Kerodon

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Definition 7.6.1.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that a collection of morphisms $\{ q_ i: Y \rightarrow Y_ i \} _{i \in I}$ in $\operatorname{\mathcal{C}}$ exhibits $Y$ as a product of the collection $\{ Y_ i \} _{i \in I}$ if the collection of homotopy classes $\{ [q_ i]: Y \rightarrow Y_ i \} _{i \in I}$ exhibits $Y$ as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched product $\{ Y_ i \} _{i \in I}$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (equipped with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment described in Construction 4.6.8.13). In other words, the collection of morphisms $\{ q_ i \} _{i \in I}$ exhibits $Y$ as a product of the collection of objects $\{ Y_ i \} _{i \in I}$ if, for every object $X \in \operatorname{\mathcal{C}}$, the induced map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \prod _{i \in I} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y_ i ) \]

is a homotopy equivalence of Kan complexes. Similarly, we say that a collection of morphisms $\{ e_ i: Y_ i \rightarrow Y \} _{i \in I}$ of $\operatorname{\mathcal{C}}$ exhibits $Y$ as a coproduct of the collection $\{ Y_ i\} _{i \in I}$ if, for every object $Z \in \operatorname{\mathcal{C}}$, the induced map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z ) \rightarrow \prod _{i \in I} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y_ i, Z) \]

is a homotopy equivalence of Kan complexes.