A remark on K-theory and S-categories

By Bertrand Toën, G. Vezzosi. 2002

68 citations recorded by the source corpus.

Abstract It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). 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? ”

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ad711d33f87e1e511c59141331c71085d23f3cc2

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Citations in the corpus

Localization theorems in topological Hochschild homology and topological cyclic homology (2008, unknown): “We take the viewpoint, first articulated by Dwyer and Kan, that enriched mapping spaces (or spectra) encode the “higher homotopy theory” of a category, and we view THH , TR, and TC as invariants of the higher homotopy theory of the category of complexes, as is K-theory [35, 2].”
On the algebraic K‐theory of higher categories (2012, methodology): “More recently, Bertrand Toën and Gabriele Vezzosi showed [70] that the Waldhausen Ktheory of many of the best-known examples of Waldhausen categories is in fact an invariant of the simplicial localization; thus Toën and Vezzosi are more explicit in identifying higher categories as a natural domain f”
A universal characterization of higher algebraic K-theory (2010, background): “It is now well understood that the K-theory spectrum is determined by the DwyerKan localization LC of the Waldhausen category C [60, 7, 10].”