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 $\alpha \in \bigoplus _{\alpha \in S}M_{\alpha }.$