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5.5.3 The Weighted Nerve

Let $\operatorname{\mathcal{C}}$ be a category which is equipped with a functor (of ordinary categories) $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cat}$. Then Definition 5.5.2.1 determines a cocartesian fibration of categories

\[ U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}, \]

whose fiber over each object $C \in \operatorname{\mathcal{C}}$ can be identified with the category $\mathscr {F}(C)$ (Proposition 5.5.2.12 and Remark 5.5.2.9). Recall that the nerve functor $\operatorname{\mathcal{E}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$ determines a fully faithful embedding from $\operatorname{Cat}$ to the category $\operatorname{Set_{\Delta }}$ of simplicial sets (Proposition 1.2.2.1). Our goal in this section is to study a variant of Definition 5.5.2.1 for $\operatorname{Set_{\Delta }}$-valued functors $\mathscr {F}:\operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ which we will refer to as the weighted nerve of $\mathscr {F}$ (Definition 5.5.3.1).

Definition 5.5.3.1 (The Weighted Nerve). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. For every integer $n \geq 0$, we let $(\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F})_{n}$ denote the collection of all pairs $( \overrightarrow {C}, \overrightarrow {\sigma })$, where $\overrightarrow {C}: [n] \rightarrow \operatorname{\mathcal{C}}$ is a functor which we view as a diagram

\[ C_0 \rightarrow C_1 \rightarrow C_2 \rightarrow \cdots \rightarrow C_{n-1} \rightarrow C_ n \]

and $\overrightarrow {\sigma }$ is a collection of simplices $\{ \sigma _{j}: \Delta ^{j} \rightarrow \mathscr {F}(C_ j) \} _{0 \leq j \leq n}$ which fit into a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{0} \ar [d]_{\sigma _0} \ar@ {^{(}->}[r] & \Delta ^{1} \ar [d]^-{\sigma _1} \ar@ {^{(}->}[r] & \Delta ^{2} \ar@ {^{(}->}[r] \ar [d]_{\sigma _2} & \cdots \ar [d] \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]_{\sigma _ n} \\ \mathscr {F}(C_0) \ar [r] & \mathscr {F}(C_1) \ar [r] & \mathscr {F}(C_2) \ar [r] & \cdots \ar [r] & \mathscr {F}(C_ n). } \]

For every nondecreasing function $\alpha : [m] \rightarrow [n]$, we define a map $\alpha ^{\ast }: (\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F})_{n} \rightarrow (\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F})_{m}$ by the formula $\alpha ^{\ast }( \overrightarrow {C}, \overrightarrow {\sigma } ) = (\overrightarrow {C} \circ \alpha , \overrightarrow {\sigma }')$, where $\overrightarrow {\sigma }'$ associates to each $0 \leq i \leq m$ the $i$-simplex $\sigma '_{i}: \Delta ^{i} \rightarrow \mathscr {F}( \alpha (i) )$ given by the composition

\[ \Delta ^{i} \xrightarrow { \alpha |_{ \{ 0 < 1 < \cdots < i \} } } \Delta ^{ \alpha (i) } \xrightarrow { \sigma _{ \alpha (i) } } \mathscr {F}( \alpha (i) ). \]

By means of these restriction maps, we regard the construction $[n] \mapsto (\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F})_{n}$ as a simplicial set. We will denote this simplicial set by $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} $ and refer to it as the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$. Note that there is an evident projection map $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, given on simplices by the construction $( \overrightarrow {C}, \overrightarrow {\sigma } ) \mapsto \overrightarrow {C}$.

Example 5.5.3.2. Let $X$ be a simplicial set, which we identify with the constant functor $\mathscr {F}: [0] \rightarrow \operatorname{Set_{\Delta }}$ taking the value $X$. Then the weighted nerve $\int ^{\mathrm{s}}_{[0]}\mathscr {F}$ is isomorphic to the simplicial set $X$.

Example 5.5.3.3. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, which we identify with a functor $\mathscr {F}: [1] \rightarrow \operatorname{Set_{\Delta }}$ (so that $X = \mathscr {F}(0)$ and $Y = \mathscr {F}(1)$). Then the weighted nerve $\int ^{\mathrm{s}}_{[1]}\mathscr {F}$ can be identified with the relative join $X \star _{Y} Y$ of Construction 5.2.4.1.

Example 5.5.3.4. Let $\operatorname{Cat}$ denote the category whose objects are (small) categories and whose morphisms are functors, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cat}$ be a functor of ordinary categories. Composing with the nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$, we obtain a functor $\operatorname{N}_{\bullet }(\mathscr {F}): \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then there is a canonical isomorphism of simplicial sets $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\operatorname{N}_{\bullet }(\mathscr {F}) \simeq \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$, where the left hand side is the weighted nerve of Definition 5.5.3.1 and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is the category of elements introduced in Definition 5.5.2.1.

Remark 5.5.3.5 (Functoriality in $\operatorname{\mathcal{C}}$). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, let $U: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $\mathscr {F}': \operatorname{\mathcal{C}}' \rightarrow \operatorname{Set_{\Delta }}$ denote the composition $\mathscr {F} \circ U$. Then there is a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}'}\mathscr {F}' \ar [r] \ar [d] & \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \ar [d] \\ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' ) \ar [r]^-{ \operatorname{N}_{\bullet }(U) } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}). } \]

Example 5.5.3.6 (The Weighted Nerve of a Constant Diagram). Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be a simplicial set, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the constant functor taking the value $X$. Then Remark 5.5.3.5 and Example 5.5.3.2 supply an isomorphism of simplicial sets $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \simeq X \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Example 5.5.3.7 (Fibers of the Weighted Nerve). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be functor. For each object $C \in \operatorname{\mathcal{C}}$, Remark 5.5.3.5 and Example 5.5.3.2 supply an isomorphism of simplicial sets

\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}. \]

Example 5.5.3.8. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. For every morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, Remark 5.5.3.5 and Example 5.5.3.3 supply an isomorphism of simplicial sets

\[ \mathscr {F}(C) \star _{ \mathscr {F}(D) } \mathscr {F}(D) \simeq \Delta ^1 \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}. \]

Variant 5.5.3.9. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\sigma $ be an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram

\[ C_0 \rightarrow C_1 \rightarrow C_2 \rightarrow \cdots \rightarrow C_ n \]

in the category $\operatorname{\mathcal{C}}$. Then the fiber product $\Delta ^ n \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ can be identified with the iterated relative join

\[ ((( \mathscr {F}(C_0) \star _{ \mathscr {F}(C_1)} \mathscr {F}(C_1)) \star _{ \mathscr {F}(C_2)} \mathscr {F}(C_2)) \star \cdots ) \star _{\mathscr {F}(C_ n)} \mathscr {F}(C_ n). \]

See Exercise 5.2.4.18.

Remark 5.5.3.10 (Functoriality in $\mathscr {F}$). Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $\mathscr {F} \mapsto \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ determines a functor from the diagram category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ to the category $( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ of simplicial sets over the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. This functor commutes with all limits and with filtered colimits.

Remark 5.5.3.11 (Vertices of the Weighted Nerve). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then vertices of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ can be identified with pairs $(C,x)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $x$ is a vertex of the simplicial set $\mathscr {F}(C)$.

Remark 5.5.3.12 (Edges of the Weighted Nerve). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, and let $(C,x)$ and $(D,y)$ be vertices of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ (see Remark 5.5.3.11). Edges of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ with source $(C,x)$ and target $(D,y)$ can be identified with pairs $(f, e)$, where $f: C \rightarrow D$ is a morphism of the category $\operatorname{\mathcal{C}}$ and $e: \mathscr {F}(f)(x) \rightarrow y$ is an edge of the simplicial set $\mathscr {F}(D)$.

Proposition 5.5.3.13. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category. Then:

$(1)$

The projection map $\pi : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration of simplicial sets.

$(2)$

Let $(f,e): (C,x) \rightarrow (D,y)$ be an edge of the simplicial set $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ (see Remark 5.5.3.12). Then $(f,e)$ is $\pi $-cocartesian if and only if $e: \mathscr {F}(f)(C) \rightarrow y$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$.

Proof of Proposition 5.5.3.13. Suppose we are given integers $0 \leq i < n$ and a lifting problem

5.39
\begin{equation} \begin{gathered}\label{equation:strict-Grothendieck-construction-fibration} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar@ {^{(}->}[d] & \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \ar [d]^-{ \pi } \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma }} \ar@ {-->}[ur]^{ \sigma } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \end{gathered} \end{equation}

Then $\overline{\sigma }$ can be identified with a composable chain of morphisms $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$, and $\sigma _0$ determines a map of simplicial sets $\tau _0: \Lambda ^{n}_{i} \rightarrow \mathscr {F}(C_ n)$. Unwinding the definitions, we see that solutions to the lifting problem (5.39) can be identified with extensions of $\tau _0$ to an $n$-simplex $\tau : \Delta ^ n \rightarrow \mathscr {F}(C_ n)$. Since $\mathscr {F}(C_ n)$ is an $\infty $-category, such an extension always exists for $0 < i < n$. This proves that $\pi $ is an inner fibration of simplicial sets. In particular, the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ is an $\infty $-category.

Suppose now that $i = 0$, $n \geq 2$, and that the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} )}$ corresponds to an edge $(f,e): (C,x) \rightarrow (D,y)$ of the simplicial set $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$. Then $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is the image of $e$ under the functor $\mathscr {F}(D) = \mathscr {F}(C_1) \rightarrow \mathscr {F}(C_ n)$. If $e$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$, then $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is an isomorphism in the $\infty $-category $\mathscr {F}(C_ n)$, so that $\tau _0$ can be extended to an $n$-simplex of $\mathscr {F}(C_ n)$ by virtue of Theorem 4.4.2.5. In particular, if $e$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$, then $(f,e)$ is a $\pi $-cocartesian morphism of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$.

Note that, for every object $(C,x)$ of the $\infty $-category $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ and every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, we can choose a morphism $(f,e): (C,x) \rightarrow (D,y)$ of the $\infty $-category $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ where $e$ is an isomorphism in $\mathscr {F}(D)$: for example, we can take $y = \mathscr {F}(f)(x)$ and $e$ to be the identity morphism $\operatorname{id}_ y$. The preceding argument then shows that $(f,e)$ is $\pi $-cocartesian. This completes the proof of $(1)$.

To complete the proof of $(2)$, it will suffice to show that if $(f,e): (C,x) \rightarrow (D,y)$ is any $\pi $-cocartesian morphism of the $\infty $-category $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$, then $e$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$. To prove this, we can assume without loss of generality that $\operatorname{\mathcal{C}}= [1]$ is the linearly ordered set with two elements $C = 0$ and $D = 1$, in which case the desired result follows from Lemma 5.2.4.15 (see Example 5.5.3.8). $\square$

Remark 5.5.3.15. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and let $\pi : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Proposition 5.5.3.13. Then the homotopy transport representation $\operatorname{hTr}_{\pi }: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 5.2.3.2 is canonically isomorphic to the composition $\operatorname{\mathcal{C}}\xrightarrow {\mathscr {F}} \operatorname{QCat}\twoheadrightarrow \mathrm{h} \mathit{\operatorname{QCat}}$. To prove this, it suffices to observe that for every morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, the induced map $\mathscr {F}(f): \mathscr {F}(C) \rightarrow \mathscr {F}(D)$ agrees with the functor

\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \{ D\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \simeq \mathscr {F}(D) \]

given by covariant transport along $f$, which follows immediately from Proposition 5.2.4.16 (together with Example 5.5.3.8).

We conclude this section by comparing the weighted nerve of Definition 5.5.3.1 with the universal mapping simplex of Notation 5.2.6.1.

Construction 5.5.3.16. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, and let $\operatorname{N}_{\bullet }^{+}( \operatorname{Set_{\Delta }})$ be the simplicial set introduced Notation 5.2.6.1. Unwinding the definitions, we see that $n$-simplices of the fiber product $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }})$ can be identified with pairs $( \overrightarrow {C}, \sigma )$, where $\overrightarrow {C} = (C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n)$ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and $\sigma $ is an $n$-simplex of the simplicial set $\mathscr {F}(C_0)$. To every such pair $(\overrightarrow {C}, \sigma )$, we can associate an $n$-simplex $( \overrightarrow {C}, \overrightarrow {\sigma } )$ of $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$, where $\overrightarrow {\sigma }$ assigns to each $0 \leq i \leq n$ the $i$-simplex $\sigma _{i}: \Delta ^{i} \rightarrow \mathscr {F}(C_ i)$ given by the composition

\[ \Delta ^{i} \hookrightarrow \Delta ^{n} \xrightarrow {\sigma } \mathscr {F}(C_0) \rightarrow \mathscr {F}(C_ i). \]

The construction $( \overrightarrow {C}, \sigma ) \rightarrow (\overrightarrow {C}, \overrightarrow {\sigma } )$ determines a morphism of simplicial sets

\[ F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}. \]

Example 5.5.3.17. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. For each object $C \in \operatorname{\mathcal{C}}$, the comparison map

\[ F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \]

of Construction 5.5.3.16 induces an isomorphism of fibers

\[ F_{C}: \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \simeq \mathscr {F}(C) \]

(see Example 5.5.3.7).

Proposition 5.5.3.18. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Assume that, for each object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category. Then the comparison map

\[ F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \]

of Construction 5.5.3.16 is a categorical equivalence of simplicial sets.

Proof. By virtue of Corollary 4.5.4.3, it will suffice to show that for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map

\[ F_{\sigma }: \Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }( \operatorname{Set_{\Delta }}) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \]

is a categorical equivalence of simplicial sets. Let us identify $\sigma $ with a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$, so that the fiber product $\Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }( \operatorname{Set_{\Delta }})$ is the mapping simplex $M( \mathscr {F}(C_0) \rightarrow \cdots \rightarrow \mathscr {F}(C_ n) )$ introduced in Construction 5.2.6.3. Let us abuse notation by identifying each $\mathscr {F}(C_ i)$ with the fiber of the projection map

\[ \pi : \Delta ^ n \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \Delta ^ n \]

over the $i$th vertex of $\Delta ^ n$. It follows from Proposition 5.5.3.13 that $\pi $ is a cocartesian fibration and that $F_{\sigma }$ is a scaffold of $\pi $, in the sense of Definition 5.2.6.12. Applying Proposition 5.2.6.19, we conclude that $F_{\sigma }$ is a categorical equivalence of simplicial sets. $\square$

Exercise 5.5.3.19. Show that the conclusion of Proposition 5.5.3.18 holds for any functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ (that is, we do not need to assume that each of the simplicial sets $\mathscr {F}(C)$ is an $\infty $-category).