Joe Moeller

Neither “Categorical product” nor “Cartesian product” are ideal descriptors imo. How are tensor products or coproducts not categorical? Cartesian is confusing when the product is not given by pairs, or when pairs gives a different monoidal structure.

I’m not sure why nobody afaict has used “limit product”.

Sep 17, 2024 at 10:37pmWeb
Show threadBenjamin RogollSep 18, 2024
@Joemoeller i think personally we should call them groupoidal categories. And in this case they should it should be called a groupoidal product.
Show threadJoe MoellerSep 18, 2024
@JewleZi groupoidal category is a great name for a 2-category with all 2-cells being invertible, a category enriched in groupoids. I’m not even sure what Cartesian product has to do with groupoids.
Show threadBenjamin RogollSep 18, 2024
@Joemoeller no i meant cartesian monoidal categories actually. Since you can define groups in them, whereas in monoidal categories you can only define monoids
Show threadJoe MoellerSep 18, 2024
@JewleZi I know, I was explaining why I didn’t like that suggestion.
Show threadEvan WashingtonSep 18, 2024
@Joemoeller consider "categorical meet" or just "meet," to indicate the generalization from preorders
Show threadJames WoodSep 18, 2024

@Joemoeller In defence of “categorical product” (even though it's not a term I really use myself), tensor products are *multi*categorical, and coproducts are not products.

I have said “limit product” in the past, when I needed to disambiguate.

Show threadTanebSep 18, 2024
@mudri @Joemoeller in dagger categories (such as at least FinDimVect is) products and coproducts coincide
Show threadJoe MoellerSep 18, 2024

@mudri coproducts do give a monoidal structure.

I’m not sure that historically that’s the distinction that “categorical” is making, but fair enough.

Show threadjuleshSep 18, 2024
@Joemoeller I was twooting about something similar to this like a month ago..... back then I realised there is no product-y version of the phrase "disjoint union”, as in “the cartesian product in Set^op is disjoint union”. Maybe we should bring back the term "ordered pairs" from set theory, as in "the coproduct in Set^op is ordered pairs”
Show threadJoe MoellerSep 18, 2024
@julesh right, but that’s not exactly the answer to my question. I mean a name for product in any category.
Show threadazulSep 18, 2024
@Joemoeller yes! i wrote about this on here some time ago, the exact same issue that "cartesian product" is not a good description for that limit in an arbitrary category. "limit product" is nice. in my thread i proposed "projection product" because it is defined by its projection morphisms.
Show threadazulSep 18, 2024
@Joemoeller oh this is all i wrote, i think ... https://gamedev.lgbt/@typeswitch/112898260454632466
lawless polymorph (@[email protected])

@[email protected] ok i don't actually hate cartesian product and CCCs, but i do have two issues: 1. cartesian closed categories are not that common in computing, or in general. 2. tuples being the main way of working with the original cartesian product ... the generalization of the cartesian product from Set to other categories, *should have been the tensor product*, not the (limit definition via projections) "product".

🏳️‍🌈 Gamedev ❤️ LGBTQIA+
Show threadZanzi @ Monoidal CafeSep 18, 2024
@typeswitch @Joemoeller I tend to use "cartesian product" just because the average programmer tends to
know just a couple of basic limits/colimits rather than limits in general. so it keeps my writing more accessible. 🙃
Show threadBoardersSep 18, 2024
@Joemoeller Cartesian product is given by pairs of generic elements, thinking it should be given by pairs of global elements is just a misintuition
Show threadJoe MoellerSep 18, 2024
@boarders it sounds like you read the opposite of what I wrote.
Show threadBoardersSep 19, 2024
@Joemoeller I re-read a couple of times, but no wiser as to what I have misinterpreted unfortunately
Show threadJoe MoellerSep 19, 2024
@boarders I may be misunderstanding your point, but I’m saying that because sometimes there are categories with a monoidal product given by pairs which is not the “limit product” as I’m suggesting to call it, the term “Cartesian product” can be conversationally confusing.
Show threadBoardersSep 19, 2024
@Joemoeller maybe I’m just confused about what “given by pairs” means, unless it is just means that it is given by pairs of elements in some underlying set or something like that?
Show threadBoardersSep 19, 2024
@Joemoeller my point was just that generic elements of the categorical product are given by pairs of generic elements and this fact means “Cartesian” is extremely appropriate (though I am not generally a fan of eponymous concepts, but neither do I see sense in top-down naming schemes since from where to do we look downwards from to take such a perspective)
Show threadJohn Carlos BaezSep 19, 2024
@boarders @Joemoeller - just in case it helps, "generic element" is what I'd call "generalized element" - i.e. a morphism going in.
Show threadEvan WashingtonSep 19, 2024
@boarders @Joemoeller Rel would be the obvious problem case here. The cartesian product of sets is monoidal, while the categorical product (limit product, meet) is a biproduct
Show threadBoardersSep 19, 2024
@eewash @Joemoeller right, that’s a good example - the inclusion of Set into Rel is not a cartesian functor. I wouldn’t say this messes up the terminology much though since I would take Cartesian not to mean product of underlying sets
Show threadJoe MoellerSep 19, 2024
@boarders @eewash right, this is precisely what I have in mind.
Show threadOmar AntolínSep 19, 2024
@Joemoeller I just call it "product". 🤷