Kerodon

Proof of Proposition 8.6.2.10. Let $\lambda = ( \lambda _{-}, \lambda _{+} ): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ denote the twisted arrow coupling of Example 8.2.0.2, and let $V: \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ denote the composition of $\lambda _{+}$ with projection onto the second factor. Note that $V$ factors as a composition

where the first map is a left fibration (since it is a pullback of $\lambda $, which is a left fibration by virtue of Proposition 8.1.1.15), and the last map is a cocartesian fibration (since it is a pullback of the projection map $\operatorname{\mathcal{E}}^{\dagger } \rightarrow \Delta ^0$). It follows that $V$ is a cocartesian fibration, and that a morphism of $\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ is $V$-cocartesian if and only if its image in $\operatorname{\mathcal{E}}^{\dagger }$ is an isomorphism. In particular, our hypotheses on $T$ guarantees that it carries $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ to $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.

Fix an object $C \in \operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{E}}^{\dagger }(C)$ denote the fiber $V^{-1} \{ C\} = \operatorname{\mathcal{E}}' \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} $, so that projection onto the middle factor gives a map $U^{\dagger }_{C}: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} $. Note that $U^{\dagger }_{C}$ is a pullback of $U^{\dagger }$. It follows that $U^{\dagger }_{C}$ is a cartesian fibration, and that a morphism of $\operatorname{\mathcal{E}}^{\dagger }(C)$ is $U^{\dagger }_{C}$-cartesian if and only if its image in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian (Remark 5.1.4.6). Let $W_ C$ denote the collection of morphisms of $\operatorname{\mathcal{E}}^{\dagger }(C)$ which satisfy this condition, so that $W = \bigcup _{C \in \operatorname{\mathcal{C}}} W_ C$. Note that $T$ restricts to a functor $T^{C}: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{\mathcal{E}}_{C}$. By virtue of Proposition 6.3.5.2, it will suffice to verify the following (for each object $C \in \operatorname{\mathcal{C}}$):

$(\ast _ C)$

The functor $T^{C}$ exhibits the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W_{C}$.

Let $\operatorname{\mathcal{K}}$ denote the full subcategory of $\operatorname{Tw}( \operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ whose objects are isomorphisms $D \rightarrow C$. By virtue of Proposition 8.1.2.1, $\operatorname{\mathcal{K}}$ can also be described as the full subcategory of $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ spanned by its initial objects. It follows that $\operatorname{\mathcal{K}}$ is a coreflective subcategory of $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ (Example 6.2.2.5). Let $\operatorname{\mathcal{E}}^{\dagger }_0(C) \subseteq \operatorname{\mathcal{E}}^{\dagger }(C)$ denote the inverse image of $\operatorname{\mathcal{K}}$ under $U^{\dagger }_{C}$, so that $\operatorname{\mathcal{E}}^{\dagger }_0(C)$ is a coreflective subcategory of $\operatorname{\mathcal{E}}^{\dagger }(C)$ (Proposition 6.2.2.24). Using Lemma 6.2.2.16, we can choose a functor $L: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{\mathcal{E}}^{\dagger }_0(C)$ and a natural transformation $\epsilon : L \rightarrow \operatorname{id}_{ \operatorname{\mathcal{E}}^{\dagger }(C)}$ which exhibits $L$ as a $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$-coreflection functor. Our assumption on $T$ guarantees that the functor $T^{C}$ carries each element of $W_{C}$ to an isomorphism in $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$, so that $\epsilon $ induces an isomorphism of functors $(T^{C}|_{ \operatorname{\mathcal{E}}^{\dagger }_{0}(C)} \circ L) \rightarrow T^ C$. Since the Kan complex $\operatorname{\mathcal{K}}$ is contractible (Corollary 4.6.7.14), the inclusion map $\{ \operatorname{id}_{C} \} \hookrightarrow \operatorname{\mathcal{K}}$ is a homotopy equivalence of Kan complexes, and therefore induces an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}^{\dagger }(C) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} \simeq \operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ (Corollary 4.5.2.29). Since the composition

is an equivalence of $\infty $-categories, we conclude that the functor $T^{C}|_{ \operatorname{\mathcal{E}}^{\dagger }_{0}(C) }$ is also an equivalence of $\infty $-categories. To complete the proof of $(2_ C)$, it will suffice to show that the functor $L$ exhibits $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W_ C$ (see Remark 6.3.1.19). Let $W^+_{C}$ denote the collection of morphisms $v$ of $\operatorname{\mathcal{E}}^{\dagger }(C)$ such that $L(v)$ is an isomorphism in $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$. By virtue of the preceding arguments, this is equivalent to the requirement that $T(v)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$; in particular, assumption $(1)$ guarantees that $W_ C$ is contained in $W^{+}_{C}$. Conversely, if $u: Y \rightarrow Z$ is a morphism of $\operatorname{\mathcal{E}}^{\dagger }(C)$ which belongs to $W^{+}_ C$, then we can choose a commutative diagram

where $u$ and $w$ exhibit $X$ as $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$-coreflections of the objects $Y$ and $Z$, respectively, and therefore belong to $W_{C}$. We are therefore reduced to showing that the functor $L$ exhibits $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W^{+}_{C}$, which is a special case of Example 6.3.3.7. $\square$