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Invariance of the \(K\)-theory for derived equivalences. (Invariance de la \(K\)-théorie par équivalences dérivées.) (French) Zbl 1230.19002

A derivable Waldhausen category is a small category of cofibrant object which admits a zero object. A “good” Waldhausen category \({\mathcal C}\), in the sense of B. Toen and G. Vezzosi [Topology 43, No. 4, 765–791 (2004; Zbl 1054.55004)], is a Waldhausen subcategory of a category of models. B. Toen and G. Vezzosi have shown that the \(K\)-theory of such a category \({\mathcal C}\) coincides with the \(K\)-theory of its simplicial localization.
In this paper it is proved that any right exact functor \(F\) between derivable Waldhausen categories, which are strongly saturated, such that \(F\) induces an equivalence after simplicial localization, gives rise to a homotopy equivalence between the corresponding \(K\)-theory spectra. It is also proved that a homotopy right exact functor induces an equivalence of homotopy categories it and only if it induces an equivalence of simplicial localizations.

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
18G55 Nonabelian homotopical algebra (MSC2010)

Citations:

Zbl 1054.55004
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References:

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