Extension-closed subcategories of triangulated categories as "almost exact" categories

Question

Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we consider these "extensions" as short exact sequences in $B$, we "almost" obtain the structure of a Quillen's exact category on it. The problem is that the corresponding admissible monomorphism are only "weak kernels" in $B$ in general (dually, admissible epimorphisms are only weak cokernels). Does some part of the theory of exact categories can be applied to this setting yet; can one associate an abelian category to $B$ (endowed with this "almost exact structure") in a reasonable way? Are there any results known for "almost exact" categories in this sense?

Upd. Possibly, it is better to consider a smaller set of exact sequences in this setting (so that the corresponding kernels and cokernels will be "strong").

Answer 1

There is a beautiful paper by Iyama and Yoshino (http://arxiv.org/abs/math/0607736). The main result in Section 4 is closely related to your question. The authors consider some extension-closed subcategory of a triangulated category which behaves almost like a Frobenius exact category. They prove that the stable category is triangulated. The proof is similar to that of Happel, with the use of push-outs and pull-balcks replaced by that of the octaedron axiom. Their main motivation for this result was to show that cluster-tilting objects in Hom-finite 2-Calabi--Yau triangulated categories have a nice theory of mutation.

This analogy with Happel's result was studied by Nakaoka in http://arxiv.org/abs/1006.1033, where he defines a common generalisation of exact categories and extension-closed subcategories of triangulated categories. However, I do not know if there exist an axiomatic definition along the lines of Quillen's.