8.6.1 Conjugate Fibrations
Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. Our goal in this section is to formalize the requirement that $U$ and $U^{\dagger }$ have “the same fibers”: that is, that there exists a family of equivalences $\{ T_ C: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C} \} _{C \in \operatorname{\mathcal{C}}}$ which in some sense depend functorially on the vertex $C \in \operatorname{\mathcal{C}}$.
Definition 8.6.1.1 (Conjugate Fibrations). Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. Let $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection maps of Notation 8.1.1.6. We say that a morphism of simplicial sets $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U$ if the following conditions are satisfied:
- $(0)$
The diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \lambda _{+} } & \operatorname{\mathcal{C}}} \]
is commutative.
- $(1)$
For every vertex $C \in \operatorname{\mathcal{C}}$, restricting $T$ to the inverse image of the vertex $\operatorname{id}_{C} \in \operatorname{Tw}(\operatorname{\mathcal{C}})$ determines an equivalence of $\infty $-categories $T_{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$.
- $(2)$
Let $e$ be an edge of the simplicial set $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$. If the image of $e$ in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian, then $T(e)$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
We say that $U^{\dagger }$ is a cartesian conjugate of $U$ if there exists a morphism $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$.
Warning 8.6.1.2 (Symmetry). Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. In §8.6.6, we will show that $U^{\dagger }$ is a cartesian conjugate of $U$ if and only if $U^{\operatorname{op}}$ is a cocartesian conjugate of $U^{\dagger ,\operatorname{op}}$ (Corollary 8.6.6.2). Beware that this is not obvious from Definition 8.6.1.1.
Example 8.6.1.3. Let $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}^{\dagger }$ be $\infty $-categories. Set $\operatorname{\mathcal{C}}= \Delta ^0$ and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}$ denote the projection maps. Then a functor
\[ T: \operatorname{\mathcal{E}}^{\dagger } \simeq \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}} \]
exhibits $U^{\dagger }$ are a cartesian conjugate of $U$ if and only if it an equivalence of $\infty $-categories. In particular, $U^{\dagger }$ is a cartesian conjugate of $U$ if and only if the $\infty $-category $\operatorname{\mathcal{E}}^{\dagger }$ is equivalent to $\operatorname{\mathcal{E}}$.
In the situation of Definition 8.6.1.1, we can regard condition $(2)$ as a formulation of the requirement that the functor $T_{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$ depends functorially on the vertex $C \in \operatorname{\mathcal{C}}$. This heuristic can be articulated more precisely as follows:
Proposition 8.6.1.5. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. Let $e: C \rightarrow C'$ be an edge of $\operatorname{\mathcal{C}}$, and let
\[ e^{\ast }: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}^{\dagger }_{C'} \quad \quad e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'} \]
be functors given by contravariant and covariant transport along $e$ for the fibrations $U^{\dagger }$ and $U$, respectively. If $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ is a morphism which satisfies conditions $(0)$ and $(2)$ of Definition 8.6.1.1, then the diagram of $\infty $-categories
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger }_{C} \ar [d]^{ T_{C} } \ar [r]^-{ e^{\ast } } & \operatorname{\mathcal{E}}^{\dagger }_{C'} \ar [d]^{ T_{C'} } \\ \operatorname{\mathcal{E}}_{C} \ar [r]^-{ e_{!} } & \operatorname{\mathcal{E}}_{C'} } \]
commutes up to isomorphism.
Proof.
The restriction of $T$ to the inverse image of the vertex $\{ e\} \subseteq \operatorname{Tw}(\operatorname{\mathcal{C}})$ determines a functor of $\infty $-categories $T_{e}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$. To complete the proof, it will suffice to verify the following pair of assertions:
- $(a)$
The functor $T_{e}$ is isomorphic to the composition $T_{C'} \circ e^{\ast }$.
- $(b)$
The functor $T_{e}$ is isomorphic to the composition $e_{!} \circ T_{C}$.
We begin by proving $(a)$. Choose a diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger }_{C} \times \Delta ^1 \ar [r]^-{H} \ar [d] & \operatorname{\mathcal{E}}^{\dagger } \ar [d]^{U^{\dagger } } \\ \Delta ^1 \ar [r]^-{e} & \operatorname{\mathcal{C}}^{\operatorname{op}} } \]
which witnesses $e^{\ast } = H_{\operatorname{\mathcal{E}}^{\dagger }_{C} \times \{ 0\} }$ as given by contravariant transport along $e$ (see Definition 5.2.2.15). Let $e_{L}: \operatorname{id}_{C} \rightarrow e$ and $e_{R}: \operatorname{id}_{C'} \rightarrow e$ denote the edges of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6, and let $\widetilde{H}$ denote the product morphism
\[ \operatorname{\mathcal{E}}^{\dagger }_{C} \times \Delta ^1 \xrightarrow {H \times e_{R}} \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}). \]
Then the composition $T \circ \widetilde{H}$ can be regarded as a natural transformation from the functor $T_{C'} \circ e^{\ast }$ to $T_{e}$. For each object $X$ of the $\infty $-category $\operatorname{\mathcal{E}}^{\dagger }_{C}$, the restriction $H|_{ \{ X\} \times \Delta ^1 }$ is a $U^{\dagger }$-cartesian edge of $\operatorname{\mathcal{E}}^{\dagger }$. Using condition $(2)$ of Definition 8.6.1.1, we conclude that $(T \circ \widetilde{H})|_{ \{ X\} \times \Delta ^1 }$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ lying over the degenerate edge $\operatorname{id}_{C'}$ of $\operatorname{\mathcal{C}}$, and is therefore an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{C'}$ (Proposition 5.1.4.12). Applying Theorem 4.4.4.4, we conclude that $T \circ \widetilde{H}$ is an isomorphism of functors from $T_{C'} \circ e^{\ast }$ to $T_{e}$.
We now prove $(b)$. Let $H'$ denote the composite map
\[ \operatorname{\mathcal{E}}^{\dagger }_{C} \times \Delta ^1 \xrightarrow {\operatorname{id}\times e_{L}} \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \xrightarrow {T} \operatorname{\mathcal{E}}, \]
We then have a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger }_{C} \times \Delta ^1 \ar [r]^-{H'} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U} \\ \Delta ^1 \ar [r]^-{e} & \operatorname{\mathcal{C}}. } \]
Condition $(2)$ of Definition 8.6.1.1 guarantees that, for each object $X \in \operatorname{\mathcal{E}}^{\dagger }_{C}$, the restriction $H'|_{ \{ X\} \times \Delta ^1 }$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$. It follows that $H'$ determines an isomorphism of $T_{e} = H'|_{ \operatorname{\mathcal{E}}^{\dagger }_{C} \times \{ 1\} }$ with the composition $e_{!} \circ (H'|_{ \operatorname{\mathcal{E}}^{\dagger }_{C} \times \{ 0\} } ) = e_{!} \circ T_{C}$.
$\square$
Corollary 8.6.1.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration having homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ (Construction 5.2.5.2), and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration having homotopy transport representation $\operatorname{hTr}_{ \operatorname{\mathcal{E}}^{\dagger } / \operatorname{\mathcal{C}}^{\operatorname{op}} }: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ (Construction 5.2.5.7). If $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U$, then $T$ induces an isomorphism of functors $\operatorname{hTr}_{\operatorname{\mathcal{E}}^{\dagger } / \operatorname{\mathcal{C}}^{\operatorname{op}}} \xrightarrow {\sim } \operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$, carrying each vertex $C \in \operatorname{\mathcal{C}}$ to (the isomorphism class of) the equivalence $T_{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$.
Corollary 8.6.1.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. If $U^{\dagger }$ is a cartesian conjugate of $U$, then the homotopy transport representations
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}^{\dagger } / \operatorname{\mathcal{C}}^{\operatorname{op}} }, \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \]
are isomorphic.
We now give some concrete examples of conjugate fibrations.
Proposition 8.6.1.8 (Conjugates of Left Fibrations). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of simplicial sets. Then:
- $(a)$
The map $H: \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ is a trivial Kan fibration of simplicial sets.
- $(b)$
Let $T_0$ be a section of $H$, and let $T: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ be the composition of $T_0$ with the projection map $\operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$. Then $T$ exhibits the opposite fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ as a cartesian conjugate of $U$.
Proof.
Note that the morphism $H$ factors as a composition
\[ \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}), \]
where the map on the left is a left fibration by virtue of Proposition 8.1.1.15, and the map on the right is pullback of $U$ (and is therefore also a left fibration). It follows that $H$ is a left fibration (Remark 4.2.1.11). To prove $(a)$, it will suffice to show that every fiber of $H$ is a contractible Kan complex (Proposition 4.4.2.14). For this, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^0$. In this case, $\operatorname{\mathcal{E}}$ is a Kan complex (Proposition 4.4.2.1) and we can identify $H$ with the projection map $\operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$, which is a trivial Kan fibration by virtue of Corollary 8.1.2.3.
Let $T: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ be as in $(b)$; we wish to show that $T$ satisfies conditions $(0)$, $(1)$, and $(2)$ of Definition 8.6.1.1. Condition $(0)$ follows from the commutativity of the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{E}}) \ar [rr] \ar [d]^{H} & & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}, } \]
and condition $(2)$ is vacuous (our assumption that $U$ is a left fibration guarantees that every edge of $\operatorname{\mathcal{E}}$ is $U$-cocartesian; see Example 5.1.1.3). To verify condition $(1)$, we may again assume that $\operatorname{\mathcal{C}}= \Delta ^0$, in which case the desired result follows from the observation that the projection maps $\operatorname{\mathcal{E}}^{\operatorname{op}} \leftarrow \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$ are homotopy equivalences of Kan complexes (see Corollary 8.1.2.3).
$\square$
Construction 8.6.1.9 (Conjugacy for Categories of Elements). Let $\operatorname{\mathcal{C}}$ be a category, let $\mathbf{Cat}$ denote the $2$-category of (small) categories, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories. Let
\[ \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F} \quad \quad \int _{ \operatorname{\mathcal{C}}} \mathscr {F} \]
denote the contravariant and covariant categories of elements of $\mathscr {F}$, respectively (see Definitions 5.6.1.4 and 5.6.1.1). Recall that objects of either category can be identified with pairs $(C,X)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is an object of the category $\mathscr {F}(C)$. However, morphisms are defined differently:
A morphism from $(C,X)$ to $(D,Y)$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is a pair $(f,u)$ where $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $u: \mathscr {F}(f)(X) \rightarrow Y$ is a morphism in the category $\mathscr {F}(D)$.
A morphism from $(C,X)$ to $(D,Y)$ in the category $\int ^{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F}$ is a pair $(g,v)$, where $g: D \rightarrow C$ is a morphism in the category $\operatorname{\mathcal{C}}$, and $v: X \rightarrow \mathscr {F}(g)(Y)$ is a morphism in the category $\mathscr {F}(C)$.
Let us identify the objects of the fiber product $( \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} ) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ with pairs $(s: C' \rightarrow C, X)$, where $s: C' \rightarrow C$ is a morphism in $\operatorname{\mathcal{C}}$ and $X$ is an object of the category $\mathscr {F}(C')$. We define a functor We define a functor
\[ T: ( \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} ) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]
as follows:
On objects, $T$ is given by the formula $T( s: C' \rightarrow C, X ) = (C, \mathscr {F}(s)(X) )$.
Let $(s: C' \rightarrow C, X)$ and $(t: D' \rightarrow D, Y)$ be objects of the category $( \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} ) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. Unwinding the definitions, we see that a morphism from $(s: C' \rightarrow C, X)$ to $(t: D' \rightarrow D, Y)$ can be identified with triples $(f, f', u)$, where $f: C \rightarrow D$ and $f': D' \rightarrow C'$ are morphisms in $\operatorname{\mathcal{C}}$ satisfying $t = f \circ s \circ f'$, and $u: X \rightarrow \mathscr {F}(f')(Y)$ is a morphism in the category $\mathscr {F}(C')$. In this case, we define $T(f,f',u)$ to be the morphism $(f,v): (C, \mathscr {F}(s)(X) ) \rightarrow (D, \mathscr {F}(t)(Y) )$, where $v$ is the morphism in $\mathscr {F}(D)$ given by the composition
\begin{eqnarray*} (\mathscr {F}(f) \circ \mathscr {F}(s))(X) & \xrightarrow { (\mathscr {F}(f) \circ \mathscr {F}(s))(u) } & (\mathscr {F}(f) \circ \mathscr {F}(s) \circ \mathscr {F}(f'))(Y) \\ & \simeq & \mathscr {F}( f \circ s \circ f')(Y) \\ & = & \mathscr {F}(t)(Y), \end{eqnarray*}
where the unlabeled isomorphism is supplied by the composition constraints for the functor $\mathscr {F}$.
Proposition 8.6.1.10. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor. Then, after passing to nerves, the functor
\[ T: ( \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} ) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]
of Construction 8.6.1.9 exhibits the forgetful functor $U^{\dagger }: \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ as a cartesian conjugate of the forgetful functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.
Proof.
Condition $(0)$ of Definition 8.6.1.1 follows immediately from the construction. To verify condition $(1)$, we observe that for each object $C \in \operatorname{\mathcal{C}}$, the functor
\[ T_{C}: ( \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} ) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} \rightarrow \{ C \} \times _{ \operatorname{\mathcal{C}}} ( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \]
can be identified with the functor $\mathscr {F}( \operatorname{id}_{C} ): \mathscr {F}(C) \rightarrow \mathscr {F}(C)$. The identity constraint of $\mathscr {F}$ supplies an isomorphism of functors $\operatorname{id}_{ \mathscr {F}(C) } \xrightarrow {\sim } T_{C}$, so that $T_{C}$ is an equivalence of categories. To verify condition $(2)$, suppose we are given a morphism $e$ in the category $( \int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F} ) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. We wish to show that, if the image of $e$ in the category $\int ^{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F}$ is $U^{\dagger }$-cartesian, then $T(e)$ is a $U$-cocartesian morphism in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Writing $e = (f,f',u)$ and $T(e) = (f,v)$ as in Construction 8.6.1.9, we are reduced to showing that if $u$ is an isomorphism, then $v$ is also an isomorphism (Proposition 5.6.1.15), which is immediate from the construction.
$\square$
Construction 8.6.1.11 (Conjugacy for Weighted Nerves). Let $\operatorname{QCat}$ denote the category of $\infty $-categories (which we regard as a full subcategory of the category of simplicial sets). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, and let $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the weighted nerve of Definition 5.3.3.1. We will identify $n$-simplices of $\operatorname{\mathcal{E}}$ with pairs $(\sigma _{+}, \tau _{+})$, where $\sigma _{+} = ( C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n )$ is an $n$-simplex of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ and $\tau _{+} = ( \tau _0, \tau _1, \cdots , \tau _ n )$ is the datum of a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{0} \ar [d]_{\tau _0} \ar@ {^{(}->}[r] & \Delta ^{1} \ar [d]^-{\tau _1} \ar@ {^{(}->}[r] & \Delta ^{2} \ar@ {^{(}->}[r] \ar [d]_{\tau _2} & \cdots \ar [d] \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]_{\tau _ n} \\ \mathscr {F}(C_0) \ar [r] & \mathscr {F}(C_1) \ar [r] & \mathscr {F}(C_2) \ar [r] & \cdots \ar [r] & \mathscr {F}(C_ n). } \]
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Corollary 5.3.3.16, given on $n$-simplices by the formula $U(\sigma _{+},\tau _{+}) = \sigma _{+}$.
Let $\mathscr {F}^{\operatorname{op}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ denote the functor given by the formula $\mathscr {F}^{\operatorname{op}}(C) = \mathscr {F}(C)^{\operatorname{op}}$, and let $\operatorname{\mathcal{E}}^{\dagger }$ denote the $\infty $-category ${\operatorname{N}_{\bullet }^{\mathscr {F}^{\operatorname{op}}}(\operatorname{\mathcal{C}})}^{\operatorname{op}}$. Unwinding the definitions, we see that $n$-simplices of $\operatorname{\mathcal{E}}^{\dagger }$ can be identified with pairs $(\sigma _{-}, \tau _{-})$, where $\sigma _{-} = (C'_ n \rightarrow C'_{n-1} \rightarrow \cdots \rightarrow C'_{0})$ is an $n$-simplex of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}}$ and $\tau _{-} = (\tau '_ n, \tau '_{n-1}, \cdots , \tau '_0 )$ is the datum of a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \{ n\} \ar [d]_{\tau '_{n}} \ar@ {^{(}->}[r] & \operatorname{N}_{\bullet }( \{ n-1 < n \} ) \ar [d]^{\tau '_{n-1}} \ar@ {^{(}->}[r] & \cdots \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]^{ \tau '_0 } \\ \mathscr {F}(C'_ n) \ar [r] & \mathscr {F}(C'_{n-1}) \ar [r] & \cdots \ar [r] & \mathscr {F}(C'_0). } \]
Corollary 5.3.3.16 supplies a cartesian fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}}$, given on $n$-simplices by the formula $U^{\dagger }( \sigma _{-}, \tau _{-} ) = \sigma _{-}$.
Let us identify $n$-simplices of the fiber product $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} } \operatorname{Tw}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) )$ with quadruples $( \sigma _{-}, \tau _{-}, \sigma _{+}, e)$, where $\sigma _{-} = (C'_ n \rightarrow \cdots \rightarrow C'_0)$ is an $n$-simplex of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}}$, $\tau _{-} = ( \tau '_{n}, \cdots , \tau '_0 )$ is as above, $\sigma _{+} = ( C_0 \rightarrow \cdots \rightarrow C_ n )$ is an $n$-simplex of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$, and $e: C'_0 \rightarrow C_0$ is a morphism in the category $\operatorname{\mathcal{C}}$. For $0 \leq i \leq n$, we let $\tau _{i}$ denote the $i$-simplex of $\mathscr {F}( C_ i )$ given by the composition
\[ \Delta ^{i} \hookrightarrow \Delta ^ n \xrightarrow { \tau '_{0} } \mathscr {F}( C'_0 ) \xrightarrow { \mathscr {F}(e) } \mathscr {F}(C_0 ) \rightarrow \mathscr {F}( C_ i ). \]
Set $\tau _{+} = ( \tau _0, \cdots , \tau _ n )$, so that the pair $( \sigma _{+}, \tau _{+} )$ determines an $n$-simplex of the simplicial set $\operatorname{\mathcal{E}}$. The construction $( \sigma _{-}, \tau _{-}, \sigma _{+}, e) \mapsto (\sigma _{+}, \tau _{+} )$ is compatible with face and degeneracy operators, and therefore determines a functor of $\infty $-categories
\[ T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} } \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{\mathcal{E}}. \]
Proposition 8.6.1.12. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ denote the cocartesian fibration of Corollary 5.3.3.16, and define $U^{\dagger }: {\operatorname{N}_{\bullet }^{\mathscr {F}^{\operatorname{op}} }(\operatorname{\mathcal{C}})}^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}}$ similarly. Then the functor
\[ T: {\operatorname{N}_{\bullet }^{\mathscr {F}^{\operatorname{op}} }(\operatorname{\mathcal{C}})}^{\operatorname{op}} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} } \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]
of Construction 8.6.1.11 exhibits $U^{\dagger }$ as a cartesian conjugate of $U$.
Proof.
Condition $(0)$ of Definition 8.6.1.1 follows immediately from the construction. Condition $(1)$ follows from the observation that, for each object $C \in \operatorname{\mathcal{C}}$, the induced map
\[ T_{C}: {\operatorname{N}_{\bullet }^{\mathscr {F}^{\operatorname{op}} }(\operatorname{\mathcal{C}})}^{\operatorname{op}} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} } \{ C\} \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]
is an isomorphism of simplicial sets (under the identifications supplied by Example 5.3.3.8, it corresponds to the identity functor from the $\infty $-category $\mathscr {F}(C)$ to itself). Condition $(2)$ follows from the characterization of $U$-cocartesian and $U^{\dagger ,\operatorname{op}}$-cocartesian morphisms given in Corollary 5.3.3.16.
$\square$
We close this section with a technical result, which will be convenient for verifying hypothesis $(2)$ of Definition 8.6.1.1. If $\operatorname{\mathcal{C}}$ is a simplicial set and $e: C \rightarrow D$ is an edge of $\operatorname{\mathcal{C}}$, we write $e_{L}: \operatorname{id}_{C} \rightarrow e$ and $e_{R}: \operatorname{id}_{D} \rightarrow e$ for the edges of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6.
Proposition 8.6.1.13. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration, and suppose we are given a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \lambda _{+} } & \operatorname{\mathcal{C}}. } \]
Then $T$ satisfies condition $(2)$ of Definition 8.6.1.1 if and only if it satisfies both of the following conditions:
- $(2')$
For every object $Y \in \operatorname{\mathcal{E}}^{\dagger }$ having image $\overline{Y} = U^{\dagger }(Y)$ and every edge $e: \overline{Y} \rightarrow \overline{X}$ of $\operatorname{\mathcal{C}}$, the morphism $T$ carries $( \operatorname{id}_{Y}, e_{L} ): (Y, \operatorname{id}_{\overline{Y}} ) \rightarrow (Y, e)$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
- $(2'')$
Let $f: X \rightarrow Y$ be a $U^{\dagger }$-cartesian edge of $\operatorname{\mathcal{E}}^{\dagger }$. Set $\overline{X} = U^{\dagger }(X)$ and $\overline{Y} = U^{\dagger }(Y)$, so that $U^{\dagger }(f)$ can be identified with an edge $e: \overline{Y} \rightarrow \overline{X}$ of the simplicial set $\operatorname{\mathcal{C}}$. Then $T$ carries $( f, e_{R} ): (X, \operatorname{id}_{ \overline{X} }) \rightarrow (Y, e )$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{ \overline{Y} }$.
Proof.
The implication $(2) \Rightarrow (2')$ is immediate from the definitions, and the implication $(2) \Rightarrow (2'')$ follows from Proposition 5.1.4.12. For the converse, suppose that conditions $(2')$ and $(2'')$ are satisfied. Let $f: X \rightarrow Y$ be a $U^{\dagger }$-cartesian edge of $\operatorname{\mathcal{E}}^{\dagger }$, and let us identify $U^{\dagger }(f)$ with an edge $e: \overline{Y} \rightarrow \overline{X}$ of the simplicial set $\operatorname{\mathcal{C}}$. Suppose we are given a lift of $e$ to an edge $\widetilde{e}: u \rightarrow v$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$, which we identify with a $3$-simplex $\sigma : \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$ depicted in the diagram
\[ \xymatrix@C =50pt@R=50pt{ \overline{X} \ar [d]^{u} & \overline{Y} \ar [l]_{e} \ar [d]^{v} \\ \overline{X}' \ar [r] & \overline{Y}'. } \]
We wish to show that $T( f, \widetilde{e} )$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
Let $\sigma '$ denote the degenerate $5$-simplex of $\operatorname{\mathcal{C}}$ given by $\gamma ^{\ast }(\sigma )$, where $\gamma : [5] \rightarrow [3]$ is given by $\gamma (0) = 0$, $\gamma (1) = \gamma (2) = \gamma (3) = 1$, $\gamma (4) = 2$, and $\gamma (5) = 3$. Let us abuse notation by identifying $\sigma '$ with the $2$-simplex of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ depicted in the diagram
8.75
\begin{equation} \begin{gathered}\label{equation:check-second-condition} \xymatrix@C =50pt@R=50pt{ \overline{X} \ar [d]^{\operatorname{id}} & \overline{X} \ar [l]_{\operatorname{id}} \ar [d]^{u} & \overline{Y} \ar [l]_{e} \ar [d]^{v} \\ \overline{X} \ar [r]^-{u} & \overline{X}' \ar [r] & \overline{Y}'. } \end{gathered} \end{equation}
Evaluating $T$ on the pair $(s^{1}_0(f), \sigma ')$, we obtain a $2$-simplex of $\operatorname{\mathcal{E}}$ depicted in the diagram
\[ \xymatrix@C =50pt@R=50pt{ & T(X, u ) \ar [dr]^{ T(f, \widetilde{e} ) } & \\ T(X, \operatorname{id}_ X) \ar [ur]^{ T( \operatorname{id}_{X}, u_{L})} \ar [rr] & & T( Y, v), } \]
where the left diagonal map is $U$-cocartesian by virtue of assumption $(2')$. Consequently, to show that the $T( f, \widetilde{e} )$ is $U$-cocartesian, it will suffice to show that the horizontal edge is $U$-cocartesian (Proposition 5.1.4.13). Note that, in the special case where $\sigma = s^{2}_0( \sigma _0 )$ for some $2$-simplex $\sigma _0$ of $\operatorname{\mathcal{C}}$, the horizontal edge coincides with $T( \operatorname{id}_{X}, v_ L )$, which is also $U$-cocartesian by virtue of $(2')$.
To handle the general case, we can replace $\sigma $ by the $3$-simplex of $\operatorname{\mathcal{C}}$ given by the outer rectangle of the diagram (8.75) (that is, by the $3$-simplex $\sigma '|_{ \operatorname{N}_{\bullet }( \{ 0 < 2 < 3 < 5 \} ) }$, and thereby reduce to the special case where $\sigma = s^{2}_1( \sigma _1)$, for some $2$-simplex $\sigma _1$ of $\operatorname{\mathcal{C}}$ (so that $\overline{X} = \overline{X}'$ and $u$ is a degenerate edge of $\operatorname{\mathcal{C}}$). In this case, we let $\sigma ''$ denote the $5$-simplex of $\operatorname{\mathcal{C}}$ given by $\beta ^{\ast }(\sigma _1)$, where $\beta : [5] \rightarrow [2]$ is given by $\beta (0) =\beta (1) = 0$, $\beta (2) = \beta (3) = \beta (4) = 1$, and $\beta (5) = 2$. Let us view $\sigma ''$ as a $2$-simplex of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ depicted in the diagram
\[ \xymatrix@C =50pt@R=50pt{ \overline{X} \ar [d]^{u} & \overline{Y} \ar [l]_{e} & \overline{Y} \ar [l]_{\operatorname{id}} \\ \overline{X}' \ar [r]^-{\operatorname{id}} & \overline{X}' \ar [r] & \overline{Y}'. } \]
Evaluating $T$ on the pair $(s^{1}_1(f), \sigma '')$, we obtain a $2$-simplex of $\operatorname{\mathcal{E}}$ depicted in the diagram
\[ \xymatrix@C =50pt@R=50pt{ & T(Y, e) \ar [dr] & \\ T(X, u) \ar [ur]^{ T(f,e_{R}) } \ar [rr]^{ T( f, \widetilde{e} ) } & & T( Y, v). } \]
Here the left diagonal edge is $U$-cocartesian by virtue of assumption $(2'')$ and the right diagonal edge is $U$-cocartesian by virtue of the special case treated above. Applying Proposition 5.1.4.13, we conclude that $T( f, \widetilde{e} )$ is also $U$-cocartesian.
$\square$
The “only if” direction follows from Proposition 8.6.1.13. To prove the converse, choose a $2$-simplex $\sigma $ of $\operatorname{\mathcal{E}}^{\dagger }$ corresponding to a diagram
\[ \xymatrix@R =50pt@C=50pt{ & X' \ar [dr]^{f'} & \\ X \ar [rr]^{f} \ar [ur]^{u} & & Y } \]
of the simplicial set $\operatorname{\mathcal{E}}^{\dagger }$, where $U^{\dagger }(\sigma )$ is a left-degenerate $2$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}$ and $f'$ is $U^{\dagger }$-cartesian. We then obtain a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & T(X',f) \ar [dr]^{ T( f', e_ R ) } & \\ T(X,f) \ar [ur]^{ T_{\overline{Y} }(u) } \ar [rr]^{ T( f, e_{R} ) } & & T(Y, \operatorname{id}_{Y} ) } \]
in the $\infty $-category $\operatorname{\mathcal{E}}_{ \overline{Y} }$. If both $T( f, e_ R )$ and $T( f', e_ R )$ are isomorphisms, then $T_{ \overline{Y} }(u)$ is an isomorphism as well. Since the functor $T_{ \overline{Y} }: \operatorname{\mathcal{E}}^{\dagger }_{ \overline{Y} } \rightarrow \operatorname{\mathcal{E}}_{ \overline{Y} }$ is an equivalence of $\infty $-categories, we conclude that $u$ is an isomorphism in the $\infty $-category$ \operatorname{\mathcal{E}}^{\dagger }_{ \overline{Y} }$, so that $f$ is also $U^{\dagger }$-cartesian (see Remark 5.1.3.8).