Kerodon

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Example 5.2.8.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category, which we regard as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes. Applying Proposition 5.2.8.7, we obtain the following:

• For every object $C \in \operatorname{\mathcal{C}}$, the corepresentable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

$\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$

is the enriched homotopy transport representation for the left fibration $\{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.

• For every object $D \in \operatorname{\mathcal{C}}$, the representable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

$\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$

is the enriched homotopy transport representation for the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ D\} \rightarrow \operatorname{\mathcal{C}}$.