# User:TakuyaMurata/S-construction

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