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

Abstract

Let (F , Σn, ) be an (n+2)-angulated Krull-Schmidt category and A ⊆ F an n-extension closed, additive and full subcategory with Hom F (ΣnA , A ) = 0. Then A naturally carries the structure (A , E A ) of an n-exact category in the sense of [Jas16, definition 4.2], arising from short (n+2)-angles in (F , Σn, ) with objects in A and there is a binatural and bilinear isomorphism YExt n (A ,E A ) (An+1, A0) ∼ = Hom F (An+1, ΣnA0) for A0, An+1 ∈ A . For n = 1 this has been shown in [Dye05] 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 in [BT13, section 4]. Additionally, we show that the axiom (F3) for pre-(n + 2)-angulated categories, stating that a commutative square can be extended to a morphism of (n + 2)-angles and defined in [GKO13, definition 2.1], implies a stronger version of itself.

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

“…It is straightforward to verify that E A is an additive subfunctor of E Σ . By [K,Lemma 3.8], we know that that s A -inflations are closed under composition. By Lemma 2.5, we have that (A , E A , s A ) is an n-exangulated category.…”

“…Theorem 3.6 [K,. Theorem 3.2] Let (C , Σ, Θ) be a Krull-Schmidt (n + 2)-angulated category and A be an n-extension closed subcategory of C .…”

Proper abelian subcategories of triangulated categories and their tilting theory

“…In particular, if Q is a finite acyclic quiver, k a field, then part (ii) of Theorem C applies to the "generic" proper abelian subcategory A = mod kQ of the negative cluster category C −n−1 (kQ), see [7, Finally, note that related work has recently appeared in [15] and [16]. The notion of proper n-exact subcategories was in effect considered in [15, thm.…”

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

“…Corollary 3.18 [K,. Theorem 3.2] Let (C , Σ, Θ) be a Krull-Schmidt (n+2)-angulated category and A be an n-extension closed subcategory ofC .…”