Intermediate categories for proper abelian subcategories

Abstract

Let $A$ be an extension-closed proper abelian subcategory of a triangulated category $T$, with no negative 1 and 2 extensions. From this, two functors from $\Sigma A⁎A$ to $A$ can be constructed giving a snake lemma mirroring that of homology without needing a t-structure.

We generalize the concept of intermediate categories, which originates from a paper by Enomoto and Saito, to the setting of proper abelian subcategories and show that under certain assumptions this collection is in bijection with torsion-free classes in $A$.