Waldhausen S-construction

Definition

Let $\mathcal{C}$ be a Waldhausen category with zero object $0$. For $n \ge 0$, let $\operatorname{Ar}[n]$ be the poset of pairs $(i, j)$ of integers with $0 \le i \le j \le n$, ordered by $(i, j) \le (i', j')$ when $i \le i'$ and $j \le j'$. The category $S_n \mathcal{C}$ has as objects the functors $A \colon \operatorname{Ar}[n] \to \mathcal{C}$, $(i, j) \mapsto A_{i,j}$, such that:

1. $A_{i,i} = 0$ for $0 \le i \le n$.
2. For $0 \le i \le j \le k \le n$, the map $A_{i,j} \to A_{i,k}$ is a cofibration, and the square formed by $A_{i,j} \to A_{i,k}$, $A_{i,j} \to A_{j,j} = 0$, $A_{i,k} \to A_{j,k}$ and $0 \to A_{j,k}$ is a pushout.

The morphisms of $S_n \mathcal{C}$ are the natural transformations. A morphism is a weak equivalence when each of its components is a weak equivalence of $\mathcal{C}$; these form the subcategory $wS_n \mathcal{C}$. An order-preserving map $[m] \to [n]$ induces a functor $\operatorname{Ar}[m] \to \operatorname{Ar}[n]$, and composition with it gives $S_n \mathcal{C} \to S_m \mathcal{C}$. So $[n] \mapsto wS_n \mathcal{C}$ is a simplicial object in categories, the S-construction $wS_\bullet \mathcal{C}$.

The K-theory space of $\mathcal{C}$ is $\Omega \abs{wS_\bullet \mathcal{C}}$, the loop space of the geometric realization of the nerves, and for $n \ge 0$ the K-groups are $K_n(\mathcal{C}) = \pi_{n+1} \abs{wS_\bullet \mathcal{C}}$.