Notation 7.6.1.11 (03EK)

Notation 7.6.1.11. In practice, we will use Definition 7.6.1.3 most often in the case where the set $I$ has exactly two elements, so that the collection $\{ Y_ i \} _{i \in I}$ can be identified with an ordered pair $(Y_0, Y_1)$ of objects of $\operatorname{\mathcal{C}}$. In this case, we say that morphisms $q_0: Y \rightarrow Y_0$ and $q_1: Y \rightarrow Y_1$ exhibit $Y$ as a product of $Y_0$ with $Y_1$ if they satisfy the requirement of Definition 7.6.1.3: that is, for every object $X \in \operatorname{\mathcal{C}}$, the induced map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y_0 ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y_1) \]

is a homotopy equivalence. If this condition is satisfied, then we will often denote the object $Y$ by $Y_0 \times Y_1$ and refer to it as the product of $Y_0$ with $Y_1$. Similarly, we say that a pair of morphisms $e_0: Y_0 \rightarrow Y$ and $e_1: Y_1 \rightarrow Y$ exhibit $Y$ as a coproduct of $Y_0$ with $Y_1$ if, for every object $Z \in \operatorname{\mathcal{C}}$, the induced map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y_0 , Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y_1, Z) \]

is a homotopy equivalence; in this case, we denote $Y$ by $Y_0 \coprod Y_1$ and refer to it as the coproduct of $Y_0$ with $Y_1$.