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• CommentRowNumber1.
• CommentAuthorJ-B Vienney
• CommentTimeNov 24th 2022

Created the entry

• CommentRowNumber2.
• CommentAuthorvarkor
• CommentTimeNov 24th 2022

Is there a particular motivation for this page, as opposed to a page on general enriched monoidal categories, for instance?

• CommentRowNumber3.
• CommentAuthorJ-B Vienney
• CommentTimeNov 24th 2022
• (edited Nov 24th 2022)

I find it more simple to have directly the definition of this particular case. I don’t know a lot general enriched monoidal categories. It’s a useful context to do ”linear” algebra in a very general context. It’s used on the page primitive element. I did first the page CMon-enriched symmetric monoidal categories which have the same role when you need the symmetry. It can be used to define differential categories, models of differential linear logic, graded bimonoids, symmetric powers… This context appears in logic when you need to sum proofs in the non-deterministic cut-elimination of differential linear logic. You have several ways to remove the cut, so you sum the possibilities. In simple linear logic, you don’t have this issue and don’t need the addition.

From a purely categorical point of view, you can say that it’s a natural context when you want to do category theory but necessarily need to sum things.

• CommentRowNumber4.
• CommentAuthorGuest
• CommentTimeNov 24th 2022
• CommentRowNumber5.
• CommentAuthorGuest
• CommentTimeNov 24th 2022

From the biproduct article on the nLab:

A semiadditive category is automatically enriched over the monoidal category of commutative monoids with the usual tensor product, as follows.

• CommentRowNumber6.
• CommentAuthorJ-B Vienney
• CommentTimeNov 24th 2022
• (edited Nov 24th 2022)

If they have products they are indeed automatically semiadditive. But it is convenient to have the enrichment in commutative monoids without having to speak of products. For instance again, for a model of the multiplicative-exponential part of differential linear logic, without the additives ie. product and coproduct, which are not the most interesting part of linear logic.

• CommentRowNumber7.
• CommentAuthorJ-B Vienney
• CommentTimeNov 24th 2022

Added that if the category possesses products, then it is automatically a semiadditive category.

• CommentRowNumber8.
• CommentAuthorJ-B Vienney
• CommentTimeNov 24th 2022
• (edited Nov 24th 2022)

Added also that terminal or initial object are automatically zero objects and that semiadditive monoidal categories are CMon-enriched monoidal categories.

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