Duke Mathematical Journal
- Duke Math. J.
- Volume 145, Number 1 (2008), 121-206.
Higher -theory via universal invariants
Using the formalism of Grothendieck's derivators, we construct the universal localizing invariant of differential graded (dg) categories. By this we mean a morphism from the pointed derivator associated with the Morita homotopy theory of dg categories to a triangulated strong derivator such that 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 from to a strong triangulated derivator that satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen's -theory becomes corepresentable in the target of the universal additive invariant. This is the first conceptual characterization of Quillen and Waldhausen's -theory (see , ) since its definition in the early 1970s. As an application, we obtain for free the higher Chern characters from -theory to cyclic homology
Duke Math. J., Volume 145, Number 1 (2008), 121-206.
First available in Project Euclid: 17 September 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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