N-extension closed subcategories of (n+2)-angulated categories

Abstract

Let C be a Krull-Schmidt (n + 2)-angulated category and A be an n-extension closed subcategory of C . Then A has the structure of an n-exangulated category in the sense of Herschend-Liu-Nakaoka. 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. As an application, our result can lead to a recent main result of Klapproth.

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

“…Corollary 3.10. [Z,Theorem 3.4] Let (C , Σ, Θ) be a Krull-Schmidt (n+2)-angulated category and A be an n-extension closed subcategory of C . Then (A , E A , t) is an n-exangulated category.…”

“…Herschend, Liu and Nakaoka [HLN,Proposition 2.35] proved that if t-inflations are closed under composition and t-deflations are closed under composition, then (A , E A , t) is an nexangulated category. Zhou [Z,Theorem 3.4] recently also proved that this result holds in an (n + 2)-angulated category, but he showed that this hypothesis (t-inflations are closed under composition and t-deflations are closed under composition) of Herschend, Liu and Nakaoka [HLN,Proposition 2.35] is automatically satisfied in an (n + 2)-angulated category. Based on this idea, we prove the first main result in the article, which is a higher couterpart of Nakaoka-Palu's result.…”