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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").

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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.

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