K-theory

Remarks

Every element of $K_0(\mathcal{C})$ is a difference $[A] - [B]$. Two objects have the same class exactly when $A \oplus C \cong B \oplus C$ for some object $C$.

Example: let $\mathcal{C}$ be the category of finite-dimensional vector spaces over a field $F$. Two such spaces are isomorphic exactly when they have the same dimension, and $\dim(V \oplus W) = \dim V + \dim W$. So $\mathrm{Iso}(\mathcal{C}) \cong \N$ and $K_0(\mathcal{C}) \cong \Z$.

Non-example: the monoid $\mathrm{Iso}(\mathcal{C})$ alone is not the K-group. For vector spaces it is $\N$, which has no negatives; the group completion is needed to get $\Z$.

The three classical instances are these. For a ring $R$ and the category of finitely generated projective $R$-modules one gets $K_0(R)$. For a compact Hausdorff space $X$ and the category of complex vector bundles on $X$ one gets the topological K-theory group $K^0(X)$. For a unital C*-algebra and its projections one gets operator K-theory.