Duke Mathematical Journal

Higher K-theory via universal invariants

Gonçalo Tabuada

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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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Duke Math. J., Volume 145, Number 1 (2008), 121-206.

First available in Project Euclid: 17 September 2008

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Zentralblatt MATH identifier

Primary: 18G55: Homotopical algebra 18F20: Presheaves and sheaves [See also 14F05, 32C35, 32L10, 54B40, 55N30] 18E30: Derived categories, triangulated categories
Secondary: 19D35: Negative $K$-theory, NK and Nil 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]


Tabuada, Gonçalo. Higher $K$ -theory via universal invariants. Duke Math. J. 145 (2008), no. 1, 121--206. doi:10.1215/00127094-2008-049. https://projecteuclid.org/euclid.dmj/1221656865

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