[nextaccountic]

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]

> 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

[knappa]

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?