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5.5.6 Comparison with the Weighted Nerve

Let $\operatorname{\mathcal{C}}$ be a category which is equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$. In §5.5.3 and §5.5.5, we described two different cocartesian fibrations associated to $\mathscr {F}$:

  • The projection map $U: \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ denotes the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$ (Definition 5.5.3.1). For each object $C \in \operatorname{\mathcal{C}}$, the fiber $U^{-1} \{ C\} $ is isomorphic (as a simplicial set) to the $\infty $-category $\mathscr {F}(C)$ (Example 5.5.3.7).

  • The projection map $U': \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ denotes the $\infty $-category of elements of the functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}}$. For each object $C \in \operatorname{\mathcal{C}}$, the fiber $U'^{-1} \{ C\} $ is equivalent (but not necessarily isomorphic) to the $\infty $-category $\mathscr {F}(C)$ (Example 5.5.4.19).

Our goal in this section is to show that these constructions are equivalent.

Construction 5.5.6.1 (The Comparison Map). Let $\operatorname{\mathcal{C}}$ be a category and let $\overrightarrow {C}$ be an $n$-simplex of the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, given by a diagram

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

in the category $\operatorname{\mathcal{C}}$. Let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets and suppose that we are given a collection of simplices $\overrightarrow {\sigma } = \{ \sigma _{j}: \Delta ^{j} \rightarrow \mathscr {F}(C_ j) \} _{0 \leq j \leq n}$ which fit into a commutative diagram

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

To this data, we can associate a commutative diagram of simplicial sets

5.41
\begin{equation} \begin{gathered}\label{equation:compare-with-weighted-nerve} \xymatrix@R =50pt@C=50pt{ \Delta ^{n} \ar [r] \ar [d]^{ \overrightarrow {C}} & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{ \Delta ^{0} / } \ar [d] \\ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \ar [r]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F}) } & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }}), } \end{gathered} \end{equation}

where the upper horizontal map can be identified with the simplicial functor $F: \operatorname{Path}[ \{ x\} \star [n] ]_{\bullet } \rightarrow \operatorname{Set_{\Delta }}$ described as follows:

  • The functor $F$ carries $x$ to the simplicial set $\Delta ^{0}$ (so that $F$ can be identified with an $n$-simplex of the coslice simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{\Delta ^{0}/}$).

  • The restriction of $F$ to the simplicial path category $\operatorname{Path}[n]_{\bullet }$ is given by the composition

    \[ \operatorname{Path}[n]_{\bullet } \rightarrow [n] \xrightarrow { \overrightarrow {C}} \operatorname{\mathcal{C}}\xrightarrow { \mathscr {F}} \operatorname{Set_{\Delta }} \]

    (as required by the commutativity of the diagram (5.41)).

  • For $0 \leq m \leq n$, the induced map of simplicial sets

    \[ \operatorname{Hom}_{\operatorname{Path}[ \{ x\} \star [n] ]}( x, m)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( F(x), F(m) )_{\bullet } = \mathscr {F}( C_ m ) \]

    is given by the composition $\operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star [n] ] }( x, m )_{\bullet } \xrightarrow {\rho } \Delta ^{m} \xrightarrow { \sigma _ m} \mathscr {F}(C_ m)$, where $\rho $ is induced by the morphism of partially ordered sets

    \[ \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star [n] }(x,m) \rightarrow [m] \quad \quad (S \subseteq \{ x\} \star [n] ) \mapsto \min ( S \setminus \{ x\} ). \]

Note that we can identify the diagram (5.41) with an $n$-simplex $\theta ( \overrightarrow {C}, \overrightarrow {\sigma } )$ of the simplicial set $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$. The construction $(\overrightarrow {C}, \overrightarrow {\sigma }) \mapsto \theta ( \overrightarrow {C}, \overrightarrow {\sigma } )$ then determines a morphism of simplicial sets $\theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$, which we will refer to as the comparison map.

Example 5.5.6.2 (The Comparison Map on Vertices). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. Let us identify vertices of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ with pairs $(C,X)$, where where $C$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is a vertex of the simplicial set $\mathscr {F}(C)$ (Remark 5.5.3.11). Under this identification, the comparison map

\[ \theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]

of Construction 5.5.6.1 is given on vertices by the construction $(C,X) \mapsto (C,X)$, where we identify $(C,X)$ with a vertex of $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ using Example 5.5.4.12. In particular, the morphism $\theta $ is bijective at the level of vertices.

Example 5.5.6.3 (The Comparison Map on Edges). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. Let $(C,X)$ and $(D,Y)$ be vertices of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$. Using Remark 5.5.3.12, we can identify edges of $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ having source $(C,X)$ and target $(D,Y)$ with pairs $(f,u)$, where $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $u: \mathscr {F}(f)(X) \rightarrow Y$ is an edge of the simplicial set $\mathscr {F}(D)$. Under this identification, the comparison map

\[ \theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]

of Construction 5.5.6.1 is given on edges by the construction $(f,u) \mapsto (f,u)$, where we identify $(f,u)$ with an edge of the simplicial set $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ using Example 5.5.4.13. In particular, the morphism $\theta $ is bijective at the level of edges.

Warning 5.5.6.4. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. The comparison map $\theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ of Construction 5.5.6.1 is generally not bijective on $n$-simplices for $n \geq 2$ (even in the special case $\operatorname{\mathcal{C}}= [0]$).

Exercise 5.5.6.5. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. Show that the comparison map $\theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ of Construction 5.5.6.1 is a monomorphism of simplicial sets.

Remark 5.5.6.6. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. Then the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \ar [rr]^-{\theta } \ar [dr] & & \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \ar [dl] \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]

is commutative, where the vertical morphisms are the projection maps of Definitions 5.5.3.1 and 5.5.4.1 and $\theta $ is the comparison morphism of Construction 5.5.6.1

Example 5.5.6.7. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. For every object $C \in \operatorname{\mathcal{C}}$, the comparison morphism $\theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ of Construction 5.5.6.1 induces a morphism of simplicial sets

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

Under the isomorphisms

\[ \mathscr {F}(C) \simeq \{ C\} \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \quad \quad \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})}( \Delta ^{0}, \mathscr {F}(C) ) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]

supplied by Examples 5.5.3.7 and 5.5.4.19, we can identify $\theta _{C}$ with the comparison map $\mathscr {F}(C) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})}( \Delta ^{0}, \mathscr {F}(C) )$ of Construction 4.6.6.3.

Proposition 5.5.6.8. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, and let

\[ \xymatrix@R =50pt@C=50pt{ \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \ar [dr]_{U} \ar [rr]^{\theta } & & \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \ar [dl]^{U'} \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]

be the commutative diagram of Remark 5.5.6.6. Then:

$(1)$

For each object $C \in \operatorname{\mathcal{C}}$, the morphism $\theta $ induces an equivalence of $\infty $-categories

\[ \theta _{C}: \{ C\} \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}). \]
$(2)$

A morphism $f$ of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ is $U$-cocartesian if and only if $\theta (f)$ is a $U'$-cocartesian morphism of the $\infty $-category $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$.

$(3)$

The functor $\theta $ is an equivalence of $\infty $-categories.

Proof. Assertion $(1)$ follows from Example 5.5.6.7 and Theorem 4.6.6.9. Assertion $(2)$ follows from Example 5.5.6.3 together with the descriptions of $U$-cocartesian and $U'$-cocartesian morphisms supplied by Proposition 5.5.3.13 and Remark 5.5.4.14. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with Theorem 5.1.5.1 (since $U$ and $U'$ are cocartesian fibrations, by virtue of Propositions 5.5.3.13 and 5.5.4.2. $\square$

Corollary 5.5.6.9. 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 5.5.4.2, 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 5.2.2.1). 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 5.5.4.19).

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$, so that $\mathscr {F}$ can be identified with a functor of ordinary categories $[1] \rightarrow \operatorname{QCat}$. In this case, the desired result follows from Proposition 5.5.6.8 and Remark 5.5.3.15. $\square$

Corollary 5.5.6.10. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets, let $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \simeq \mathrm{h} \mathit{\operatorname{QCat}}$ be the induced functor between homotopy categories, and let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ be the cocartesian fibration of Proposition 5.5.4.2. Then the homotopy transport representation $\operatorname{hTr}_{U}$ of Construction 5.2.3.2 is isomorphic to $\mathrm{h} \mathit{\mathscr {F}}$ (as an object of the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{QCat}})$).