S-construction

In mathematics, specifically in algebraic K-theory, the Waldhausen S-construction produces from a Waldhausen category C a sequence of Kan complexes ${\displaystyle S_{n}(C)}$, which forms a spectrum. Let ${\displaystyle K(C)}$ denote the loop space of the geometric realization ${\displaystyle |S_{*}(C)|}$ of ${\displaystyle S_{*}(C)}$. Then the group

${\displaystyle \pi _{n}K(C)=\pi _{n+1}|S_{*}(C)|}$

is the n-th K-group of C. Thus, it gives a way to define higher K-groups. Another approach for higher K-theory is Quillen's Q-construction.

The construction is due to Friedhelm Waldhausen.

See also

• Complete Segal space

References

• F. Waldhausen, Algebraic K-theory of spaces, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318–419, pdf.
• Lurie, J., Higher K-Theory of ∞-Categories (Lecture 16)

External links

• https://ncatlab.org/nlab/show/Waldhausen+S-construction