[knappa]

There are different sum- and product-like operations for different types of things: For sets (and topological spaces), the sum operation is the disjoint union and the product is the Cartesian product. For vector spaces, the sum operation is the Cartesian product and the product operation is the tensor product.

If you think of finite dimensional vector spaces as functions on finite sets, then these all match up. i.e. if V (vector space) is the functions on S (set) and W (vector space) is the functions on T (set), then the functions on S⊔T are V⊕W because you just list the values of the functions on S and then on T. Further, you can see that functions on S×T are V⊗W since the natural basis for V⊗W consists of vectors e_s⊗e_t that pick out elements (s,t) ∈ S×T.

[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?

[bmacho]

> For vector spaces, the sum operation is the Cartesian product and the product operation is the tensor product.

For vector spaces the sum operation is the Cartesian product, and the product operation is also the Cartesian product (at least for finite many vector spaces).

C.f. https://math.stackexchange.com/questions/1910808/prove-that-...