Comment by gjm11 | Hacker News Reader

Comment by gjm11 | Hacker News Reader

Unfortunately, a critically important point in this article is rather misleadingly expressed. The author's first axiom is: "The vectors are v⊗w for v in V, w in W" -- but (as the author clearly realises, and does state elsewhere, but never forcefully enough) the huge majority of elements of V⊗W are not of the form v⊗w.

For this reason, I think the author's attempt to present V⊗W as a sort of modified version of V×W is ill-judged. Yes, those are both called products, but they're very different sorts of product; indeed (dropping briefly into the categorical machinery the author introduces at the end) in the category of vector spaces V×W is a coproduct rather than a product.

Again, it's clear that the author knows all this. But you wouldn't think so from the early parts of the article.

I assume the reason for all this is that the author's trying to relate (scary) tensor products to (nice comfortingly familiar) cartesian products. A laudable aim, but I fear the result will be confusion.

Double-plus nitpick: in Vect, VxW can be seen as both a product (which is usually called cartesian product) and a co-product (which is usually called direct sum). So, I guess I would not say "rather than" but "in addition to".

It's been a while since I studied category theory so maybe my brain is failing me here, but unless I'm super-confused (1) products are unique up to isomorphism and (2) the product in Vect is the tensor product.

The cartesian product is the categorical product in some other categories you can see vector spaces as belonging to -- sets, groups -- but there is only one product in the category of vector spaces and it's the tensor product.

What am I missing?

(2) is incorrect in that the tensor product is not the category theoretical product in Vect.

The product of Vect is given by forming the cartesian product of the vector spaces (with the usual vector space structure on it; and the usual projections into the components are the morphisms you need for the product).

The coproduct is the direct sum of vector spaces: In the product construction, take all those vectors where only finitely many coordinates are non-zero (but the morphisms you need for the coproduct are the embeddings of the components into the direct sum).

Of course, if you take products or coproducts of finitely many vector spaces, the two coincide.

The tensor product is neither a product nor a coproduct in Vect, as the universal property it satisfies is rather different from the product/coproduct ones.

(If you consider tensor products of algebras over a commutative base ring R, then the tensor product is the coproduct in the category of R-algebras, but this is probably not what you had in mind.)

> the tensor product is not the category theoretical product in Vect.

Oh, you're right and I'm a twit. (Not simply because I got it wrong, but because if I'd spent 30 seconds to think what the product construction does in Vect rather than relying on my plainly-unreliable memory, it would have been obvious that it isn't the tensor product: there are obviously no candidates for the required projection morphisms.)

Thanks! (And apologies to Jeremy for having made an invalid argument, though in fact I think the arguments "the product is the tensor product, the cartesian product is actually a coproduct, so they're very different" and "the product is the cartesian product, the tensor product is an entirely different kind of product, so they're very different" are about equally convincing modulo the fact that the first one's key premise turns out to be false.)

Author here. It's a good point, and I updated that snippet of text to account for it.

But seriously, you don't think that saying things like "a nasty abomination of a product space," and "they’re only related in that their underlying sets of vectors are built from pairs of vectors in V and W" is enough to drive that point home?

I do struggle in balancing the amount of overly deep/technical bits on my blog with the easier to understand mathematics. If only my audience was primarily HN! :)

To me, "a nasty abomination of a product space" reads like "a product space which is a nasty abomination". The "only related ..." bit is good, but it comes much later in the article; it's the early bits that I think are liable to confuse.

I do appreciate, by the way, that explaining nontrivial mathematics to anything other than an audience of very good mathematicians is really bloody hard!

That's a good point. Maybe the word "adulteration" or "bastardization" would be better? Or maybe to avoid odd connotations I should just say it in long form: taking a mathematical machete to a product space and making it nearly unrecognizable.

I do still think, despite what many have said, that a comparison and distinction needs to be made between usual products and tensor products. I think "modifying a product space" happens to be a different way of thinking about it (maybe not the best way), and for what it's worth it helps me keep track of the damn things.

Reply on news.ycombinator.com