Let $\mathscr{ A}$ be an extension closed proper abelian subcategory of a triangulated category $\mathscr{ T}$, with no negative 1 and 2 extensions. From this, two functors from $\Sigma\mathscr{ A}\ast\mathscr{ A}$ to $\mathscr{ A}$ 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 $\mathscr{ A}$.
Categories: math.RT