Non connective K-theory via universal invariants

In this article, we further the study of higher K-theory of dg categories via universal invariants, initiated by the second named author. Our main result is the co-representability of non-connective K-theory by the base ring in the universal localizing motivator. As an application, we obtain for free higher Chern characters, resp. higher trace maps, e.g. from non-connective K-theory to cyclic homology, resp. to topological Hochschild homology.

💡 Research Summary

This paper develops a motivic framework for the higher K‑theory of differential graded (dg) categories, focusing on universal invariants introduced in earlier work by the second author. The authors distinguish two fundamental types of invariants: additive invariants, which preserve filtered homotopy colimits, the terminal object, and split exact sequences (sending them to direct sums); and localizing invariants, which satisfy the same conditions but replace split exactness with the stronger requirement that all exact sequences become distinguished triangles. Both notions are formalized using Grothendieck derivators, allowing the authors to avoid model‑category technicalities and to work directly at the level of homotopy theories.

The paper first constructs the universal additive invariant (U_{\mathrm{add}}^{\mathrm{dg}}\colon \mathrm{HO}(\mathrm{dgcat})\to \mathrm{Mot}{\mathrm{add}}^{\mathrm{dg}}) and proves that for any dg category (\mathcal A) the spectrum of morphisms (\mathbb R!\operatorname{Hom}(U{\mathrm{add}}^{\mathrm{dg}}(k),U_{\mathrm{add}}^{\mathrm{dg}}(\mathcal A))) is naturally equivalent to Waldhausen’s K‑theory spectrum (K(\mathcal A)). This recovers the known co‑representability of connective K‑theory in the additive motivator.

To treat non‑connective K‑theory, the authors introduce a hierarchy of “α‑additive” invariants (for a regular cardinal (\alpha)) and then localize them with respect to strict exact sequences, obtaining a universal α‑localizing invariant (U_{\mathrm{loc}}^{\alpha}). By a careful analysis of Schlichting’s construction of non‑connective K‑theory for Frobenius pairs, they adapt it to dg categories, defining a functor (V_{\ell}) which is shown to be α‑localizing. For sufficiently large (\alpha), they prove \