By Thorsten Holm
During the last few many years triangulated different types became more and more vital, to the level that they could now be considered as a unifying conception underlying significant elements of contemporary arithmetic. This number of survey articles, written by means of prime specialists, covers primary facets of triangulated different types, in addition to purposes in algebraic geometry, illustration conception, commutative algebra, microlocal research and algebraic topology. those self-contained articles are an invaluable advent for graduate scholars coming into the sector and a precious reference for specialists.
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Extra info for Triangulated Categories
There are still some portions of an abelian pro-Lie group G which we do not fully control: 18 Panoramic Overview – The factor group G/G1 ∼ = H / comp G is prodiscrete and has no compact subgroups but is otherwise uncharted. – comp(G) = comp(H ) is a pro-Lie group that is a directed union of compact subgroups. We do not know much more in the absence of local compactness. If we impose certain natural additional hypothesis that are traditionally invoked in topological group theory, the situation is at once much better.
The full subcategory of locally compact abelian groups in the category of abelian pro-Lie groups has a celebrated structure theory that is primarily due to the highly effective and elegant duality theory going back to L. S. Pontryagin  and E. R. van Kampen  in the early thirties of the 20th century. For any topological abelian group G we let G = Hom(G, T) denote its dual with the compact open topology. g. ) There is a natural morphism of abelian groups ηG : G → G given by ηG (g)(χ) = χ (g) which may or may not be continuous; information regarding this issue is to be found for instance in [102, pp.
If an abelian topological group is isomorphic to the additive group of a weakly complete topological vector space, that is, to RJ for some set J , then we shall call it a weakly complete vector group. The abelian pro-Lie groups we know best are the compact abelian groups and the weakly complete vector groups. So it is very pleasing that we can state the following result. 12). Any abelian almost connected pro-Lie group is isomorphic to the direct product of a weakly complete vector group and a compact abelian group.
Triangulated Categories by Thorsten Holm