Every category containing O and P must address this question. In the usual category of math functions, if P has only those two pairs then the source object of P is exactly {4,5}, so O and P can’t be composed. In the category of relations, that is arbitrary sets of pairs between the source and target sets, O and P would compose to the empty relation between letters and countries.
I would suspect there are rules how it works that way but now it is not intuitive for me why that would be the result. Why it would not produce the empty function? And if you have a empty relation isn’t it a relation of any type to any type at the same time? Would it or why it would not be an empty relation between letter-shapes and country-dances? But apparently you can have different kinds of empty morphisms based on what their source and target objects are.
I didn’t also realise that composing is relative to how you view the objects.
Categories are what we call it when each arrow remembers its source and target. When they don’t, and you can compose anything, it’s called a monoid. The difference is the same as between static and dynamic type systems. The more powerful your system is, the less you can prove about it, so whenever we can, we express that particular arrows can’t be composed, using definitions of source and target.
You mean object.
Every category containing O and P must address this question. In the usual category of math functions, if P has only those two pairs then the source object of P is exactly {4,5}, so O and P can’t be composed. In the category of relations, that is arbitrary sets of pairs between the source and target sets, O and P would compose to the empty relation between letters and countries.
I would suspect there are rules how it works that way but now it is not intuitive for me why that would be the result. Why it would not produce the empty function? And if you have a empty relation isn’t it a relation of any type to any type at the same time? Would it or why it would not be an empty relation between letter-shapes and country-dances? But apparently you can have different kinds of empty morphisms based on what their source and target objects are.
I didn’t also realise that composing is relative to how you view the objects.
Categories are what we call it when each arrow remembers its source and target. When they don’t, and you can compose anything, it’s called a monoid. The difference is the same as between static and dynamic type systems. The more powerful your system is, the less you can prove about it, so whenever we can, we express that particular arrows can’t be composed, using definitions of source and target.