$n$-exact categories arising from $n$-exangulated categories

Abstract: Let $\mathscr{C}$ be a Krull-Schmidt $n$-exangulated category and $\mathscr{A}$ be an $n$-extension closed subcategory of $\mathscr{C}$. Then $\mathscr{A}$ inherits the $n$-exangulated structure from the given $n$-exangulated category in a natural way. This construction gives $n$-exangulated categories which are not $n$-exact categories in the sense of Jasso nor $(n+2)$-angulated categories in the sense of Geiss-Keller-Oppermann in general. Furthermore, we also give a sufficient condition on when an $n$-exangulated category $\mathscr{A}$ is an $n$-exact category. These results generalize work by Klapproth and Zhou.