Corollary 8.1.1.14 (03JS)

Corollary 8.1.1.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the projection maps $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ are cocartesian fibrations of $\infty $-categories. Moreover, a morphism $f$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is $\lambda _{-}$-cocartesian if and only if $\lambda _{+}(f)$ is an isomorphism, and $\lambda _{+}$-cocartesian if and only if $\lambda _{-}(f)$ is an isomorphism.

Proof. Let $\pi _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\pi _{+}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denote the projection maps. Then $\pi _{-}$ and $\pi _{+}$ are cocartesian fibrations of simplicial sets. Moreover, a morphism $(e_{-}, e_{+})$ of $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ is $\pi _{-}$-cocartesian if and only if $e_{+}$ is an isomorphism in $\operatorname{\mathcal{C}}$, and $\pi _{+}$-cocartesian if and only if $e_{-}$ is an isomorphism in $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (this follows immediately from Remark 5.1.4.6 and Example 5.1.1.4). Corollary 8.1.1.14 now follows by applying Proposition 8.1.1.11 to left and right sides of the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [dl]_{ \lambda _{-} } \ar [dr]^{ \lambda _{+} } \ar [d]^{ ( \lambda _{-}, \lambda _{+} ) } & \\ \operatorname{\mathcal{C}}^{\operatorname{op}} & \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [l]_-{\pi _{-}} \ar [r]^-{ \pi _{+} } & \operatorname{\mathcal{C}}, } \]

since the vertical map in the center is a left fibration (Proposition 8.1.1.11). $\square$