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Let $\CC$ be a locally bounded $\k$-category, where $\k$ is a field. It is proved that $\CC$ is pure-semisimple, i.e., every object of $\Mod \CC$ is pure-projective, if and only if every family of morphisms between indecomposable finitely generated $\CC$-modules is noetherian. Our formalism establishes the pure-semisimplicity of Galois coverings, that is, if $\CC$ is a $G$-category with a free $G$-action on $\ind \CC$, then $\CC$ is pure-semisimple if and only if $\CC/G$ is so.
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