Joe MoellerSep 17, 2024

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”.

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 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 threadBoarders
@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
Sep 19, 2024 at 1:41amWeb
Show threadJoe MoellerSep 19, 2024
@boarders @eewash right, this is precisely what I have in mind.