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

Abstract

Let $\mathcal{F}$ be an $(n+2)$-angulated Krull-Schmidt category and $\mathcal{A}\subset \mathcal{F}$ an $n$-extension closed, additive and full subcategory with ${\mathrm{Hom}}_{\mathcal{F}}({\Sigma }_n\mathcal{A},\mathcal{A})=0$. Then $\mathcal{A}$ naturally carries the structure of an $n$-exact category in the sense of Jasso, arising from short $(n+2)$-angles in $\mathcal{F}$ with objects in $\mathcal{A}$ and there is a binatural and bilinear isomorphism ${\mathrm{YExt}}_{(\mathcal{A},{\mathcal{E}}_{\mathcal{A}})}^n(A_{n+1},A_0)\cong {\mathrm{Hom}}_{\mathcal{F}}(A_{n+1},{\Sigma }_n{A}_0)$ for $A_0,A_{n+1}\in \mathcal{A}$. 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.

Added reference to Oppermann-Thomas-2012 and Fedele-2019 to minimal (n+2)-angles

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