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nLab: Ring objects as monoid objects

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  1. In https://ncatlab.org/nlab/show/ring+object , it is stated that under reasonable assumptions on a given cartesian monoidal category, a ring object in it is the same as a monoid object in its category of abelian groups, with a suitable "tensor product", just as it is the case for usual rings.

    What are these "reasonable assumptions" that allow us to build such a tensor product on abelian group objects?
    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeApr 6th 2016

    I don’t know if I count it as “reasonable”, but it would be enough to have an elementary topos with NNO. I think it boils down to having free abelian groups, effective quotients, and well-behaved cartesian products (or maybe cartesian-closedness).

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeApr 7th 2016

    Yeah, something like that. On the other hand, abelian group objects in any cartesian monoidal category at all form a multicategory, and a ring object is always the same as a monoid in that multicategory.

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