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.
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.
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).
$\square$
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).