Remark 8.1.10.4 (04AF): Contravariant Transport for Cospan Fibrations

Remark 8.1.10.4 (Contravariant Transport for Cospan Fibrations). In the situation of Theorem 8.1.10.3, let $C$ and $C'$ be objects of $\operatorname{\mathcal{C}}$, so that Proposition 8.1.7.6 supplies equivalences of $\infty $-categories

\[ \rho _{+}^{C}: \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}_ C) = V^{-1} \{ C\} \quad \quad \rho _{+}^{C'}: \operatorname{\mathcal{E}}_{C'} \hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{E}}_{C'}) = V^{-1} \{ C'\} . \]

Let $e$ be a morphism from $C$ to $C'$ in the $(\infty ,2)$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, which we identify with a pair of morphisms $C \xrightarrow {f} B \xleftarrow {f'} C'$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Choose functors $f^{\ast }: \operatorname{\mathcal{E}}_{B} \rightarrow \operatorname{\mathcal{E}}_{C}$, $f'_{!}: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{B}$ and diagrams

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}_{B} \ar [r]^-{ H } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} & \Delta ^1 \times \operatorname{\mathcal{E}}_{C'} \ar [l]_{H'} \ar [d] \\ \Delta ^1 \ar [r]^-{ f } & \operatorname{\mathcal{C}}& \Delta ^{1} \ar [l]_{f'} } \]

which exhibit $f^{\ast }$ and $f'_{!}$ as given by contravariant and covariant transport along $f$ and $f'$, respectively (see Definitions 5.2.2.4 and 5.2.2.15). Then the composition

\begin{eqnarray*} \operatorname{Tw}(\Delta ^1 \times \operatorname{\mathcal{E}}_{C'} ) & \simeq & \operatorname{Tw}(\Delta ^1) \times \operatorname{Tw}(\operatorname{\mathcal{E}}_{C'} ) \\ & \rightarrow & \operatorname{Tw}(\Delta ^1) \times \operatorname{\mathcal{E}}_{C'} \\ & \simeq & ( \operatorname{N}_{\bullet }( \{ (0,0) < (0,1) \} ) \amalg _{ \operatorname{N}_{\bullet }( \{ (0,1) \} ) } \operatorname{N}_{\bullet }( \{ (1,1) < (0,1) \} ) ) \times \operatorname{\mathcal{E}}_{C'} \\ & \xrightarrow {(H \circ (\operatorname{id}\times f'_{!}), H')} & \operatorname{\mathcal{E}}\end{eqnarray*}

can be identified with a functor $T: \Delta ^1 \times \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{E}})$ which fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}_{C'} \ar [d] \ar [r]^-{T} & \operatorname{Pith}(\operatorname{Cospan}^{\mathrm{all},W}(\operatorname{\mathcal{E}})) \ar [d]^{V} \\ \Delta ^1 \ar [r]^-{e} & \operatorname{Pith}(\operatorname{Cospan}(\operatorname{\mathcal{C}})). } \]

For each object $X \in \operatorname{\mathcal{E}}_{C'}$, the characterization of $V$-cartesian morphisms given in Theorem 8.1.10.3 shows that $T|_{ \Delta ^1 \times \{ X\} }$ is a $V$-cartesian morphism of $\operatorname{Pith}(\operatorname{Cospan}^{\mathrm{all},R}(\operatorname{\mathcal{E}}))$. It follows that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{C'} \ar [r]^-{ f'_{!}} \ar [d]^{ \rho ^{C'}_{+} } & \operatorname{\mathcal{E}}_{B} \ar [r]^-{ f^{\ast } } & \operatorname{\mathcal{E}}_{C} \ar [d]^{ \rho ^{C}_{+} } \\ V^{-1} \{ C'\} \ar [rr]^{ e^{\ast } } & & V^{-1} \{ C\} } \]

commutes up to homotopy, where the functor $e^{\ast }$ is given by contravariant transport along $e$ (for the cartesian fibration $V$).