nLab
K-theory spectrum

The K-theory spectrum in the strict sense is the spectrum that represents the generalized (Eilenberg-Steenrod) cohomology theory topological K-theory. For complex topological K-theory this is periodic with period 2 (reflect Bott periodicity) of the form

$$\mathbb{Z}\times BU,U,\cdots .$$\mathbb{Z} \times B U ,\; U ,\; \cdots \,.

More generally, to every stable (infinity,1)-category $C$ is associated a K-theory space which in good cases, such as when the category is presented by a Waldhausen category is the degree 0 piece of a corresponding algebraic K-theory spectrum. The detailed construction is known as the Waldhausen S-construction.

Domenico Fiorenza: Is it possible to say that the K-theory spectrum is the Waldhausen construction on the stabilization of ${\mathrm{Vect}}_{\mathrm{fin}.\dim .}$ (seen as an $(\mathrm{\infty }\mathrm{,1})$-category)? Since $\mathbb{N}$ is the decategorification of ${\mathrm{Vect}}_{\mathrm{fin}.\dim .}$ by taking dimension, one can expect a map of spectra induced by $\dim$, and this could be cojecturally be the Chern character at a spectra level (the fact that the leading therm of $\mathrm{Ch}$ is $\dim$ seems to support this point of view). I’ll try to exand this on my personal area.