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5.3.3 The Weighted Nerve

Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$. In §5.3.2, we introduced the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$, which is a simplicial set equipped with a projection map $U: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. If $\mathscr {F}$ carries each morphism of $\operatorname{\mathcal{C}}$ to a Kan fibration, then the projection map $U$ is a left fibration of simplicial sets (Exercise 5.3.2.17). Beware that $U$ is not a left fibration in general. In this section, we introduce a variant of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ which we will refer to as the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$ and denote by $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (Definition 5.3.3.1). The weighted nerve is equipped with a projection map $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which is a left fibration provided that $\mathscr {F}$ is a diagram of Kan complexes (Corollary 5.3.3.19). In §5.3.5, we will construct a comparison map $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (Construction 5.3.4.11) which is a categorical equivalence of simplicial sets (Corollary 5.3.5.9); in particular, it is a weak homotopy equivalence.

Definition 5.3.3.1 (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 $\operatorname{N}^{\mathscr {F}}_{n}( \operatorname{\mathcal{C}})$ denote the collection of all pairs $( \sigma , \tau )$, where $\sigma : [n] \rightarrow \operatorname{\mathcal{C}}$ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which we identify with a diagram

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

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

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

For every nondecreasing function $\alpha : [m] \rightarrow [n]$, we define a map $\alpha ^{\ast }: \operatorname{N}^{\mathscr {F}}_{n}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\mathscr {F}}_{m}( \operatorname{\mathcal{C}})$ by the formula $\alpha ^{\ast }( \sigma , \tau ) = (\sigma \circ \alpha , \tau ')$, where $\tau ' = \{ \tau '_{i}: \Delta ^{i} \rightarrow \mathscr {F}( \alpha (i) ) \} _{0 \leq i \leq m}$ is determined by the requirement that each $\tau '_{i}$ is equal to the composition

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

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

Example 5.3.3.2. 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 $\operatorname{N}_{\bullet }^{\mathscr {F}}([0])$ can be identified with the simplicial set $X$.

Remark 5.3.3.3 (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 $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ 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 5.3.3.4 (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 $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Remark 5.3.3.3). Edges of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ 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 $\mathscr {F}(D)$.

Remark 5.3.3.5. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Let $K$ be an auxiliary simplicial set, and define $\mathscr {F}^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {F}^{K}(C) = \operatorname{Fun}(K, \mathscr {F}(C) )$. Then the weighted nerves of $\mathscr {F}$ and $\mathscr {F}^{K}$ are related by a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }^{ \mathscr {F}^{K} }(\operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}(K, \operatorname{N}_{\bullet }^{\mathscr {F} }(\operatorname{\mathcal{C}})) \ar [d] \\ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}(K, \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) ). } \]

Example 5.3.3.6. Let $\operatorname{\mathcal{C}}$ be a category, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the functor given on objects by the formula $\mathscr {F}(C) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} )$. Then there is a canonical isomorphism of simplicial sets

\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}([1], \operatorname{\mathcal{C}}) ) = \operatorname{Fun}( \Delta ^1, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ). \]

Remark 5.3.3.7 (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

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

Example 5.3.3.8 (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 5.3.3.7 and Example 5.3.3.2 supply an isomorphism of simplicial sets

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

Example 5.3.3.9 (The Weighted Nerve of a Constant Diagram). Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be a simplicial set, and let $\underline{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the constant functor taking the value $X$. Then Remark 5.3.3.7 and Example 5.3.3.2 supply an isomorphism of simplicial sets $\operatorname{N}_{\bullet }^{\underline{X}}(\operatorname{\mathcal{C}}) \simeq X \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Remark 5.3.3.10 (Functoriality in $\mathscr {F}$). Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $\mathscr {F} \mapsto \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ 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.

Exercise 5.3.3.11. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Show that, if $\alpha $ is a levelwise trivial Kan fibration, then the induced map of weighted nerves $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})$ is a trivial Kan fibration of simplicial sets.

Example 5.3.3.12 (The Weighted Nerve of a Cone). Let $\operatorname{\mathcal{C}}$ be a category and let $\operatorname{\mathcal{C}}^{\triangleright }$ denote the right cone on $\operatorname{\mathcal{C}}$ (Example 4.3.2.5), and let ${\bf 1} \in \operatorname{\mathcal{C}}^{\triangleright }$ denote the final object. Suppose we are given a diagram of simplicial sets $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$ and $Y = \overline{\mathscr {F}}( {\bf 1} )$, so that $\overline{\mathscr {F}}$ determines a natural transformation $\alpha : \mathscr {F} \rightarrow \underline{Y}$ (where $\underline{Y}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denotes the constant functor taking the value $Y$). Combining Remark 5.3.3.10 with Example 5.3.3.9, we obtain morphisms of simplicial sets

\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \xrightarrow {\alpha } \operatorname{N}_{\bullet }^{\underline{Y}}(\operatorname{\mathcal{C}}) \simeq Y \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow Y. \]

Unwinding the definitions, there is a canonical isomorphism of simplicial sets

\[ \operatorname{N}_{\bullet }^{ \overline{\mathscr {F}} }(\operatorname{\mathcal{C}}^{\triangleright }) \simeq \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \star _{Y} Y, \]

where the right hand side denotes the relative join of Construction 5.2.3.1.

Example 5.3.3.13. 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 Example 5.3.3.12 supplies an isomorphism of simplicial sets $\operatorname{N}_{\bullet }^{\mathscr {F}}([1]) \simeq X \star _{Y} Y$.

Example 5.3.3.14. 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 5.3.3.7 and Example 5.3.3.13 supply an isomorphism of simplicial sets

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

Proposition 5.3.3.15. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Assume that:

  • For each object $C \in \operatorname{\mathcal{C}}$, the morphism $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a cocartesian fibration of simplicial sets.

  • For each morphism $u: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the morphism $\mathscr {F}(u): \mathscr {F}(C) \rightarrow \mathscr {F}(D)$ carries $\alpha _{C}$-cocartesian edges of $\mathscr {F}(C)$ to $\alpha _ D$-cocartesian edges of $\mathscr {F}(D)$.

Then:

$(1)$

The induced map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})$ is a cocartesian fibration of simplicial sets.

$(2)$

Let $(f,e): (C,x) \rightarrow (D,y)$ be an edge of the simplicial set $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Remark 5.3.3.4). Then $(f,e)$ is $U$-cocartesian if and only if $e: \mathscr {F}(f)(x) \rightarrow y$ is an $\alpha _{D}$-cocartesian edge of the simplicial set $\mathscr {F}(D)$.

Proof. By virtue of Proposition 5.1.4.7 and Remark 5.3.3.7, we may assume without loss of generality that $\operatorname{\mathcal{C}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ for some nonnegative integer $n$. We proceed by induction on $n$. If $n = 0$, then $U$ can be identified with the cocartesian fibration $\alpha _0: \mathscr {F}(0) \rightarrow \mathscr {G}(0)$ (Example 5.3.3.2), so that assertions $(1)$ and $(2)$ are immediate. Let us therefore assume that $n > 0$, so that $\operatorname{\mathcal{C}}$ can be identified with the cone $\operatorname{\mathcal{C}}_{0}^{\triangleright }$ for $\operatorname{\mathcal{C}}_0 = [n-1]$. Set $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ and $\mathscr {G}_0 = \mathscr {G}|_{ \operatorname{\mathcal{C}}_0 }$. It follows from our inductive hypothesis that $U$ restricts to a cocartesian fibration $U_0: \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_0}(\operatorname{\mathcal{C}}_0)$ is a cocartesian fibration of $\infty $-categories, and that an edge of $\operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0)$ is $U_0$-cocartesian if and only if it satisfies the criterion described in $(2)$. It follows that the functor $\operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \rightarrow \mathscr {F}(n)$ described in Example 5.3.3.12 carries $U_0$-cocartesian morphisms to $\alpha _{n}$-cocartesian morphisms of the $\infty $-category $\mathscr {F}(n)$. Unwinding the definitions, we can identify $U$ with the map of relative joins

\[ \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \star _{\mathscr {F}(n)} \mathscr {F}(n) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_0}(\operatorname{\mathcal{C}}_0) \star _{\mathscr {G}(n)} \mathscr {G}(n). \]

Assertions $(1)$ and $(2)$ now follow from Lemma 5.2.3.17. $\square$

Corollary 5.3.3.16. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$. Then:

$(1)$

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

$(2)$

Let $(f,e): (C,x) \rightarrow (D,y)$ be an edge of the simplicial set $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Remark 5.3.3.4). Then $(f,e)$ is $U$-cocartesian if and only if $e: \mathscr {F}(f)(x) \rightarrow y$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$.

In particular, $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is an $\infty $-category.

Proof. Apply Proposition 5.3.3.15 in the special case where $\mathscr {G}$ is the constant diagram taking the value $\Delta ^0$. $\square$

Exercise 5.3.3.17. Let $\operatorname{\mathcal{C}}$ be a category, let $n$ be an integer, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$. Assume that, for each object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $(n,1)$-category (Definition 4.8.1.8). Show that the cocartesian fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is $n$-categorical, in the sense of Definition 4.8.6.23.

In particular, if each of the simplicial sets $\mathscr {F}(C)$ is (isomorphic to) the nerve of an ordinary category, then the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is also isomorphic to the nerve of an ordinary category. For a more precise statement, see Example 5.6.1.8.

Corollary 5.3.3.18. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the morphism $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a left fibration of simplicial sets. Then the induced map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})$ is also a left fibration of simplicial sets.

Corollary 5.3.3.19. 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 a Kan complex. Then the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration.

Proof. Apply Corollary 5.3.3.18 in the special case where $\mathscr {G}$ is the constant diagram taking the value $\Delta ^0$. $\square$

Corollary 5.3.3.20. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {F}'$ be a natural transformation between functors $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then $\alpha $ is a levelwise categorical equivalence if and only if the induced map $T: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$ is a categorical equivalence of simplicial sets.

Proof. Assume first that $\alpha $ is a levelwise categorical equivalence. To prove that $T$ is a categorical equivalence of simplicial sets, it will suffice to show that for every simplex $\sigma : \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map $T_{\sigma }: \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$ is a categorical equivalence of simplicial sets (Corollary 4.5.7.3). Using Remark 5.3.3.7, we can reduce to the special case where $\operatorname{\mathcal{C}}$ is the linearly ordered $[n] = \{ 0 < 1 < \cdots < n \} $ for some $n \geq 0$. We now proceed by induction on $n$. If $n = 0$, the result is immediate from Example 5.3.3.2. The inductive step follows by combining Example 5.3.3.12 with Corollary 5.2.4.7.

We now prove the converse. Using Proposition 4.1.3.2, we can choose a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F} \ar [r]^{\alpha } \ar [d] & \mathscr {F}' \ar [d] \\ \mathscr {G} \ar [r]^{\beta } & \mathscr {G}' } \]

in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$, where the vertical maps are levelwise categorical equivalences and the simplicial sets $\mathscr {G}(C)$ and $\mathscr {G}'(C)$ are $\infty $-categories for each $C \in \operatorname{\mathcal{C}}$. Using the first part of the proof, we can replace $\alpha $ by $\beta $ and thereby reduce to the special case where $\mathscr {F}$ and $\mathscr {F}'$ are diagrams of $\infty $-categories. In this case, the projection maps $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \leftarrow \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$ are cocartesian fibrations of $\infty $-categories (Corollary 5.3.3.16). It then follows from Theorem 5.1.6.1 (together with Example 5.3.3.8) that if $T$ is an equivalence of $\infty $-categories, then $\alpha $ is a levelwise categorical equivalence. $\square$

Example 5.3.3.21. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category, so that the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration (Corollary 5.3.3.16). Define $\mathscr {F}^{\simeq }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {F}^{\simeq }( C) = \mathscr {F}(C)^{\simeq }$. Then $\operatorname{N}_{\bullet }^{\mathscr {F}^{\simeq }}(\operatorname{\mathcal{C}})$ can be identified with with simplicial subset of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ spanned by those $n$-simplices which carry each edge of $\Delta ^ n$ to a $U$-cocartesian edge of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. That is, the projection map $U^{\simeq }: \operatorname{N}_{\bullet }^{\mathscr {F}^{\simeq }}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the underlying left fibration of the cocartesian fibration $U$ (see Corollary 5.1.4.15).

Remark 5.3.3.22 (The Homotopy Transport Representation). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Corollary 5.3.3.16. Then the homotopy transport representation

\[ \operatorname{hTr}_{\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \]

of Construction 5.2.5.2 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 functor

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

is given by covariant transport along $f$, which follows immediately from Proposition 5.2.3.15 and Example 5.3.3.14.

We conclude this section by showing that the weighted nerve can be characterized by a universal mapping property.

Notation 5.3.3.23. Let $\operatorname{\mathcal{C}}$ be a category and suppose we are given a morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. For every object $C \in \operatorname{\mathcal{C}}$, let $\mathscr {G}_{\operatorname{\mathcal{E}}}(C)$ denote the fiber product $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$. The construction $C \mapsto \mathscr {G}_{\operatorname{\mathcal{E}}}(C)$ then determines a functor $\mathscr {G}_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$.

Suppose we are given an $n$-simplex $\sigma $ of $\operatorname{\mathcal{E}}$. Then $U( \sigma )$ is an $n$-simplex of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we can identify with a diagram

\[ C_0 \xrightarrow { f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow { f_ n } C_ n \]

in the category $\operatorname{\mathcal{C}}$. For $0 \leq m \leq n$, we can view the diagram

\[ C_0 \xrightarrow { f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow { f_ m } C_ m \xrightarrow { \operatorname{id}} C_ m \]

as an $m$-simplex $\overline{\tau }_ m$ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ / C_ m } )$. The pair $( \overline{\tau }_ m, U(\sigma )|_{ \Delta ^ m} )$ can then be viewed as an $m$-simplex $\tau _ m$ of $\mathscr {G}_{\operatorname{\mathcal{E}}}(C_ m)$. Setting $\tau = ( \tau _0, \tau _1, \cdots , \tau _ n )$, we observe that the pair $(U(\sigma ), \tau )$ can be regarded as an $n$-simplex $u_{\operatorname{\mathcal{E}}}( \sigma )$ of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {G}_{\operatorname{\mathcal{E}}}}(\operatorname{\mathcal{C}})$. Allowing $n$ to vary, the construction $\sigma \mapsto u_{\operatorname{\mathcal{E}}}(\sigma )$ determines a morphism of simplicial sets $u_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_{\operatorname{\mathcal{E}}}}(\operatorname{\mathcal{C}})$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{ u_{\operatorname{\mathcal{E}}} } \ar [dr]^{U} & & \operatorname{N}_{\bullet }^{\mathscr {G}_{\operatorname{\mathcal{E}}}}(\operatorname{\mathcal{C}}) \ar [dl] \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]

is commutative.

Proposition 5.3.3.24. Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets, and let $u_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }^{ \mathscr {G}_{\operatorname{\mathcal{E}}} }(\operatorname{\mathcal{C}})$ be the morphism of Notation 5.3.3.23. For every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, precomposition with $u_{\operatorname{\mathcal{E}}}$ induces a bijection

\[ T_{\operatorname{\mathcal{E}}}: \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {G}_{\operatorname{\mathcal{E}}}, \mathscr {F} ) \rightarrow \operatorname{Hom}_{ ( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } }( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) ). \]

Corollary 5.3.3.25. Let $\operatorname{\mathcal{C}}$ be a category. Then the weighted nerve functor

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow ( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \quad \quad \mathscr {F} \mapsto \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]

has a left adjoint, given by the construction $\operatorname{\mathcal{E}}\mapsto \mathscr {G}_{\operatorname{\mathcal{E}}}$ of Notation 5.3.3.23.

Proof of Proposition 5.3.3.24. The construction $\operatorname{\mathcal{E}}\mapsto T_{\operatorname{\mathcal{E}}}$ carries colimits in the category $(\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to limits in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$. We can therefore assume without loss of generality that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex, so that the morphism $U$ determines a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$. Unwinding the definitions, we see that the codomain of $T_{\operatorname{\mathcal{E}}}$ can be identified with the set of tuples $\tau = (\tau _0, \tau _1, \cdots , \tau _ n)$, where $\tau _ i: \Delta ^{i} \rightarrow \mathscr {F}(C_ i)$ are simplices for which the diagram

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

is commutative. Let us regard $\tau $ as fixed; we wish to prove that there is a unique natural transformation $\alpha : \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ) = \tau $.

Let $D$ be an object of $\operatorname{\mathcal{C}}$ and let $m \geq 0$ be an integer. Then $m$-simplices of the simplicial set $\mathscr {G}_{\operatorname{\mathcal{E}}}(D) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/D} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$ can be identified with pairs $(f, g)$, where $g: [m] \rightarrow [n]$ is a nondecreasing function and $f: C_{ g(m) } \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$. Let $\alpha _{D}(f,g)$ denote the $m$-simplex of $\mathscr {F}(D)$ given by the composition

\[ \Delta ^{m} \xrightarrow { g } \Delta ^{g(m) } \xrightarrow { \tau _{g(m)} } \mathscr {F}( C_{g(m)} ) \xrightarrow { \mathscr {F}(f) } \mathscr {F}(D). \]

The construction $(f,g) \mapsto \alpha _{D}(f,g)$ determines a morphism of simplicial sets $\alpha _{D}: \mathscr {G}_{\operatorname{\mathcal{E}}}(D) \rightarrow \mathscr {F}(D)$. The assignment $D \mapsto \alpha _{D}$ determines a natural transformation of functors $\alpha : \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ) = \tau $. This proves existence.

We now prove uniqueness. Suppose we are given another natural transformation $\alpha ': \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ') = \tau $; we wish to show that $\alpha = \alpha '$. Fix an object $D \in \operatorname{\mathcal{C}}$ and an $m$-simplex of the simplicial set $\mathscr {G}_{\operatorname{\mathcal{E}}}(D)$, which we identify with a pair $(f,g)$ as above. We wish to verify that $\alpha _{D}(f,g)$ and $\alpha '_{D}(f,g)$ coincide (as $m$-simplices of the simplicial set $\mathscr {F}(D)$). Set $n' = g(m)$, so that the function $g$ factors as a composition $[m] \xrightarrow {g} [n'] \xrightarrow {\iota } [n]$, where $\iota : [n'] \hookrightarrow [n]$ is the inclusion map. Since $\alpha _ D$ and $\alpha '_{D}$ are morphisms of simplicial sets, it will suffice to prove that $\alpha _{D}( f, \iota )$ and $\alpha '_{D}(f, \iota )$ coincide (as $n'$-simplices of the simplicial set $\mathscr {F}(D)$). Since both $\alpha _{D}$ and $\alpha '_{D}$ are natural in $D$, we may assume without loss of generality that $D = C_{n'}$ and that $f$ is the identity morphism. In this case, the identities $T_{\operatorname{\mathcal{E}}}( \alpha ) = \tau = T_{\operatorname{\mathcal{E}}}( \alpha ' )$ give $\alpha _{D}(f,\iota ) = \tau _{n'} = \alpha '_{D}(f,\iota )$. $\square$

Variant 5.3.3.26. Let $\operatorname{\mathcal{C}}$ be a category, and let us regard $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ as equipped with the simplicial enrichment described in Example 2.4.2.2. For every morphism of simplicial sets $\operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, precomposition with the morphism $u_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_{\operatorname{\mathcal{E}}} }(\operatorname{\mathcal{C}})$ of Notation 5.3.3.23 induces an isomorphism of simplicial sets

\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {G}_{\operatorname{\mathcal{E}}}, \mathscr {F} )_{\bullet } \rightarrow \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) ). \]

To see that this map is bijective on $m$-simplices, we can replace $\operatorname{\mathcal{E}}$ by the product $\Delta ^ m \times \operatorname{\mathcal{E}}$ to reduce to the case $m = 0$, in which case it follows from Proposition 5.3.3.24.