Definition 13.37.1. Let $\mathcal{D}$ be an additive category with arbitrary direct sums. A *compact object* of $\mathcal{D}$ is an object $K$ such that the map

\[ \bigoplus \nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}}(K, E_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}}(K, \bigoplus \nolimits _{i \in I} E_ i) \]

is bijective for any set $I$ and objects $E_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ parametrized by $i \in I$.

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