Definition 5.3.4.1. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$, and let $e$ be an edge of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$. Let us identify $e$ with a pair $(f, \overline{e} )$, where $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $\overline{e}$ is an edge of the simplicial set $\mathscr {F}(C)$. We will say that the edge $e = (f, \overline{e} )$ is horizontal if $\overline{e}$ is a degenerate edge of $\mathscr {F}(C)$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
5.3.4 Scaffolds of Cocartesian Fibrations
Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$. Our goal in this section is to show that the diagram $\mathscr {F}$ can be recovered, up to equivalence, from the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Definition 5.3.3.1. More precisely, we will show that there exists a levelwise categorical equivalence from $\mathscr {F}$ to the strict transport representation $\operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}}$ of Construction 5.3.1.5 (Corollary 5.3.4.19).
We begin with some general remarks. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be any cocartesian fibration of simplicial sets and let $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be the weak transport representation of $U$ (Construction 5.3.1.1). Every levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ can be viewed as a natural transformation from $\mathscr {F}$ to the weak transport representation $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, which we can identify (using Corollary 5.3.2.23) with a morphism from the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ into $\operatorname{\mathcal{E}}$. Our first goal is to give an explicit characterization of the collection of morphisms $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ which arise in this way, which we will refer to as scaffolds of the cocartesian fibration $U$ (Definition 5.3.4.2 and Remark 5.3.4.10).
Definition 5.3.4.2. Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. We will say that a morphism of simplicial sets $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ is a scaffold if it satisfies the following conditions:
The diagram of simplicial sets
is commutative (where the left vertical map is the projection map of Construction 5.3.2.1).
The morphism $\lambda $ carries horizontal edges of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ to $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.
For every object $C \in \operatorname{\mathcal{C}}$, the induced map
is a categorical equivalence of simplicial sets.
\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [rr]^{\lambda } \ar [dr] & & \operatorname{\mathcal{E}}\ar [dl]_{U} \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]
\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow {\lambda } \{ C \} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}} \]
Example 5.3.4.3. Let $n$ be a nonnegative integer and let $\operatorname{\mathcal{E}}$ denote the nerve of the partially ordered set $Q = \{ (i,j) \in [n] \times [n]: j \leq i \} $. Then there is a cocartesian fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$, given on vertices by the formula $U(i,j) = i$. Let $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$ denote the functor given by $\mathscr {F}(i) = \Delta ^ i$, so that vertices of the homotopy colimit can be identified with elements of $Q$. There is a unique morphism of simplicial sets $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ which is the identity at the level of vertices, which is a scaffold of the cocartesian fibration $U$. Moreover, $\lambda $ is a monomorphism, and an $n$-simplex $(i_0, j_0) \leq (i_1, j_1) \leq \cdots \leq (i_ n, j_ n)$ belongs to the image of $\lambda $ if and only if $j_ n \leq i_0$. The case $n=3$ is depicted in the following diagram, where the image of $\lambda $ is indicated with solid arrows:
\[ \xymatrix@C =30pt@R=30pt{ & & & (3,3) \\ & & (2,2) \ar@ {-->}[ur] \ar [r] & (3,2) \ar [u] \\ & (1,1) \ar@ {-->}[ur] \ar [r] & (2,1) \ar [u] \ar [r] & (3,1) \ar [u] \\ (0,0) \ar@ {-->}[ur] \ar [r] & (1,0) \ar [r] \ar [u] & (2,0) \ar [u] \ar [r] & (3,0). \ar [u] } \]
Example 5.3.4.4. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category equipped with a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ having fibers $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_{1} = \{ 1\} \times _{ \Delta ^1} \operatorname{\mathcal{E}}$. Choose a functor $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ and a morphism $h: \Delta ^1 \times \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}$ which witnesses $F$ as given by covariant transport along the nondegenerate edge of $\Delta ^1$, in the sense of Definition 5.2.2.4. Then $F$ can be identified with a diagram $\mathscr {F}: [1] \rightarrow \operatorname{QCat}$, and the map is a scaffold.
\[ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) = ( \Delta ^1 \times \operatorname{\mathcal{E}}_0 ) {\coprod }_{ ( \{ 1\} \times \operatorname{\mathcal{E}}_0)} \operatorname{\mathcal{E}}_1 \xrightarrow {(h,\operatorname{id})} \operatorname{\mathcal{E}} \]
Remark 5.3.4.5 (Isomorphism Invariance). In the situation of Definition 5.3.4.2, suppose that we are given a pair of morphisms $\lambda , \lambda ': \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ which are isomorphic when viewed as objects of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}})$. Then $\lambda $ is a scaffold if and only if $\lambda '$ is a scaffold (see Corollary 5.1.2.5 and Remark 4.5.1.15).
Remark 5.3.4.6 (Change of $\operatorname{\mathcal{E}}$). Suppose we are given a commutative diagram where the vertical maps are cocartesian fibrations and $T$ is an equivalence of cocartesian fibrations over $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then a morphism $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ is a scaffold of the cocartesian fibration $U$ if and only if $T \circ \lambda $ is a scaffold of the cocartesian fibration $U'$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^{T} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{ U' } \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), & } \]
We now describe two important examples of scaffolds, both of which can be regarded as generalizations of Example 5.3.4.3.
Construction 5.3.4.7 (The Universal Scaffold). Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories, and let $\operatorname{sTr}_{ \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ denote the strict transport representation of $U$ (Construction 5.3.1.5). For each $n \geq 0$, we can identify $n$-simplices of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$ with pairs $(\sigma , \tau )$, where $\sigma $ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (given by a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$) and $\tau $ is an $n$-simplex of the $\infty $-category $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C_0) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0 / } ), \operatorname{\mathcal{E}})$, which we identify with a morphism of simplicial sets $\Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ C_0/ }) \rightarrow \operatorname{\mathcal{E}}$. Let us identify the diagram $C_0 \xrightarrow {\operatorname{id}} C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ with an $n$-simplex $\widetilde{\sigma }$ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0/} )$, and let $\lambda _{u}( \sigma , \tau )$ denote the $n$-simplex of $\operatorname{\mathcal{E}}$ given by the composite map The construction $(\sigma , \tau ) \mapsto \lambda _{u}( \sigma , \tau )$ determines a morphism of simplicial sets which we will refer to as the universal scaffold of the cocartesian fibration $U$.
\[ \Delta ^ n \xrightarrow { (\operatorname{id}, \widetilde{\sigma }) } \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0/} ) \xrightarrow {\tau } \operatorname{\mathcal{E}}. \]
\[ \lambda _{u}: \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{\mathcal{E}}, \]
Proposition 5.3.4.8. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories. Then the morphism $\lambda _{u}: \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{\mathcal{E}}$ of Construction 5.3.4.7 is a scaffold, in the sense of Definition 5.3.4.2.
Proof. It is clear that the composition $U \circ \lambda _{u}$ coincides with the projection map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Let $e$ be a horizontal edge of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$, determined by a morphism $\overline{e}: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$ together with a degenerate edge $\operatorname{id}_{T}$ of the simplicial set $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( C)$. Identifying $T$ with an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C / } ), \operatorname{\mathcal{E}})$, we see that $\lambda _{u}(e)$ coincides with the morphism $T( \overline{e} )$ and is therefore a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$. To complete the proof, we observe that for every object $C \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \xrightarrow {\lambda } \{ C \} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}} \]
agrees with the map $\operatorname{ev}_{C}: \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C / } ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$ given by evaluation on the initial object $\operatorname{id}_{C} \in \operatorname{\mathcal{C}}_{C/}$, and is therefore a trivial Kan fibration of simplicial sets (Proposition 5.3.1.7). $\square$
Corollary 5.3.4.9. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories. Then there exists a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$.
Remark 5.3.4.10 (Universality). Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Applying Corollary 5.3.2.23, we obtain a bijection from the set of morphisms $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ in the category $(\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to the set of natural transformations $\alpha : \mathscr {F} \rightarrow \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. Unwinding the definitions, we see that $\alpha $ factors through the subfunctor $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \subseteq \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ if and only if $\lambda $ satisfies condition $(1)$ of Definition 5.3.4.2. If this condition is satisfied, then $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is a levelwise categorical equivalence if and only if $\lambda $ satisfies condition $(2)$ of Definition 5.3.4.2. We therefore obtain a bijection Concretely, this bijection carries a levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ to the composite map where $\lambda _{u}$ is the universal scaffold of Construction 5.3.4.7.
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Levelwise categorical equivalences $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$} \} \ar [d]^{\Phi } \\ \{ \textnormal{Scaffolds $\lambda : \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$} \} . } \]
\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow { \alpha } \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \xrightarrow { \lambda _ u } \operatorname{\mathcal{E}}, \]
Construction 5.3.4.11 (The Taut Scaffold). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. By definition, an $n$-simplex of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ is a pair $(\sigma , \tau )$, where $\sigma $ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (given by a diagram $C_0 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$) and $\tau $ is an $n$-simplex of the simplicial set $\mathscr {F}(C_0)$. For $0 \leq i \leq n$, let $\tau _{i}$ denote the composite map We then have a commutative diagram of simplicial sets Consequently, we can view the pair $( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ as an $n$-simplex of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. The construction $(\sigma , \tau ) \mapsto ( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ determines a morphism of simplicial sets $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. In the special case where $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is a diagram of $\infty $-categories, we will refer to $\lambda _{t}$ as the taut scaffold of the cocartesian fibration $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
\[ \Delta ^{i} \hookrightarrow \Delta ^{n} \xrightarrow { \tau } \mathscr {F}(C_0) \rightarrow \mathscr {F}(C_ i). \]
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{0} \ar [d]_{\tau _0} \ar@ {^{(}->}[r] & \Delta ^{1} \ar [d]^-{\tau _1} \ar@ {^{(}->}[r] & \Delta ^{2} \ar@ {^{(}->}[r] \ar [d]_{\tau _2} & \cdots \ar [d] \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]_{\tau _ n} \\ \mathscr {F}(C_0) \ar [r] & \mathscr {F}(C_1) \ar [r] & \mathscr {F}(C_2) \ar [r] & \cdots \ar [r] & \mathscr {F}(C_ n). } \]
Remark 5.3.4.12. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Then the diagram of simplicial sets commutes, where $\lambda _{t}$ is the morphism of Construction 5.3.4.11 and the vertical morphism are the projection maps of Construction 5.3.2.1 and Definition 5.3.3.1.
\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \ar [rr]^{\lambda _{t}} \ar [dr] & & \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [dl] \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]
Example 5.3.4.13. Let $X$ be a simplicial set, which we identify with a diagram $\mathscr {F}: [0] \rightarrow \operatorname{Set_{\Delta }}$. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ and the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}([0])$ can both be identified with $X$ (see Examples 5.3.2.2 and 5.3.3.2). Under these identifications, the taut scaffold $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}([0])$ of Construction 5.3.4.11 corresponds to the identity map $\operatorname{id}_{X}$.
Remark 5.3.4.14 (Functoriality). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the morphism of Construction 5.3.4.11. If $T: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ is any functor between categories, then $\lambda $ induces a morphism Setting $\mathscr {F}' = \mathscr {F} \circ T$, we can use Remarks 5.3.2.3 and 5.3.3.7 to identify $\lambda '_{t}$ with a morphism from the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' )$ to the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}')$. This morphism coincides with the map obtained by applying Construction 5.3.4.11 to the diagram $\mathscr {F}'$.
\[ \lambda '_{t}: \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}). \]
Example 5.3.4.15 (Comparison of Fibers). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the morphism of Construction 5.3.4.11. Combining Example 5.3.4.13 with Remark 5.3.4.14, we see that for every object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers is an isomorphism of simplicial sets (under the identifications provided by Remark 5.3.2.3 and Example 5.3.3.8, it corresponds to the identity morphism $\operatorname{id}: \mathscr {F}(C) \rightarrow \mathscr {F}(C)$).
\[ \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]
Example 5.3.4.16. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, which we identify with a diagram $\mathscr {F}: [1] \rightarrow \operatorname{Set_{\Delta }}$. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ can be identified with the mapping cylinder $(\Delta ^1 \times X) {\coprod }_{ (\{ 1\} \times X) } Y$ (Example 5.3.2.13), and the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}([1])$ can be identified with the relative join $X \star _{Y} Y$ (Example 5.3.3.13). Under these identifications, Construction 5.3.4.11 corresponds to a morphism of simplicial sets Unwinding the definitions, we see that this map classifies the commutative diagram In particular, the morphism $\lambda _{t}$ is an isomorphism if and only if (5.27) is a pushout square of simplicial sets.
\[ \lambda _{t}: (\Delta ^1 \times X) {\coprod }_{ (\{ 1\} \times X) } Y \rightarrow X \star _{Y} Y. \]
5.27
\begin{equation} \begin{gathered}\label{equation:comparison-map-over-edge} \xymatrix@R =50pt@C=50pt{ \emptyset \star _{X} X \ar [r] \ar [d] & \emptyset \star _{Y} Y \ar [d] \\ X \star _{X} X \ar [r] & X \star _{Y} Y. } \end{gathered} \end{equation}
Proposition 5.3.4.17. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories indexed by a category $\operatorname{\mathcal{C}}$. Then the morphism $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Construction 5.3.4.11 is a scaffold of the cocartesian fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
Proof. Condition $(0)$ of Definition 5.3.4.2 follows from Remark 5.3.4.12, condition $(2)$ from Example 5.3.4.15, and condition $(1)$ from the characterization of $U$-cocartesian morphisms supplied by Corollary 5.3.3.16. $\square$
Corollary 5.3.4.18. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then there exists a cocartesian fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$.
Proof. Using Proposition 4.1.3.2, we can choose a diagram of $\infty $-categories $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \mathscr {F}'$. We can then take $\lambda $ to be the composition $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow { \alpha } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \xrightarrow {\lambda _{t}} \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$, where $\lambda _{t}$ is the taut scaffold of Proposition 5.3.4.17. $\square$
Corollary 5.3.4.19. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$, 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 there exists a levelwise categorical equivalence from $\mathscr {F}$ to the strict transport representation $\operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}}$.
Proof. Combine Proposition 5.3.4.17 with Remark 5.3.4.10 (for a more precise statement, see Construction 7.5.3.3). $\square$
In certain cases, one can improve on Example 5.3.4.15.
Proposition 5.3.4.20. 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 morphism $u: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the image $\mathscr {F}(u): \mathscr {F}(C) \rightarrow \mathscr {F}(D)$ is a left covering map (Definition 4.2.3.8). Then the morphism $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Construction 5.3.4.11 is an isomorphism.
Proof. Let $( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ be an $n$-simplex of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. We identify $\sigma $ with a diagram $C_0 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$, and each $\tau _ i$ with an $i$-simplex of the simplicial set $\mathscr {F}(C_ i)$. We wish to show that there is a unique $n$-simplex $\tau $ of $\mathscr {F}(C_0)$ satisfying $\lambda _{t}(\sigma , \tau ) = ( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$. Note that, for this condition to be satisfied, the simplex $\tau $ must be a solution to the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [d] \ar [r] & \mathscr {F}(C_0) \ar [d] \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r]^{ \tau _ n } & \mathscr {F}(C_ n). } \]
Since the inclusion $\{ 0\} \hookrightarrow \Delta ^ n$ is left anodyne (Example 4.3.7.11), our assumption that the right vertical map is a left covering guarantees that this lifting problem has a unique solution $\tau : \Delta ^ n \rightarrow \mathscr {F}(C_0)$ (Corollary 4.2.4.12). This proves uniqueness. To prove existence, write $\lambda _{t}(\sigma , \tau ) = ( \sigma , \{ \tau '_{i} \} _{0 \leq i \leq n} )$. We wish to prove that $\tau _ i = \tau '_ i$ for $0 \leq i \leq n$. For this, we observe that both $\tau _ i$ and $\tau '_ i$ can be viewed as solutions to a common lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [r]^{ \tau _ i(0) } \ar [d] & \mathscr {F}(C_ i) \ar [d] \\ \Delta ^{i} \ar@ {-->}[ur] \ar [r] & \mathscr {F}( C_ n ). } \]
Since the inclusion $\{ 0\} \hookrightarrow \Delta ^ i$ is left anodyne (Example 4.3.7.11) and the right vertical map is a left covering, the solution to this lifting problem is uniquely determined (Corollary 4.2.4.12). $\square$
Example 5.3.4.21 (Set-Valued Functors). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a diagram of sets, and let us abuse notation by identifying $\mathscr {F}$ with a diagram of discrete simplicial sets. Then the taut scaffold $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is an isomorphism. It follows that $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ can be identified with the nerve of the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (see Example 5.3.2.5).
Corollary 5.3.4.22. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor which carries each morphism of $\operatorname{\mathcal{C}}$ to an isomorphism of simplicial sets. Then the morphism $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Remark 5.3.4.12 is an isomorphism.
Corollary 5.3.4.23. Let $\operatorname{\mathcal{C}}$ be a groupoid and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is a Kan complex.
Proof. Using Corollaries 5.3.4.22 and 5.3.3.19, we see that the map $U: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration. Since $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex (Proposition 1.3.5.2), it follows that $U$ is a Kan fibration (Corollary 4.4.3.8), so that $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ is also a Kan complex (Remark 3.1.1.11). $\square$
Example 5.3.4.24 (Homotopy Quotients). Let $G$ be a group and let $BG$ denote the associated groupoid (consisting of a single object with automorphism group $G$). Let $X$ be a simplicial set equipped with an action of $G$, which we identify with a functor $\mathscr {F}: BG \rightarrow \operatorname{Set_{\Delta }}$. Applying Corollary 5.3.4.22, we obtain an isomorphism of simplicial sets $X_{\mathrm{h}G} \xrightarrow {\sim } \operatorname{N}_{\bullet }^{\mathscr {F}}(BG)$, where $X_{\mathrm{h}G}$ is the homotopy quotient of $X$ by the action of $G$ (Example 5.3.2.15). If $X$ is a Kan complex, then Corollary 5.3.4.23 guarantees that $X_{\mathrm{h}G}$ is also a Kan complex.