compact object

Let us consider an additive category $ \mathcal{A}$ with arbitrary direct sums (also called coproducts).

Definition 0.1   An object $ X$ of $ \mathcal{A}$ is called compact if, for an arbitrary set of objects of $ \mathcal{A}$ and a morphism

$\displaystyle f : X \to \bigoplus_{\alpha \in I} M_{\alpha},$

there exists some finite set $ S \subset I$ such that $ Im ~f$ is a subobject of $ \alpha \in \bigoplus_{\alpha \in S} M_{\alpha}.$



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