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I already this question here but I didn't get any satisfactory answer, so I will try in MO now.

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?

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  • $\begingroup$ The category which you describe in the second paragraph is additive (if you allow the objects to be all nonnegative integers rather than only strictly positive ones). In fact, this category is equivalent to the category of finitely generated free right $R$-modules. $\endgroup$
    – Leonid Positselski
    Commented Jun 18, 2020 at 2:17
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    $\begingroup$ What is wrong with the answers you got at MSE? You don't seem to have responded to any of the answers there, to explain to the authors of those answers what more you are looking for $\endgroup$
    – Yemon Choi
    Commented Jun 18, 2020 at 2:18

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I am not sure if you consider these creative but some typical examples of additive categories are

  • the category $\mathcal{R}$-mod, of modules over a ring or a $k$-algebra $\mathcal{R}$,
  • the category $\mathcal{Comp(\mathcal{R}-mod)}$, of chain complexes of $\mathcal{R}$-modules,
  • the category $\mathcal{Comp(\mathcal{A})}$, of chain complexes in an additive category $\mathcal{A}$,
  • the localization $\mathcal{S}^{-1}\mathcal{A}$, where $\mathcal{A}$ is an additive category and $\mathcal{S}$ is a localizing class of morphisms,
  • the homotopy category $\mathcal{K(A)}$ (with $\mathcal{A}$ an additive category). This is equivalent to the localization of $\mathcal{Comp(\mathcal{A})}$ with respect to the chain homotopy equivalences,
  • the derived category $\mathcal{D(A)}$ of $\mathcal{A}$. This is equivalent to localizing $\mathcal{Comp(\mathcal{A})}$ with respect to quasi-isomorphisms of $\mathcal{Comp(\mathcal{A})}$ (or to localizing $\mathcal{K(A)}$ wrt to quasi-isomorphisms in $\mathcal{K(A)}$),
  • the category $\mathcal{Ab}$ of abelian groups,
  • the category $\mathcal{H}$ of commutative, cocommutative hopf algebras, over an algebraically closed field of characteristic zero

Details on their structure can be found in most Category theory textbooks (and the linked answer above).

On the other hand, examples of non-additive categories are: the category of sets, the category of fields, the category of $k$-algebras, etc.

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