Kerodon

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

Construction 5.5.3.3 (The Homotopy Transport Representation: Cartesian Case). Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets and let $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ denote the homotopy category of $\infty $-categories (Construction 4.5.1.1). It follows from Proposition 5.5.3.1 and Example None that there is a unique morphism of simplicial sets $\operatorname{hTr}_{q}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Cat}_{\infty } } )$ with the following properties:

  • For each vertex $X$ of the simplicial set $\operatorname{\mathcal{D}}$, $\operatorname{hTr}_{q}(X)$ is the $\infty $-category $\operatorname{\mathcal{C}}_{X} = \{ X\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ (regarded as an object of $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$.

  • For each edge $e: X \rightarrow Y$ in the simplicial set $\operatorname{\mathcal{D}}$, $\operatorname{hTr}_{q}(e)$ is the (isomorphism class of) the contravariant transport functor $[e^{\ast }]$ of Notation 5.5.2.5, regarded as an element of $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }} }( \operatorname{\mathcal{C}}_{Y}, \operatorname{\mathcal{C}}_{X} ) = \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}_{Y}, \operatorname{\mathcal{C}}_{X})^{\simeq } )$.

Let $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ denote the homotopy category of the simplicial set $\operatorname{\mathcal{D}}$ (Notation 1.2.5.3). Then the morphism $\operatorname{hTr}_{q}$ determines a functor of ordinary categories $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$, which we will denote also by $\operatorname{hTr}_{q}$ and will refer to as the homotopy transport representation of the cartesian fibration $q$.