On the definition of 2-rigs

I am reading the nLab entry on 2-rigs. In its list of definitions, it says that a 2-rig category can be defined as a $Ab$-enriched category which is enriched monoidal. Why is the enrichment in $Ab$? Wouldn't it be more natural to enrich it on commutative monoids instead of abelian groups? Doesn't this contradict the fact that we are defining 2-rigs instead of 2-riNgs?

• The nLab doesn't propose a unique definition, and there are lots of variants you could consider, including the one where you only enrich in commutative monoids. Most examples that people care about are enriched in abelian groups. This doesn't not make them rigs: the analogue of addition here is direct sum of objects, not addition of morphisms. – Qiaochu Yuan Mar 17 '16 at 17:13
• @QiaochuYuan What's the point of the enrichment in abelian groups then? – zxv Mar 17 '16 at 17:55
• It's just convenient. For example, it allows you to reduce the computation of equalizers and coequalizers to kernels and cokernels, respectively. It also gets you endomorphism rings, and not just endomorphism rigs. – Qiaochu Yuan Mar 17 '16 at 17:56