# User:TakuyaMurata/S-construction

In algebra, the **S-construction** is a construction in algebraic K-theory that produces a model that can be used to define higher K-groups. It is due to Friedhelm Waldhausen and concerns a category with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category (cofibrations are admissible monos and weak equivalences isomorphisms.) According to Waldhausen, "S" is for Segal as in Graeme B. Segal.^{[1]}

Unlike the Q-construction, which produces a space, the S-construction produces a simplicial set.

## Details

The arrow category of a category *C* is a category whose objects are morphisms in *C* and whose morphisms are squares in *C*. We shall view the ordered set as a category in the usual way. Let *C* be a category with cofibrations and weak equivalences and let be a category whose objects are functors such that, for , , is a cofibration, and is the pushout of and . Then form the simplicial category . In fact, with a little modification, we see

## References

- Gunnar Carlsson, Deloopings in Algebraic K-theory, 2003
- Bjørn Ian Dundas, Thomas G. Goodwillie and Randy McCarthy,
*The local structure of algebraic K-theory*, 2004

## Further reading

- For the recent ∞-category approach, see