Example 11.9.1.7 (04EH): Path Fibrations

Example 11.9.1.7 (Path Fibrations). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let

\[ \operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}} \]

denote the functors given by evaluation at the vertices $0,1 \in \Delta ^1$. Then $\operatorname{ev}_0$ is a cartesian fibration, and $\operatorname{ev}_1$ is a cocartesian fibration (Example 5.3.7.5). Let $L$ denote the collection of $\operatorname{ev}_1$-cartesian morphisms of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (that is, the collection of morphisms $f$ such that $\operatorname{ev}_0(f)$ is an isomorphism of $\operatorname{\mathcal{C}}$), and let $R$ denote the collection of $\operatorname{ev}_0$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ (that is, the collection of morphisms $f$ such that $\operatorname{ev}_1(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$). Applying the construction of Notation 8.6.3.1 to the cocartesian fibration $\operatorname{ev}_1$, we obtain an $\infty $-category $\operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}})$. The morphism $\Xi $ of Construction 11.9.1.3 determines a morphism $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}})$ which fits into a commutative diagram

\[ \xymatrix@R =50pt@C=40pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{\Xi } \ar [d] & \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})/\operatorname{\mathcal{C}}) \ar [d] \ar [r] & \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \ar [d]^{\operatorname{ev}_0, \operatorname{ev}_1} \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r] & \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}}) \times \operatorname{\mathcal{C}}\ar [r] & \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}) \times \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}). } \]

Here the right half of the diagram is a pullback square, the vertical maps are left fibrations (Proposition 8.1.1.11 and Lemma 11.9.1.2), the lower horizontal maps are equivalences of $\infty $-categories (Proposition 8.1.7.6). Applying Proposition 11.9.1.5 (and Corollary 4.5.2.29), we deduce that the $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})/\operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories.