By Theo Bühler
It's a common opinion between specialists that (continuous) bounded cohomology can't be interpreted as a derived functor and that triangulated equipment holiday down. the writer proves that this can be unsuitable. He makes use of the formalism of actual different types and their derived different types on the way to build a classical derived functor at the type of Banach $G$-modules with values in Waelbroeck's abelian type. this offers us an axiomatic characterization of this idea at no cost, and it's a easy subject to reconstruct the classical semi-normed cohomology areas out of Waelbroeck's type. the writer proves that the derived different types of correct bounded and of left bounded complexes of Banach $G$-modules are such as the derived classification of 2 abelian different types (one for every boundedness condition), a end result of the speculation of summary truncation and hearts of $t$-structures. furthermore, he proves that the derived different types of Banach $G$-modules will be built because the homotopy different types of version buildings at the different types of chain complexes of Banach $G$-modules, therefore proving that the speculation suits into yet one more normal framework of homological and homotopical algebra
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Extra resources for On the algebraic foundations of bounded cohomology
Let (T , Σ, Δ) be a pre-triangulated category. 11: for a morphism A − triangle f g h →B− →C− → ΣA A− and the isomorphism class of C is uniquely determined by f . It is therefore natural to ask how this isomorphism class behaves under composition of morphisms and the octahedral axiom gives an answer to precisely this question. 1. 1. It is often convenient to depict distinguished triangles diagrammatically as actual triangles C _❅ ⑦⑦ ❅❅❅ g ⑦ ❅❅ ⑦⑦ ❅❅ • ⑦⑦ ⑦ f /B A h where the bullet indicates that h is a morphism of degree 1 in that it is not an arrow from C to A but rather from C to ΣA.
31 31 33 36 . 36 . 38 . 1. 1. Recognizing Abelian Subcategories. In order to motivate the approach, we look at the “classical case” of the derived category D (A ) of an abelian category A . We consider A as a full subcategory of D (A ) by viewing an object of A as a complex concentrated in degree zero. We start with the following easy observation. 1. The subcategory A of D (A ) satisﬁes: (i) It is a full additive subcategory. (ii) Homi (X, Y ) := HomD (A ) (X, Y [i]) = 0 for all i < 0 and all X, Y ∈ A .
1. Provided that K / T and (F, α) exist, a morphism u of K is mapped to an isomorphism of K / T under F if and only if its cone is in T . Thus, let W be the class of morphisms whose cone is in T . 8]). The class W satisﬁes: (i) 1X ∈ W for all X ∈ K. (ii) The class W is closed under composition. f s −X − → Y with s ∈ W there (iii) (Left Ore Condition) Given a diagram X ← exists a commutative diagram f X s X f /Y /Y t with t ∈ W . f t − Y with t ∈ W there (iii ) (Right Ore Condition) Given a diagram X −→ Y ← exists a commutative diagram op f X s X f /Y /Y t with s ∈ W .