Electronic ISBN:  9781470406233 
Product Code:  MEMO/214/1006.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 214; 2011; 97 ppMSC: Primary 18; Secondary 57; 20; 46;
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

Chapters

Introduction and Main Results

1. Triangulated Categories

1. Triangulated Categories

2. The Derived Category of an Exact Category

3. Abstract Truncation: $t$Structures and Hearts

2. Homological Algebra for Bounded Cohomology

4. Categories of Banach Spaces

5. Derived Categories of Banach $G$Modules

A. Appendices

A. Mapping Cones, Homotopy PushOuts, Mapping Cylinders

B. PullBack of Exact Structures

C. Model Categories

D. Standard Borel $G$Spaces are Regular

E. The Existence of Bruhat Functions


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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.

Chapters

Introduction and Main Results

1. Triangulated Categories

1. Triangulated Categories

2. The Derived Category of an Exact Category

3. Abstract Truncation: $t$Structures and Hearts

2. Homological Algebra for Bounded Cohomology

4. Categories of Banach Spaces

5. Derived Categories of Banach $G$Modules

A. Appendices

A. Mapping Cones, Homotopy PushOuts, Mapping Cylinders

B. PullBack of Exact Structures

C. Model Categories

D. Standard Borel $G$Spaces are Regular

E. The Existence of Bruhat Functions