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From the viewpoint of higher homological algebra, we intro-duce pure semisimple n-abelian categories, which are analogs of pure semisimple abelian categories. Let ï¿½?be an Artin al-gebra and Mbe an n-cluster tilting subcategory of Mod-ï¿½?. We show that Mis pure semisimple if and only if each mod-ule in Mis a direct sum of finitely generated modules. Let mbe an n-cluster tilting subcategory of mod-ï¿½?. We show that Add(m)is an n-cluster tilting subcategory of Mod-ï¿½?if and only if mhas an additive generator if and only if Mod(m)is locally finite. This generalizes Auslanderï¿½??s classical results on pure semisimplicity of Artin algebras
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