# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Notation 5.7.5.14 (The Covariant Transport Representation). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having essentially small fibers. We let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ denote a covariant transport representation of $U$, regarded as an object of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ (which exists by virtue of Corollary 5.7.5.12). We write $[ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ]$ for the isomorphism class of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, regarded as an object of the set $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})^{\simeq } )$. By virtue of Corollary 5.7.5.13, the isomorphism class $[ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ]$ is well-defined: that is, it depends only on the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Beware that $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is not uniquely determined: in fact, any diagram isomorphic to $\operatorname{Tr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$ is also a covariant transport representation of $U$ (Remark 5.7.5.9). Nevertheless, it will be convenient to abuse terminology and refer to $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the covariant transport representation of $U$, with the caveat that it is well-defined only up to isomorphism.