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Whoa, thanks! That's confusing. And actually.. since a vector space has an underlying set, it seems to me that to do a direct sum on a vector space, you must f

Whoa, thanks! That's confusing. And actually.. since a vector space has an underlying set, it seems to me that to do a direct sum on a vector space, you must first do a direct sum on the set (that is, do a disjoint union), and then do other things to map the rest of the structure of a vector space into the sum. So, somehow, a disjoint union (of the underlying set) becomes a cartesian product (of the whole vector space)? Also: the direct sum of an abelian group is also a cartesian product, right? Of any algebraic structure, not only vectors? Why are sets so special to have a different direct sum than algebraic structures built on top of sets? (Is there somewhere to read about this to gain intuition?)

bmacho 4y ago
> And actually.. since a vector space has an underlying set, it seems to me that to do a direct sum on a vector space, you must first do a direct sum on the set (that is, do a disjoint union), and then do other things to map the rest of the structure of a vector space into the sum. No. > So, somehow, a disjoint union (of the underlying set) becomes a cartesian product (of the whole vector space)? No. If anything it is the other way: the cartesian product of the underlying sets becomes the (equivalent of) "disjoint union" of the vector spaces. > (Is there somewhere to read about this to gain intuition?) I don't think you want that. But there definitely are some category theory textbooks, and introductory pdfs. E.g. Tom Leinster Basic Category Theory is a textbook, and it aims to give you intuition. https://arxiv.org/abs/1612.09375 https://arxiv.org/abs/1612.09375
knappa 4y ago
No, there is nothing all that specific about vector spaces. This is true about general algebraic structures. What you want to care about here is preserving structures: How would you define addition on a disjoint union? e.g. If you have V⊔W you can add two things in V and you can add two things in W, but what about something in V with something in W? Which zero vector is "the" zero vector? If you don't have all those properties that make a vector space, then it's just a set. Then, yes, if for some reason you want to do the sum of vector spaces V and W as sets, you will get the disjoint union of these sets. That would be a pretty odd thing to do though; in that case, why are they vector spaces?

bmacho 4y ago

> And actually.. since a vector space has an underlying set, it seems to me that to do a direct sum on a vector space, you must first do a direct sum on the set (that is, do a disjoint union), and then do other things to map the rest of the structure of a vector space into the sum. No. > So, somehow, a disjoint union (of the underlying set) becomes a cartesian product (of the whole vector space)? No. If anything it is the other way: the cartesian product of the underlying sets becomes the (equivalent of) "disjoint union" of the vector spaces. > (Is there somewhere to read about this to gain intuition?) I don't think you want that. But there definitely are some category theory textbooks, and introductory pdfs. E.g. Tom Leinster Basic Category Theory is a textbook, and it aims to give you intuition. https://arxiv.org/abs/1612.09375 https://arxiv.org/abs/1612.09375

knappa 4y ago

No, there is nothing all that specific about vector spaces. This is true about general algebraic structures. What you want to care about here is preserving structures: How would you define addition on a disjoint union? e.g. If you have V⊔W you can add two things in V and you can add two things in W, but what about something in V with something in W? Which zero vector is "the" zero vector? If you don't have all those properties that make a vector space, then it's just a set. Then, yes, if for some reason you want to do the sum of vector spaces V and W as sets, you will get the disjoint union of these sets. That would be a pretty odd thing to do though; in that case, why are they vector spaces?