Examples of additive categories

Question

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.

Answer 1

The best ones, in my experience, are the ones that are also Abelian categories, so I'd make sure these make it onto your list.

For example, the category of Abelian groups and Abelian group homomorphisms, but also the category of left (resp. right) $R$-modules and left (resp. right) $R$-module homomorphisms for a fixed ring $R$. These are the most important examples in my opinion, as additive categories are usually introduced to build to Abelian categories, of which these are the most important (they are used very often in homological algebra).

Answer 2

At this point in my study of categories I am intrigued or enamored with "functor categories". I was delighted to find out that there are categories of functors (though I should have probably guessed).

Apparently, this is fairly natural, sort of like saying a group theorist is interested in the symmetric group. Indeed, according to the Yoneda lemma, every category embeds into a functor category. Shades of Cayley's theorem.

Anyway, apparently the functor category $D^C$ associated to any additive/abelian category $D$ is abelian. And thus additive. Abelian categories are additive because they have a zero object.

Answer 3

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.