Recall that a [preadditive category](https://en.wikipedia.org/wiki/Preadditive_category) is just a category $\mathcal{C}$ enriched in the category of abelian groups such that composition is linear with respect to the various group operations, so $$f\circ(g+h)=f\circ g+f\circ h,$$ $$(g+h)\circ f=g\circ f+h\circ f,$$ and a preadditive category with one object is a categorification of the notion of a noncommutative ring where we interpret the ring addition as the addition inherited from enrichment and multiplication as composition of morphisms.



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> Is a preadditive groupoid with one object a categorification of the notion of a skew field, and if so where can I read more about it?
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Naively it seems like we now have additive inverses and multiplicative inverses for each object; we only seem to be missing multiplicative commutativity to categorify fields. This leads to a followup question,



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> Is there a notion of a category where composition of endoarrows is always commutative? That is, a category $\mathcal{C}$ such that for all endoarrows $f,g:X\rightrightarrows X$ we have $f\circ g=g\circ f$?
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This seems like a bad definition since it involves equality, so perhaps a better request is a category $\mathcal{C}$ such that composition of endoarrows is commutative up to isomorphism in a (co?)slice category, with the above notion recovered as the 'strict' version.



One object preadditive groupoids as a categorification of skew fields
There aren't any nontrivial preadditive groupoids; a preadditive category always has zero morphisms, and if zero morphisms are invertible then every object is a zero object.


If you think of rings as one-object preadditive categories, then commutative rings can be thought of as one-object preadditive monoidal categories, by the Eckmann-Hilton argument. This is a more natural way to enforce commutativity since it's related to thinking of monoidal categories themselves as one-object $2$-categories and delooping etc.


