of Elements). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$ be a morphism of simplicial sets and let $\sigma _0: \operatorname{\partial \Delta }^2 \rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ be a morphism of simplicial sets, which we depict informally as a diagram
\[ \xymatrix@R =50pt@C=50pt{ & (D,Y) \ar [dr]^{ (g,v) } & \\ (C,X) \ar [ur]^{(f,u)} \ar [rr]^-{ (h,w) } & & (E,Z). } \]
Extensions of $\sigma _0$ to a $2$-simplex of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ can be identified with pairs $(\mu , \theta )$, where $\mu : \mathscr {F}(g) \circ \mathscr {F}(f) \rightarrow \mathscr {F}(h)$ is an edge of the simplicial set $\operatorname{Fun}( \mathscr {F}(C), \mathscr {F}(E) )$, and $\theta : \operatorname{\raise {0.1ex}{\square }}^{2} \rightarrow \mathscr {F}(E)$ is a morphism of simplicial sets whose restriction to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ (\mathscr {F}(g) \circ \mathscr {F}(f))(X) \ar [r]^-{\mu (X)} \ar [d]^-{ \mathscr {F}(g)(u)} & \mathscr {F}(h)(X) \ar [d]^-{w} \\ \mathscr {F}(g)(Y) \ar [r]^-{v} & Z } \]
(see Example 5.5.6.17). Moreover, the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is given on $2$-simplices by the construction $U( \mu , \theta ) = \mu $.
Example 5.6.2.17. Let $\operatorname{\mathcal{E}}$ be a simplicial set, which we identify with the morphism of simplicial sets $\Delta ^{0} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$ taking the value $\operatorname{\mathcal{E}}$. Then the simplicial set $\int _{\Delta ^{0}} \operatorname{\mathcal{E}}$ can be identified with the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}})$. In particular, Construction 4.6.8.3 supplies a comparison morphism
\[ \theta _{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}= \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{0}, \operatorname{\mathcal{E}})_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}}) = \int _{\Delta ^{0}} \operatorname{\mathcal{E}}. \]
If $\operatorname{\mathcal{E}}$ is an $\infty $-category, then $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}})$ is also an $\infty $-category, and the comparison morphism $\rho $ is an equivalence of $\infty $-categories (Theorem 4.6.8.9). Beware that $\theta _{\operatorname{\mathcal{E}}}$ is generally not an isomorphism (though it is always a monomorphism which is bijective on simplices of dimension $\leq 1$). For example, Example 5.6.2.16 implies that $2$-simplices of $\int _{\Delta ^0} \operatorname{\mathcal{E}}$ can be identified with morphisms of simplicial sets $\rho : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{E}}$ for which the restriction $\rho |_{ \Delta ^1 \times \{ 0\} }$ is a degenerate edge of $\operatorname{\mathcal{E}}$, as indicated in the diagram
\[ \xymatrix@R =10pt@C=10pt{ X \ar [rrrr]^{\operatorname{id}_ X} \ar [dddd]_{u} \ar [ddddrrrr] & & & & X \ar [dddd]^{w} \\ & & & \sigma & \\ & & & & \\ & \tau & & & \\ Y \ar [rrrr]_{v} & & & & Z. } \]
The corresponding $2$-simplex of $\int _{\Delta ^{0}} \operatorname{\mathcal{E}}$ belongs to the image of $\theta _{\operatorname{\mathcal{E}}}$ if and only if $\sigma $ is a left-degenerate $2$-simplex of $\operatorname{\mathcal{E}}$ (in which case it is given by $\theta _{\operatorname{\mathcal{E}}}(\tau )$).
Example 5.6.2.19 (Fibers of the $\infty $-Category of Elements). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ be a morphism of simplicial sets. For each vertex $C \in \operatorname{\mathcal{C}}$, Remark 5.6.2.18 and Example 5.6.2.17 supply a canonical isomorphism
\[ \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \simeq \operatorname{Hom}^{\mathrm{L}}_{\operatorname{ \pmb {\mathcal{QC}} }}( \Delta ^{0}, \mathscr {F}(C) ). \]
In particular, Construction 4.6.8.3 supplies a comparison functor $\theta _{C}: \mathscr {F}(C) \rightarrow \{ C \} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is an equivalence of $\infty $-categories (Theorem 4.6.8.9), but generally not an isomorphism of simplicial sets.
Example 5.6.2.20 (Elements of Diagram $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram. Fix another simplicial set $K$, and let $\mathscr {F}^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ denote the composition of $\mathscr {F}$ with the functor
\[ \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}\quad \quad \operatorname{\mathcal{D}}\mapsto \operatorname{Fun}(K, \operatorname{\mathcal{D}}). \]
Let $\underline{K}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{S}})$ denote the constant map taking the value $K$, so that Remark 5.6.2.18 and Example 5.6.2.17 supply a comparison map $\rho : K \times \operatorname{\mathcal{C}}\rightarrow \int _{\operatorname{\mathcal{C}}} \underline{K}_{\operatorname{\mathcal{C}}}$. The composition
\begin{eqnarray*} K \times \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{K} & \xrightarrow { \rho \times \operatorname{id}} & \int _{\operatorname{\mathcal{C}}} \underline{K}_{\operatorname{\mathcal{C}}} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{K} \\ & \simeq & \int _{\operatorname{\mathcal{C}}} (\underline{K}_{\operatorname{\mathcal{C}}} \times \mathscr {F}^{K}) \\ & \xrightarrow {\operatorname{ev}} & \int _{\operatorname{\mathcal{C}}} \mathscr {F} \end{eqnarray*}
is determines a commutative diagram
\[ \xymatrix { \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{K} \ar [dr]_{U} \ar [rr]^{T} & & \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \ar [dl]^{V} \\ & \operatorname{\mathcal{C}}. & } \]
Here $U$ and $V$ are cocartesian fibrations, and Remark 5.6.2.14 guarantees that $T$ carries $U$-cocartesian edges to $V$-cocartesian edges. Moreover, for each vertex $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $T_{C}: U^{-1} \{ C\} \rightarrow V^{-1} \{ C\} $ fits into a commutative diagram
\[ \xymatrix { \mathscr {F}^{K}(C) \ar@ {=}[r] \ar [dr] & \operatorname{Fun}(K, \mathscr {F}(C) ) \ar [d] \\ \{ C\} \times _{ \operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{K} \ar [r]^{T_ C } & \operatorname{Fun}(K, \{ C \} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} ), } \]
where the vertical maps are equivalences by virtue of Example 5.6.2.19. It follows that $T$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ (see Proposition 5.1.7.15).
Proposition 5.6.2.21. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be diagrams, and let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ and $U': \int _{\operatorname{\mathcal{C}}} \mathscr {F}' \rightarrow \operatorname{\mathcal{C}}$ be the projection maps. If $\mathscr {F}$ and $\mathscr {F}'$ are isomorphic as objects of the diagram $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$, then $U$ and $U'$ are equivalent as cocartesian fibrations over $\operatorname{\mathcal{C}}$ (in the sense of Definition 5.1.7.1).
Proof.
Apply Proposition 5.1.7.11 to the cocartesian fibration $\operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ of Proposition 5.5.6.11.
$\square$
Proposition 5.6.2.22. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor of $\infty $-categories, let $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the $\infty $-category of elements of $\mathscr {F}$, and let
\[ \operatorname{hTr}_{ \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \]
denote the enriched homotopy transport representation associated to the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ (see Construction 5.2.8.9). Then there is a canonical isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors $\theta : \mathrm{h} \mathit{\mathscr {F}} \rightarrow \operatorname{hTr}_{ \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$, which carries each object $C \in \operatorname{\mathcal{C}}$ to the comparison map
\[ \theta _{C}: \mathscr {F}(C) \rightarrow \operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}(C) = \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]
of Example 5.6.2.19.
Proof.
By virtue of Remarks 5.2.8.10 and 5.6.2.18, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}$ and that $\mathscr {F}$ is the identity functor. In this case, the desired result is a restatement of Proposition 5.5.6.14.
$\square$
Corollary 5.6.2.23. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets, let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ be the cocartesian fibration of Proposition 5.6.2.2, and let
\[ f_{!}: \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \{ D\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]
be a functor which is given by covariant transport along an edge $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ (Definition 5.2.2.4). Then the diagram
\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}(C) \ar [d]^{ [\mathscr {F}(f)]} \ar [r]^-{\sim } & \{ C\} \times _{ \operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [d]^{ [ f_{!} ] } \\ \mathscr {F}(D) \ar [r]^-{\sim } & \{ D\} \times _{ \operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} } \]
commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ (where the horizontal maps are the equivalences described in Example 5.6.2.19).
Proof.
Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$, in which case the desired result reduces to Proposition 5.6.2.22.
$\square$
Corollary 5.6.2.24. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor of $\infty $-categories and set $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Then there is a canonical isomorphism $\mathrm{h} \mathit{\mathscr {F}} \xrightarrow {\sim } \operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$ in the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{\mathcal{QC}}})$, which carries each vertex $C \in \operatorname{\mathcal{C}}$ to the comparison map
\[ \theta _{C}: \mathscr {F}(C) \rightarrow \operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}(C) = \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]
of Example 5.6.2.19.