Higher K-theory via universal invariants, by Gonçalo Tabuada

Using the formalism of Grothendieck's derivators, we construct `the universal localizing invariant of dg categories'. By this, we mean a morphism Ul from the pointed derivator associated with the Morita homotopy theory of dg categories to a triangulated strong derivator Mloc 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. Similary, we construct `the universal additive invariant of dg categories', i.e. the universal morphism of derivators Ua to a strong triangulated derivator Madd 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.

Gonçalo Tabuada <tabuada@math.jussieu.fr>