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Niel de Beaudrap (Oct 12 2021 at 15:38):I find myself working on something where I would like to describe the differences between two maps in a common Hom set of a symmetric monoidal category C, which makes sense if C is enriched over Ab. Taking a cue from Wikipedia [https://en.wikipedia.org/wiki/Abelian_category], this would seem to make C a "preadditive SMC". This terminology is direct enough as it goes, but these categories seem to have a nice enough structure (seemingly an example of a category of R-modules) that perhaps it has a pithier name or a nice characterisation. Is there standard terminology / a standard result which can be used to describe such SMCs?
John Baez (Oct 12 2021 at 15:40):I'd call it a symmetric monoidal Ab-category, which while lengthy has the advantage that anyone who knows enriched category theory will instantly understand it.