# nLab K-theory spectrum

### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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

$ℤ×BU,\phantom{\rule{thickmathspace}{0ex}}U,\phantom{\rule{thickmathspace}{0ex}}\cdots \phantom{\rule{thinmathspace}{0ex}}.$\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.

Revised on August 29, 2012 00:52:19 by Urs Schreiber (82.113.121.75)