K-theory spectrum

Remarks

Waldhausen proved that the adjoint maps $\abs{wS^{(n)} \mathcal{C}} \to \Omega \abs{wS^{(n+1)} \mathcal{C}}$ are homotopy equivalences for $n \ge 1$. Hence $K(\mathcal{C})$ is an $\Omega$-spectrum above degree $0$, its homotopy groups are $\pi_n K(\mathcal{C}) = \pi_{n+1} \abs{wS_\bullet \mathcal{C}} = K_n(\mathcal{C})$ for $n \ge 0$, and $\pi_n K(\mathcal{C}) = 0$ for $n < 0$.

Example: for a field $k$ and $\mathcal{C}$ the finite-dimensional $k$-vector spaces with isomorphisms as weak equivalences, $\pi_0 K(\mathcal{C}) = K_0(k) \cong \Z$, through the dimension.

Non-example: the map in degree $0$, $\abs{w\mathcal{C}} \to \Omega \abs{wS_\bullet \mathcal{C}}$, is in general not a homotopy equivalence. In the example above, $\pi_0 \abs{w\mathcal{C}}$ is the set of isomorphism classes of vector spaces, which is $\N$, while $\pi_0$ of the loop space is $\Z$. In this example the map is a group completion.

The spectrum $K(\mathcal{C})$ is connective. Bass's negative K-groups are captured by a nonconnective variant, built by Thomason for rings and schemes.