nLab Waldhausen K-theory of a dg-category

Idea

The Waldhausen K-theory of a dg-category is defined as the Waldhausen K-theory of a Waldhausen category associated to it. It is an additive invariant in the sense of noncommutative motives. There is a nonconnective variant studied by Marco Schlichting, which is a localizing invariant (again in the sense of noncommutative motives).

On the other hand, one could define the K-theory of a pretriangulated dg-category as the algebraic K-theory of its dg-nerve, which is a stable infinity-category. This should coincide with the Waldhausen K-theory (presumably).

Definition

Let $A$ be a dg-category. Consider the triangulated category $D(A)$ of dg-presheaves on $A$, and let $perf(A) \subset D(A)$ denote its full subcategory of compact objects. There is the structure of a Waldhausen category on $perf(A)$ where the weak equivalences are objectwise quasi-isomorphisms and the cofibrations are degreewise split monomorphisms. The Waldhausen K-theory of $A$ is the Waldhausen K-theory of $perf(A)$ with this Waldhausen structure.

References

• Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Zürich, 2006, pdf.

• Marco Schlichting, Negative K-theory of derived categories, Math. Z. 253 (2006), no. 1, 97 - 134, pdf.

• Goncalo Tabuada, Higher K-theory via universal invariants, Duke Mathematical Journal, 145 (2008), no. 1, 121-206, arXiv.