TITLE: Examples of additive categories
QUESTION [5 upvotes]: There are a lot of interesting and creative examples of categories, such as for example, the category whose objects are the positive integers and the set of morphisms from $n$ to $m$ is the set of $m \times n$ matrices with values in some ring with identity $R$.
Therefore, my question is: are there some nice (creative) examples of additive categories? In this case, could you describe their structures?
I would be grateful for some references on this.

REPLY [4 votes]: Here's a surprising one (at least it was to me when I first heard about it).
The Spanier-Whitehead category, $\newcommand\SW{\mathbf{SW}}\SW$ has objects $(n,X)$, where $n\in\newcommand\Z{\mathbb{Z}}\Z$, and $X$ is a finite, pointed CW complex.
The morphisms are defined by $$\SW((n,X),(m,Y)) = \newcommand\colim{\operatorname{colim}}\colim_{k\to \infty} [\Sigma^{k+n}X,\Sigma^{k+m}Y],$$
where $[X,Y]$ denotes homotopy classes of maps, and $\Sigma$ is the suspension functor, and $k$ large enough that $k+n,k+m \ge 2$. That way the hom sets are all abelian groups,
using the fact that for any spaces $X$ and $Y$, $[\Sigma^2X,Y]$, is always an abelian group, since $\Sigma^2X = S^2\wedge X$, and $S^2$ has an abelian h-cogroup structure (the same one that we use to define the abelian group structure on $\pi_2(X)=[S^2,X]$.
The zero object is $(0,*)$, and if $(n,X)$ and $(m,Y)$ are objects, then their biproduct is $(l,\Sigma^{n-l}X\vee \Sigma^{m-l}Y)$, where $l=\min\{n,m\}$.
The motivation is that the functor 
$X\mapsto (0,X)$ from finite pointed CW complexes to the Spanier-Whitehead category should turn the suspension functor into an autoequivalence. We think of $(n,X)$ as a formal version of $\Sigma^n X$, where now $n$ can be negative.