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\(K\)-theory for triangulated categories. III(A): The theorem of the heart. (English) Zbl 0937.19001

This is the author’s fourth article of a series on the \(K\)-theory of triangulated categories [continuing A. Neeeman, Part I(A), Asian J. Math. 1, No. 2, 330-417 (1997; Zbl 0906.19002), Part I(B), Asian J. Math. 1, No. 3, 435-529 (1997; Zbl 0906.19003), and Part II, Asian J. Math. 2, No. 1, 1-125 (1998; Zbl 0923.19002)]. He who has introduced in the previous articles a particular type of simplicial sets here is mainly concerned with the nature and structure of the homotopies between them.
The following is quoted from his introduction: “there are two types of homotopies that I know, for the simplicial sets that come up in triangulated \(K\)-theory. The first type is the trivial homotopies. These are the triangulated analogues of contractions to an initial or terminal object. The second type of homotopy is the non-trivial homotopies. And one of the key features of this theory is that there is really only one of the non-trivial homotopies... We make this very precise, showing with explicit examples how to reduce a typical non-trivial homotopy in this theory to a blueprint”.

MSC:

19D06 \(Q\)- and plus-constructions
18E30 Derived categories, triangulated categories (MSC2010)
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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