Memoirs of the American Mathematical Society 2011; 97 pp; softcover Volume: 214 ISBN10: 0821853112 ISBN13: 9780821853115 List Price: US$70 Individual Members: US$42 Institutional Members: US$56 Order Code: MEMO/214/1006
 It is a widespread opinion among experts that (continuous) bounded cohomology cannot be interpreted as a derived functor and that triangulated methods break down. The author proves that this is wrong. He uses the formalism of exact categories and their derived categories in order to construct a classical derived functor on the category of Banach \(G\)modules with values in Waelbroeck's abelian category. This gives us an axiomatic characterization of this theory for free, and it is a simple matter to reconstruct the classical seminormed cohomology spaces out of Waelbroeck's category. The author proves that the derived categories of right bounded and of left bounded complexes of Banach \(G\)modules are equivalent to the derived category of two abelian categories (one for each boundedness condition), a consequence of the theory of abstract truncation and hearts of \(t\)structures. Moreover, he proves that the derived categories of Banach \(G\)modules can be constructed as the homotopy categories of model structures on the categories of chain complexes of Banach \(G\)modules, thus proving that the theory fits into yet another standard framework of homological and homotopical algebra. Table of Contents  Introduction and main results
Part 1. Triangulated categories  Triangulated categories
 The derived category of an exact category
 Abstract truncation: \(t\)structures and hearts
Part 2. Homological algebra for bounded cohomology  Categories of Banach spaces
 Derived categories of Banach \(G\)modules
Part 3. Appendices  Appendix A. Mapping cones, homotopy pushouts, mapping cylinders
 Appendix B. Pullback of exact structures
 Appendix C. Model categories
 Appendix D. Standard Borel \(G\)spaces are regular
 Appendix E. The existence of Bruhat functions
 Bibliography
