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1 October 2008 Higher K-theory via universal invariants
Gonçalo Tabuada
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Duke Math. J. 145(1): 121-206 (1 October 2008). DOI: 10.1215/00127094-2008-049


Using the formalism of Grothendieck's derivators, we construct the universal localizing invariant of differential graded (dg) categories. By this we mean a morphism Ul from the pointed derivator HO(dgcat) associated with the Morita homotopy theory of dg categories to a triangulated strong derivator Mdgloc such that Ul commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle, and is universal for these properties.

Similarly, we construct the universal additive invariant of dg categories, that is, the universal morphism of derivators Ua from HO(dgcat) to a strong triangulated derivator Mdgadd that satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen's K-theory becomes corepresentable in the target of the universal additive invariant. This is the first conceptual characterization of Quillen and Waldhausen's K-theory (see [34], [43]) since its definition in the early 1970s. As an application, we obtain for free the higher Chern characters from K-theory to cyclic homology


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Gonçalo Tabuada. "Higher K-theory via universal invariants." Duke Math. J. 145 (1) 121 - 206, 1 October 2008.


Published: 1 October 2008
First available in Project Euclid: 17 September 2008

zbMATH: 1166.18007
MathSciNet: MR2451292
Digital Object Identifier: 10.1215/00127094-2008-049

Primary: 18E30 , 18F20 , 18G55
Secondary: 19D35 , 19D55

Rights: Copyright © 2008 Duke University Press


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Vol.145 • No. 1 • 1 October 2008
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