nLab K-theory of a stable (∞,1)-category

Definition

There is a Waldhausen S-construction $w_\bullet S_\bullet C$ for stable (infinity,1)-categories. One defines the algebraic K-theory of $C$ as

$$K(C) = \Omega | w_\bullet S_\bullet C |$$

in the usual way.

Properties

There is a universal characterization of the construction of the K-theory spectrum $K(A)$ of a stable $(\infty,1)$-category $A$:

there is an $(\infty,1)$-functor

$$U : (\infty,1)StabCat \to N$$

to a stable $(\infty,1)$-category which is universal with the property that it respects filtered colimits and exact sequences in a suitable way. Given any stable $(\infty,1)$-category $A$, its (connective or non-connective, depending on details) algebraic K-theory spectrum is the hom-object

$$K(A) \simeq Hom(U(Sp), U(A)) \,,$$

where $Sp$ denotes the stable $(\infty,1)$-category of compact spectra.

See (Blumberg-Gepner-Tabuada 10).

References

• Andrew Blumberg, David Gepner, Gonçalo Tabuada, A universal characterization of higher algebraic K-theory (arXiv:1001.2282).