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just a minimum for the moment, in order to record the definition in:
Thanks for the additions.
One remark: What you seem to call the “Cartesian product” of quantum sets is known as the external tensor product (as noted in the original material in the entry which has become section 3.) It’s not Cartesian – and that is a key property which makes quantum sets be “quantum”! Therefore I’d suggest to change the terminology.
Probably what you want to say instead is that the quantum analog of the Cartesian product of ordinary sets is the external tensor product of quantum sets.
I don’t think this is an issue of the nPOV (nor of the nature of the nLab as you suggest by email), but just of standard category theory (cf. also Wikipedia): By a “Cartesian product” one understands a monoidal product that explicitly is the categorical product. Whence one speaks of “Cartesian monoidal categories” in contrast to general monoidal categories.
I understand that you want to highlight the analogy to ordinary sets, which is clearly a good cause. To achieve this one could speak of the “product of quantum sets”, which would both invoke the analogy to the product of sets while retainign the flexibility to speak of more general monoidal products.
But I also understand now that the contested terminology is the one used in your published articles, which probably means that it’s too late to change it.
Let’s just be sure to have a remark right after Def. 2.2 in the entry, to clarify the situation. As it stands now without accompanying commentary, it looks superficially like a technical mistake, and an unfortunate one.
“Cartesian product” one understands a monoidal product that explicitly is the categorical product
I disagree with this. There are categories where the objects are sets, and the cartesian product is a monoidal structure, but it is not the categorical product. Not least the groupoid of finite sets and bijections! But also other ones I have used as examples, namely the category of sets (or replete subcategories thereof) where all functions are “small” in that they have (regular cardinal) bounds on the sizes of their fibres, or surjective functions with bounds on the size of the fibres (with the bound “<2” being the case of the core of the category of sets in the latter example), and then analogous things like focusing just on finite sets, sets in a weak universe and so on.
If we are going to stick with things that are justified by being in print as much as possible, Andre’s remark about the usage in the literature should also carry a lot of weight, no?
David R., you haven’t even read to the end the message which you quote from (#9).
I asked Andre to at least add a clarifying comment, and meanwhile he did something in this direction.
Just for the record, I still think it’s unfortunate:
The hallmark of quantum information is that its tensor is non-Cartesian.
Now you rename the tensor product of Hilbert spaces to “Cartesian product”!
It’s like renaming “quantum physics” to “classical physics”. Concretely:
It’s technically wrong. (As opposed to your supposed counter examples, the tensor product of Hilbert spaces is not Cartesian with respect to any notion of morphisms between them.)
It’s substantially misleading, bound to confuse newcomers.
There is no need for it. The only purpose is to alert people that the tensor of Hilbert spaces is the quantum analog of the Cartesian product of sets — which is a fine purpose in itself but is much better served by just saying so explicitly in prose.
A good article that does so is Baez’s “Quantum Quandaries” [arXiv:quant-ph/0404040]. Also Abramsky’s No-Cloning in categorical quantum mechanics [arXiv:0910.2401].
Respectfully, I disagree with the quoted statement as a standalone statement about mathematical naming and practice, regardless of the rest of the message conceding what is going to happen on the nLab page. “One” here is too general, because I don’t understand this, and apparently neither do others I’ve seen who have the same usage as me.
Granting the claim in the statement, though, what should one call the cartesian product of finite sets in the groupoid of finite sets and bijection, since it is not a categorical product? I’m interested to hear suggestions.
David R., I think you are trolling, but I’ll reply for young bystanders.
There is a default category of sets, and in that the Cartesian product of sets is the Cartesian tensor product. This is the very origin of the terminology “Cartesian” in category theory.
By common usage, speaking of the “X of sets” is to tacitly refer to the default category of sets (otherwise it would have to be specified otherwise), and hence “the Cartesian product of sets” is quite unambiguous, no matter what else the context.
I am not trolling, I am serious. We still call the product of sets in Rel, and in Core(Set), the cartesian product, and that is not a categorical product. Ergo, I do not “understand ” that “cartesian product” should be synonymous with categorical product in those categories.
It’s not worth getting too worked up about, I just wanted to point out I am not included among your “one” who understands cartesian=categorical.
In support of David’s point, there’s also the general term cartesian bicategory where the tensor product is not cartesian in the categorical or universal property sense.
In any case, let’s please not escalate by personal accusations of trolling.
It seems to me that the article has it right: the nomenclature “cartesian” seems well-established and is consonant with similar such usages elsewhere, but a warning about the nomenclature was appropriately put in. Maybe we can leave it at that.
The tensor products of cartesian bicategories have diagonals and projections, which is the key property of cartesian over non-cartesian tensor products when it comes to quantum theory, their absence being the no-cloning/no-deleting hallmarks of quantum states, or equivalently the sub-structural property of the corresponding quantum logic.
So this example does not either support the case of calling the tensor product of Hilbert spaces a Cartesian product. It just isn’t, or else the term would be meaningless.
What the tensor product of Hilbert spaces is — remarkably — is the quantum analog (and semantically the multiplicative linear logic analog) of the Cartesian product of sets, it’s quantization really, with the basic quantization map
ℂ[−]:(FinSet,×)⟶(Hilb,⊗)being a strong monoidal functor from the Cartesian to the non-cartesian product. That’s a deep story, told in parts by the articles by Baez and Abramsky referenced above, and elsewhere.
Conflating this means to conflate quantum with classical, which defeats the whole point, it’s like renaming quantum physics to classical physics with the argument that this will help newcomers get acquainted. No, it will just mislead them.
Eventually the entry can say this clearly, I am sure.
The tensor products of cartesian bicategories have diagonals and projections
(As Urs knows) what is missing there is the naturality of diagonals and projections; instead there is a lax naturality. This is enough to rule out “cartesian” according to the standard set out as to what the “one” in comment #9 understands.
I do agree that “cartesian monoidal” has a very standard meaning, and this warrants inclusion of a remark.
I would not be unhappy with “quantum-cartesian product”, as a compromise. The fact that there are (according to the article) different equivalent categories modelling (these) quantum sets where families of hilbert spaces are just one model, then it would be nice to have a single abstract term for the equivalent monoidal product among all of them. Saying something like ’external tensor’ for the abstract, model-independent notion is not meaningful to me, when someone might be working with the C*-algebra model, for example.
Furthermore, its monoidal product X×Y satisfies the uniqueness condition in the definition of the categorical product.
do you mean something like: given two maps f,g:Z→X×Y such that pri∘f=pri∘g for i=1,2, then f=g?
I removed the “only” in “while qRel fails to be an allegory only because the relevant modular law fails”. This “only” could make it seem that qRel just barely misses being an allegory, whereas in fact the Freyd modular law is a crucial hypothesis that gives the theory of allegories its special character.
Similarly, although I didn’t edit this, the “only” in “Overall, qSet is unlike an elementary topos in only two respects” could give a misleading impression, that qSet barely misses being a topos. One of those respects is lack of a subobject classifier, which is huge.
Well, an infinitary Π-pretopos is very nearly a Grothendieck topos, missing “only” a subobject classifier, and satisfying nearly all of the conditions of Giraud’s theorem, and it really is actually remarkably like a topos. But I get the point that one could instead say something like “imagine this category with finite limits … it is only unlike an elementary topos in that is lacks power objects”. And then you really don’t have something close to being a topos.
I’ve removed the second occurrence of “only.”
I’ve reworded the last paragraph of section 2, somewhat reverting the edit. Hopefully, the meaning is clear. I understood the phrase “carries the structure” to refer to the other structure of a monoidal product, but I feel that this is too technical a point for this part of the article. The reader is just getting an initial sense for the roles of these basic operations. Similarly, it would be more correct to say that X+Y carries the structure of a coproduct, since a coproduct is more than a binary operation on objects, but I think that saying so here would not help the reader.
I’d like to replace “bicategory” in Remark 2.4 with “category.” The comparison between Rel and qRel is the main anchor point for developing intuition about quantum sets. I don’t want the casual reader to get the impression that they need to understand bicategories. Of course, both Rel and qRel are bicategories.
Mainly I didn’t like “the” monoidal product, so I’ve now put in “designated”; “chosen” would also be acceptable.
It’s better (but not essential) to speak of the bicategory (or locally ordered category) of sets and relations. But seeing that this is considered bothersome, I’ve adjusted the wording to circumvent the dreaded “bi”.
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