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A remark on K-theory and S-categories

Bertrand Toen·Gabriele Vezzosi
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Abstract

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (see [Schlichting]). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: ``To which extent $K$-theory is not an invariant of triangulated derived categories ?''

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Bertrand Toen, Gabriele Vezzosi. 2003-10-02. A remark on K-theory and S-categories. https://arxiv.org/abs/math/0210125

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