PlanetPhysics/Compact Object

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

An object $$X$$ of $$\mathcal{A}$$ is called compact if, for an arbitrary set of objects of $$\mathcal{A}$$ and a morphism $$f : X \to \bigoplus_{\alpha \in I} M_{\alpha},$$ there exists some finite set $$S \subset I$$ such that $$Im ~f$$ is a subobject of $$\alpha \in \bigoplus_{\alpha \in S} M_{\alpha}.$$