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Here's one answer. One way to define a group is as a category with one object and all morphisms invertible. If you want to find a way to define an abelian group as some kind of category but you don't want to say the word "commutative," there's a way: you can define an abelian group as a 2-category with one object and one morphism where all endofunctors are invertible. John Baez has a lot to say on this subject.

Edit: And now I'm going to throw out some words and hope they make sense to somebody. If groups naturally act on sets (via functors from G to Set), then abelian groups naturally act on categories (via 2-functors from G to Cat). At this point it's natural to replace "group" with "monoid" and make the following analogy: monoids are to monoid actions as commutative monoids are to symmetric monoidal categories.
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Here's one answer. One way to define a group is as a category with one object and all morphisms invertible. If you want to find a way to define an abelian group as some kind of category but you don't want to say the word "commutative," there's a way: you can define an abelian group as a 2-category with one object and one morphism where all endofunctors are invertible. John Baez has a lot to say on this subject.

Edit: And now I'm going to throw out some words and hope they make sense to somebody. If groups naturally act on sets (via functors from G to Set), then abelian groups naturally act on categories (via 2-functors from G to Cat). At this point it's natural to replace "group" with "monoid" and make the following analogy: monoids are to monoid actions as commutative monoids are to symmetric monoidal categories.
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Here's one answer. One way to define a group is as a category with one object and all morphisms invertible. If you want to find a way to define an abelian group as some kind of category but you don't want to say the word "commutative," there's a way: you can define an abelian group as a 2-category with one object and one morphism where all endofunctors are invertible. John Baez has a lot to say on this subject.