K-theory of a stable (∞,1)-category


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

K(C)=Ω|w S C| K(C) = \Omega | w_\bullet S_\bullet C |

in the usual way.


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

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

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

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

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

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

See (Blumberg-Gepner-Tabuada 10).