$n$-Exact categories arising from $(n+2)$-angulated categories

Carlo Klapproth

Published 2021-08-10Version 1

Let $\mathscr{F}$ be an $(n+2)$-angulated Krull-Schmidt category and $\mathscr{A} \subset \mathscr{F}$ an $n$-extension closed, additive and full subcategory with $\operatorname{Hom}_{\mathscr{F}}(\Sigma_n \mathscr{A}, \mathscr{A}) = 0$. Then $\mathscr{A}$ naturally carries the structure of an $n$-exact category in the sense of Jasso, arising from short $(n+2)$-angles in $\mathscr{F}$ with objects in $\mathscr{A}$ and there is a binatural and bilinear isomorphism $\operatorname{YExt}^{n}_{(\mathscr{A},\mathscr{E}_{\mathscr{A}})}(A_{n+1},A_0) \cong \operatorname{Hom}_{\mathscr{F}}(A_{n+1}, \Sigma_n A_{0})$ for $A_0, A_{n+1} \in \mathscr{A}$. For $n = 1$ this has been shown by Dyer and we generalize this result to the case $n > 1$. On the journey to this result, we also develop a technique for harvesting information from the higher octahedral axiom (N4*) as defined by Bergh and Thaule. Additionally, we show that the axiom (F3) for pre-$(n+2)$-angulated categories, introduced by Geiss, Keller and Oppermann and stating that a commutative square can be extended to a morphism of $(n+2)$-angles, implies a stronger version of itself.