Definition 9.2.0.2. Let $\operatorname{\mathcal{C}}$ be a category which admits small filtered colimits. We say that an object $C \in \operatorname{\mathcal{C}}$ is compact if the functor $D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ preserves small filtered colimits.
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