zbMATH — the first resource for mathematics

The \(K\)-theory of a triangulated derivator. (La \(K\)-théorie d’un dérivateur triangulé.) (French) Zbl 1136.18002
Davydov, Alexei (ed.) et al., Categories in algebra, geometry and mathematical physics. Conference and workshop in honor of Ross Street’s 60th birthday, Sydney and Canberra, Australia, July 11–16/July 18–21, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3970-6/pbk). Contemporary Mathematics 431, 341-373 (2007).
Neeman introduced the \(K\)-theory of triangulated categories. In the case of the bounded derived category of an abelian category his definition coincides with the classical \(K\)-theory. But M. Schlichting showed [Invent. Math. 150, No. 1, 111–116 (2002; Zbl 1037.18007)] that one cannot define a \(K\)-theory functor for triangulated categories which satisfies both the localization axiom (exact sequences of triangulated categories yield long exact sequences), and the comparison axiom for all exact categories (Neeman’s definition works for abelian ones). It seems that one needs a richer structure to achieve this and the interesting idea here is to work with triangulated derivators. Roughly speaking, instead of looking at a triangulated category, one also includes triangulated categories of diagrams and the relationship between them given by functors between the indexing categories. The author gives a precise definition of a triangulated derivator and its \(K\)-theory. He conjectures that this notion should satisfy both the localization axiom and the comparison axiom, which he proves holds at the level of \(K_0\). In an appendix, Keller shows how to associate a triangulated derivator to an exact category: the value the derivator takes on a small category \({\mathcal A}\) is the bounded derived category of presheaves on \({\mathcal A}\).
For the entire collection see [Zbl 1116.18001].

18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
18E10 Abelian categories, Grothendieck categories
18E30 Derived categories, triangulated categories (MSC2010)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18G50 Nonabelian homological algebra (category-theoretic aspects)
18G55 Nonabelian homotopical algebra (MSC2010)
19A99 Grothendieck groups and \(K_0\)
19D06 \(Q\)- and plus-constructions
19D10 Algebraic \(K\)-theory of spaces
55P42 Stable homotopy theory, spectra