My motivation comes from How to see that a one-object pre-additive category is a ring? and this master thesis. Basically, I want to see concrete examples of preadditive categories and what structure they induce.

The basic example

A preadditive category $\mathcal{A}$ with $obj(\mathcal{A}) = \{X\}$ induces a ring structure in the sets of morphisms $(\mathcal{A}(X,X),+,\circ)$.

More advanced example

For a finite number of objects in the category this article Category Theoretic Interpretation of Rings seems to construct a category equivalence with certain kinds of rings.

The infinite case?

However, there are still cases to discuss. What happens if I want to interpret a preadditive category with an infinite number of objects? Someone told me I should obtain a ring without 1. Can you confirm this point?

  • $\begingroup$ This question is rather unclear. In particular when you're talking about the induced structure it isn't clear on what object are you actually inducing a structure. Your first example is either wrong or contains a typo: if $\mathcal A$ and $X=\mathcal A$ it doesn't make any sense the data $\mathcal A(X,X)=\mathcal A(\mathcal A,\mathcal A)$. Could you add some details and try to be a little more specific? Otherwise it is very difficult to address your problem. $\endgroup$ – Giorgio Mossa May 1 '18 at 15:22
  • $\begingroup$ Updated comment: Your second example is false. Preadditvity requires the hom sets to be abelian groups, and the empty set is not an abelian group. This means there is at least one morphism between any two objects, and that is the zero morphism. (your comment about the union is completely correct, I was confused. It's still not a possible scenario) $\endgroup$ – Jo Be May 2 '18 at 14:03
  • $\begingroup$ It is still unclear to me what are you exactly looking for. In particular it is not clear what kind of structure should be induced on what. $\endgroup$ – Giorgio Mossa May 2 '18 at 20:53
  • $\begingroup$ I feel a bit confused about the post -- all of the motivation and exposition seems to be about how to express various features of ring theory in the language of preadditive categories, but the questions seem to be about the opposite: trying to recast the theory of preadditive categories in terms of ring theory. $\endgroup$ – Hurkyl May 18 '18 at 15:10
  • $\begingroup$ Incidentally, another example that seems to be missing from the source material is that if Z is an abelian group, then a Z-graded ring $A$ can be expressed as a preadditive category whose objects are the elements of Z and $\hom(z, z')$ is the subgroup of $A$ of elements of degree $z' - z$, with composition given by the product in $A$. This can furthermore be made into a monoidal category, with the monoidal operation given on objects by addition in $Z$ and on morphisms by the product in $A$. $\endgroup$ – Hurkyl May 18 '18 at 15:12

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