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Matteo Capucci (he/him) (Nov 27 2023 at 10:21):I'm entertaining using 'ambicartesian' to refer to categories with finite biproducts, since 'bicartesian' is taken by categories with finite products & coproducts, which however might not coincide.
The prefix 'ambi' denotes something which is 'both' two things at once, so it fits like a glove. Moreover it hints at the fact that having finite biproducts means having an [[ambidextrous+adjoint]] to the diagonal functor.
Thoughts?
Damiano Mazza (Nov 27 2023 at 10:28):Aren't these usually called semiadditive categories?
Damiano Mazza (Nov 27 2023 at 10:30):Maybe you don't like very much the name "semiadditive" and are looking for an alternative? I think the terminology is pretty well-established though, I'm not sure it's a good idea to introduce a new one...
Matteo Capucci (he/him) (Nov 27 2023 at 10:31):Semiadditive categories are enriched in commutative monoids. All categories with finite biproducts are semiadditive, but I don't think the converse is true!
Matteo Capucci (he/him) (Nov 27 2023 at 10:31):In fact, that's why I'm here
Damiano Mazza (Nov 27 2023 at 11:09):I'm pretty sure that the converse is true. I don't have time to look for a reference now, I'll come back to this later! (Or someone else will correct me or beat me to it, it's more likely :big_smile:)
Damiano Mazza (Nov 27 2023 at 11:17):(By the converse being true I mean that a category enriched in commutative monoids and with finite products is the same as a category with finite biproducts. Of course enrichement alone does not automatically give products).
Nathanael Arkor (Nov 27 2023 at 12:01):Matteo Capucci (he/him) said:
Semiadditive categories are enriched in commutative monoids. All categories with finite biproducts are semiadditive, but I don't think the converse is true!
A semiadditive category is defined to be a category with finite biproducts: see [[biproduct]]. It is equivalent to asking for a CommMon-enriched category with products/coproducts/absolute (co)limits, as spelled out later on that nLab page.
Nathanael Arkor (Nov 27 2023 at 12:03):That being said, I do like the prefix "ambi-" over "bi-" for coincident structure, since "bi-" is often ambiguous.
Nathanael Arkor (Nov 27 2023 at 12:04):If you use the terminology "ambiproduct" rather than "biproduct" (so long as you mention that such things are also called biproducts), then I think "ambicartesian category" is reasonable.
Matteo Capucci (he/him) (Nov 27 2023 at 12:55):Nathanael Arkor said:
Matteo Capucci (he/him) said:
Semiadditive categories are enriched in commutative monoids. All categories with finite biproducts are semiadditive, but I don't think the converse is true!
A semiadditive category is defined to be a category with finite biproducts: see [[biproduct]]. It is equivalent to asking for a CommMon-enriched category with products/coproducts/absolute (co)limits, as spelled out later on that nLab page.
Damn, I got properly confused! Probably because I was reading a paper that defined semiadditve as cmon-enriched.
Matteo Capucci (he/him) (Nov 27 2023 at 12:56):Nathanael Arkor said:
If you use the terminology "ambiproduct" rather than "biproduct" (so long as you mention that such things are also called biproducts), then I think "ambicartesian category" is reasonable.
Funnily, the bi in biproducts makes sense because a biproduct has a product and coproduct structure which don't coincide in general
Matteo Capucci (he/him) (Nov 27 2023 at 12:57):Damiano Mazza said:
(By the converse being true I mean that a category enriched in commutative monoids and with finite products is the same as a category with finite biproducts. Of course enrichement alone does not automatically give products).
Yeah I wasn't thinking of requirjng products
Nathanael Arkor (Nov 27 2023 at 15:22):Matteo Capucci (he/him) said:
Nathanael Arkor said:
If you use the terminology "ambiproduct" rather than "biproduct" (so long as you mention that such things are also called biproducts), then I think "ambicartesian category" is reasonable.
Funnily, the bi in biproducts makes sense because a biproduct has a product and coproduct structure which don't coincide in general
What do you mean by "don't coincide"?
Matteo Capucci (he/him) (Nov 27 2023 at 16:15):They're just different: the projections and the injections are different morphisms. But now that I say it again, I don't think it makes much sense, since with this attitude I'd end up also saying a category with finite biproducts has different cartesian and cocartesian structures...
Mike Shulman (Nov 27 2023 at 16:18):The bigger problem with "bi-" is that it's sometimes used to denote things that are bicategorical, in place of "2-" for strict 2-categorical.