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.
Unlike the Q-construction, which produces a space, the S-construction produces a simplicial set.
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
- 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
- For the recent ∞-category approach, see