Higher K-theory via universal invariants

By Gonçalo Tabuada. 2007

95 citations recorded by the source corpus.

Using the formalism of Grothendieck's derivators, we construct `the universal localizing invariant of dg categories'. By this, we mean a morphism U_l from the pointed derivator associated with the Morita homotopy theory of dg categories to a triangulated strong derivator M^loc such that U_l commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle and is universal for these properties. Similary, we construct the `the universal additive invariant of dg categories', i.e. the universal morphism of derivators U_a to a strong triangulated derivator M^add which satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen K-theory appears as a mapping space in the target of the universal additive invariant. This is the first conceptual characterization of Quillen-Waldhausen's K-theory since its definition in the early 70's. As an application we obtain for free the higher Chern characters from K-theory to cyclic homology.

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

View Paper: https://www.semanticscholar.org/paper/477bd25082fd4af7c9f50d35e3cfd81656897c84

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Citations in the corpus

Symmetric monoidal structure on non-commutative motives (2010, background): “In [42], the second named author proved that there exists a localizing invariant U loc dg : HO(dgcat) −→ Mot loc dg , with values in a strong triangulated derivator (see §A.”
A universal characterization of higher algebraic K-theory (2010, background): “In this vein, over the past few years the third author of this paper has carried out a program [54] of providing universal characterizations of algebraic K-theory via the formalism of derivators.”
Chow motives versus non-commutative motives (2011, background): “localizing invariant of dg categories was constructed in [35].”
On the algebraic K‐theory of higher categories (2012, background): “Even more recent work of Andrew Blumberg, David Gepner, and Gonçalo Tabuada [14] has built upon the brilliant work of the last of these authors in the context of DG categories [66] to produce another universal characterization of the algebraic K-theory of stable ∞-categories.”
A Guided Tour Through the Garden of Noncommutative Motives (2011, background): “(Quillen model) The additive and the localizing motivator admit natural Quillen models given in terms of a Bousfield localization of presheaves of (symmetric) spectra; consult [41] for details.”
Higher traces, noncommutative motives, and the categorified Chern character (2015, background): “24, and the proof consists instead in an adaptation of the arguments of [29] and [9] to the enriched setting.”