nLab
K-theory spectrum

Idea

The K-theory spectrum $\mathrm{KU}$ (for complex K-theory) or $\mathrm{KO}$ (for real K-theory) 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.

Related concepts

• Eilenberg-MacLane spectrum

• tmf

• cohomology theory