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.

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

Details

The arrow category $Ar(C)$ 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 $[n]=\{1<2<\cdots <n\}$ as a category in the usual way. Let C be a category with cofibrations and weak equivalences and let $S_{n}C$ be a category whose objects are functors $f:Ar[n]\to C$ such that, for $i\leq j\leq k$, $f(i=i)=*$, $f(i\leq j)\to f(i\leq k)$ is a cofibration, and $f(j\leq k)$ is the pushout of $f(i\leq j)\to f(i\leq k)$ and $f(i\leq j)\to f(j=j)=*$. Then $S_{n}C$ form the simplicial category $SC$. 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

• http://arxiv.org/pdf/1001.2282v4.pdf
• http://arxiv.org/pdf/1212.3563v1.pdf

External links

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