Remark 8.3.7.5 (06C5)

Remark 8.3.7.5. The composite map

\[ \operatorname{\mathcal{E}}_{+} \hookrightarrow \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) = \operatorname{Fun}(\Delta ^1, \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \]

fits into a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{+} \ar [r] \ar [d]^{V_{+}} & \operatorname{Fun}( \Delta ^1, \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \ar [d] \\ \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \times _{ \operatorname{\mathcal{C}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) } \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}), } \]

where the right vertical map is induced by the projection functor $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ of Notation 8.1.1.6. The functor $\lambda _{+}$ is a cocartesian fibration (Corollary 8.1.1.14) and therefore an isofibration (Proposition 5.1.4.9). Applying Proposition 4.4.5.1 (and Remark 4.5.6.11), we deduce that $V_{+}$ is also an isofibration. Note that the functor $\pi _{+}$ factors as a composition

\[ \operatorname{\mathcal{E}}_{+} \xrightarrow {V_{+}} \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \times _{\operatorname{\mathcal{C}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}_{+} \times _{\operatorname{\mathcal{C}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}), \]

where the second map is a cocartesian fibration (Corollary 8.1.1.14) and the third map is a pullback of $F_{+}$. It follows that, if the functor $F_{+}$ is an isofibration, then the functor $\pi _{+}: \operatorname{\mathcal{E}}_{+} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$ is also an isofibration. Similarly, if $F_{-}$ is an isofibration, then $\pi _{-}: \operatorname{\mathcal{E}}_{-} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$ is an isofibration.