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Scott Lingran (Aug 21 2020 at 13:19):If multiplication and addition are specific instances of limits and colimits, what are subtraction and division?
Apologies for the basic question, but I'm genuinely stumped.
Thomas Read (Aug 21 2020 at 14:56):As far as I know the short answer is that it's much harder to apply the same perspective to subtraction and division, and I don't think there's anything as simple and universal as "multiplication <=> categorical product". But there certainly are some very interesting approaches you can take.
There's a nice discussion of this in @John Baez and James Dolan's From Finite Sets to Feynman Diagrams - you want to get up to around p12-16. They start by explaining how to categorify addition and multiplication by replacing natural numbers with finite sets, addition with coproducts, multiplication with products. They then give some citations to approaches for subtraction, and give a brief overview of one (though it looks pretty advanced unless you have a background in homotopy theory...). They then spend longer describing an approach to division - roughly speaking, they look at a group acting on a finite set , and suggest that " divided by " might correspond to the quotient (i.e. the set of orbits under the group action). In fact that doesn't quite work - you need to use the "weak quotient" , and you need to be a bit careful to make the analogy precise.
fosco (Aug 21 2020 at 21:33):More or less:
product : categorical product = division : internal hom
Scott Lingran (Aug 22 2020 at 04:06):@Thomas Read thanks, I think that's exactly what I'm looking for!
@fosco, hmm not sure I follow with the internal hom? As I understand it, it's more akin to exponentials with arithmetics?
fosco (Aug 22 2020 at 08:37):Scott Lee said:
Thomas Read thanks, I think that's exactly what I'm looking for!
fosco, hmm not sure I follow with the internal hom? As I understand it, it's more akin to exponentials with arithmetics?
If you consider the category of sets, yes; but sometimes adjoints to a monoidal structure are denoted ad "divisions", on the left or on the right.
fosco (Aug 22 2020 at 08:37):There's nothing precise in this claim tho :grinning:
Amar Hadzihasanovic (Aug 22 2020 at 13:51):I would say that internal hom corresponds to division when you see monoidal product as corresponding to product...
For example in a monoid seen as a monoidal category whose morphisms are all identities, left/right homs are exactly division on the left/right (when it exists).
Amar Hadzihasanovic (Aug 22 2020 at 13:53):If your starting point is "product as limit" then what Thomas suggests about quotients may be more promising.
Cole Comfort (Aug 22 2020 at 23:30):Amar Hadzihasanovic said:
I would say that internal hom corresponds to division when you see monoidal product as corresponding to product...
For example in a monoid seen as a monoidal category whose morphisms are all identities, left/right homs are exactly division on the left/right (when it exists).
The notation for the internal hom in pregroup grammars (which are rigid monoidal) is given by the symbols /, \ for exactly this reason.