Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Definition 5.5.2.9. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, let $e: X \rightarrow Y$ be an edge of the simplicial set $\operatorname{\mathcal{D}}$, and let $F: \operatorname{\mathcal{C}}_{X} \rightarrow \operatorname{\mathcal{C}}_{Y}$ be a functor. We say that $F$ is given by covariant transport along $e$ if there exists a morphism of simplicial sets $h: \Delta ^1 \times \operatorname{\mathcal{C}}_{X} \rightarrow \operatorname{\mathcal{C}}$ which satisfies the following conditions:

$(1)$

The diagram of simplicial sets

\[ \xymatrix { \Delta ^1 \times \operatorname{\mathcal{C}}_{X} \ar [r]^-{ h } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^1 \ar [r]^{e} & \operatorname{\mathcal{D}}} \]

commutes.

$(2)$

The restriction $h|_{ \{ 0\} \times \operatorname{\mathcal{C}}_{X} }$ is equal to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}_{X}}$, and the restriction $h|_{ \{ 1\} \times \operatorname{\mathcal{C}}_{X} \} }$ is equal to $F$.

$(3)$

For each object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}_{X}$, the restriction $h|_{ \Delta ^1 \times \{ C\} }$ is a locally $q$-cocartesian edge of $\operatorname{\mathcal{C}}$.

If these conditions are satisfied, we say that the morphism $h$ witnesses $F$ as given by covariant transport along $e$.