The characteristic cohomology class of a triangulated category
  This is the final version of a series of papers uploaded in May 25, 2005. We have splitted the long last paper of the previous version in two parts to make it easier to understand. The results are essentially the same, although the presentation has changed substantially. The first three papers have not changed.   This is a collection of five papers on the foundation of triangulated categories in the context of groupoid-enriched categories, termed track categories, and characteristic cohomology classes. As a main result it is shown that given an additive category A with a translation functor t: A --> A and a class V in translation cohomology H^3(A,t) then two simple properties of V imply that (A,t) is a triangulated category. The cohomology class V yields an equivalence class (B,[s]) where B is a track category with homotopy category A and [s] is the homotopy class of a pseudofunctor s: B --> B inducing t. The two properties of V correspond to natural axioms on B and s which again imply that (A,t) is a triangulated category.   The five papers of this volume depend on each other by cross references, but each paper can be read independently of the others so that the reader is free to choose one of the papers to start. Each paper has its own abstract, introduction and literature. 
H.-J. Baues and F. Muro