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September 23, 2021

Axioms for the Category of Hilbert Spaces (bis)

Posted by Tom Leinster

Guest post by Chris Heunen

Dusk. The alley is secreted in mist. The miserly reflection of the single lamp on the cobblestones makes you nervous. The stranger approaches, hands inside trenchcoat. Why on Earth did you agree to meet here? The transfer happens, the stranger walks away without a word. The package, it’s all about the package. You have it now. You’d been promised it was the category of Hilbert spaces. But how can you be sure? You can’t just ask it. It didn’t come with a certificate of authenticity. All you can check is how morphisms compose. You leg it home and verify the Axioms for the category of Hilbert spaces!

  • Axiom 1: the category has to be equipped with a dagger.

  • Axiom 2: the category has to be equipped with a dagger symmetric monoidal structure, and the tensor unit II has to be simple and a monoidal separator. This means that II has exactly two subobjects, and that f,g:HKLf,g \colon H \otimes K \to L are the same as soon as f(hk)f \circ (h \otimes k) and g(hk)g \circ (h \otimes k) are the same for all h:IHh \colon I \to H and k:IKk \colon I \to K.

  • Axiom 3: the category has to have finite dagger biproducts.

  • Axiom 4: the category has to have finite dagger equalisers.

  • Axiom 5: any dagger monomorphism - that is, any morphism ff satisfying f f=idf^\dagger \circ f = \mathrm{id} - has to be kernel of some morphism.

  • Axiom 6: the subcategory of dagger monomorphisms has to have directed colimits.

All checks pass. It’s the genuine article: the theorem guarantees that you have in your hands the category of all Hilbert spaces and continuous linear maps between them! Well, the fine print says you have one of two versions of it. If any morphism z:IIz \colon I \to I equals z z^\dagger, then you have in your possession the category of real Hilbert spaces and continuous linear maps, otherwise you have the category of complex Hilbert spaces and continuous linear maps. Fair play, your prize may only be equivalent to one of those fabled categories, but that’s good enough for you, because the equivalence preserves all the (co)limits, dagger, and monoidal structure.

You can’t help but feel somewhat amazed. The axioms were purely categorical. They never mentioned anything like probabilities, real or complex numbers, convexity, continuity, or dimensions. How can this be? The article itself is short and sweet, so you remind yourself to read it properly, but for now you content yourself with this sketch of the proof.

First, you remember the scalars z:IIz \colon I \to I in any monoidal category form a commutative monoid under composition by the Eckmann-Hilton argument. Axiom 1 gives it an involution, and Axiom 3 an addition, making it a commutative semiring. Axioms 2 and 4 conspire to give multiplicative inverses. You knew from semiring theory that the scalars must now either form a field or be zerosumfree, meaning that w+z=0w+z=0 implies w=z=0w=z=0; but the latter contradicts Axiom 5. So the scalars form an involutive field.

Next, you try to remember what you know about projections, those endomorphisms pp satisfying p p=pp^\dagger \circ p = p. They are ordered by pqp \leq q if and only if qp=pq \circ p = p. But you prefer to work with dagger monomorphisms, which are order isomorphic to projections by playing around with mostly Axiom 5. Now, those carry an orthocomplement given by f =ker(f )f^\perp = \mathrm{ker}(f^\dagger), and Axioms 3, 4, and 6 makes it a complete lattice. So projections must be a complete lattice too, and clearly p =idpp^\perp = \mathrm{id}-p make those into a complete ortholattice.

Then you realise that hom(I,H)\mathrm{hom}(I,H) is a vector space, and the projection lattice of HH is isomorphic to the closed subspaces of hom(I,H)\mathrm{hom}(I,H). Here, a subspace Vhom(I,H)V \subseteq \mathrm{hom}(I,H) is closed when V =VV^{\perp\perp}=V, where the orthocomplement of a subspace is taken with respect to the sesquilinear form fg=g f\langle f \mid g \rangle = g^\dagger \circ f, just like you are used to in Hilbert space. You even quickly prove that hom(I,H)\mathrm{hom}(I,H) is orthomodular in the sense that it is a direct sum VV V \oplus V^\perp for any closed subspace VV.

Combining Axioms 3 and 6 make you think about looking at the object I AI^A which consists of AA many copies of the tensor unit II for any set AA. You think about this as a sort of standard object like the standard Hilbert space 2(A)\ell^2(A) of dimension |A||A|. And indeed, you find an orthonormal basis of cardinality |A||A| for hom(I,I A)\mathrm{hom}(I,I^A) with some fiddling. Now Solèr’s theorem tells you the scalars must be \mathbb{R} or \mathbb{C}, and it quickly follows that hom(I,H)\mathrm{hom}(I,H) must be a Hilbert space for any object HH. Now you’re on a roll. It’s all coming back to you: hom(I,)\mathrm{hom}(I,-) is a functor which you already know is essentially surjective, and Axioms 2 and 3 make sure it’s full and faithful. Some more symbol pushing shows that the equivalence is monoidal and preserves the dagger. Done!

You stop for a moment to appreciate all the work that led to this. All these structures - scalars, biproducts, projections, orthomodular lattices, orthomodular spaces, etc - were studied by a long line of people, including Von Neumann, Mackey, Jauch, Piron, Keller, Solèr, Rump, Bénabou, Mac Lane, Mitchell, Freyd, Abramsky, Coecke, and many more. You have to be grateful that it all comes together so nicely!

Now a philosophical mood takes you. This enterprise reminds you of Lawvere’s Elementary Theory of the Category of Sets. That also characterises a category, that of sets and functions, and the axioms are of a very similar nature. That category is clearly very important, and those axioms gave rise to the powerful logical methods of topos theory. You’re similarly reminded of the categories of modules that are so very important in algebra, and that the axioms of abelian categories give rise to the powerful method of diagram chasing through Mitchell’s embedding theorem.

The category of Hilbert spaces is also fundamental to several parts of mathematics, and you wonder if these six axioms can also lead to similarly powerful and similarly general methods. You make a mental note to look again at quantum logic in dagger kernel categories, or maybe even effectus theory. Clearly the dagger is a crucial ingredient that lets you treat much of the analysis of Hilbert space algebraically, and you should probably take dagger limits more seriously.

Your inner physicist voice pipes up. Hilbert spaces are the mathematical foundation for quantum theory, but people always wonder why. Some spend their lives trying to reconstruct them from physical first principles. Can you interpret these axioms physically? Axiom 1 seems to say something about conservation of information, Axiom 2 about compound systems. Axiom 3 might have to do with measurement or superselection. But what about the other axioms? Can you reformulate them to make physical sense? Maybe you could use symmetry arguments, or tomographic principles.

The night is young and the stars inviting. Can you do characterise the category of finite-dimensional Hilbert spaces? The category of Hilbert modules, maybe using sheaf techniques? C*-categories? You feel full of hope, and get to work.

Posted at September 23, 2021 1:47 PM UTC

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Excerpt: Chris Heunen and Andre Kornell's axiomatic characterization of the category of Hilbert spaces.
Tracked: September 23, 2021 1:57 PM

Re: Axioms for the Category of Hilbert Spaces (bis)

I loved the book and I’m enjoying the ‘movie’ now. Beautiful!

Posted by: Matteo Capucci on September 23, 2021 3:16 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

I’ve recently been investigating dagger-categories in the context of understanding Evil. Based on the “ategory” analogy from this MO answer and comments, I’ve developed the following explanation, which passes the explain-it-to-roommates test:

Suppose that we think of an arrow in a category as having an ordered pair of objects which designates its source and target. Then arrows in a dagger-category merely have an unordered pair instead.

An immediate consequence is that composition is also unordered, although the objects still must line up. This causes the normal definition of “functor” to give dagger-functors, “equivalence of categories” to give dagger-equivalence, etc. It is immediately obvious why span categories and Rel are dagger-categories, since their arrows in their corresponding traditional categories are naturally unordered. This also suggests that the relationship between Cat and DagCat is similar to the one between Grp and Ab, so that there could be a “daggerizing” 2-functor with an adjoint which takes quotients.

Objects of DagCat are dagger-categories, not categories. Applying all of this to your amazing result, I think that the daggering should be seen as a more primitive operation which suffuses the axioms of Hilb and not just as a dagger structure. This lets us restate the axioms:

  1. ‘Tis a dagger-category.
  2. The dagger-category is equipped with monoidal structure, and the tensor unit is simple and a monoidal separator.
  3. The dagger-category has binary products.
  4. The dagger-category has binary equalizers.
  5. Monic arrows are kernels.
  6. The subcategory of monic arrows has directed colimits.

While I’ve been terse and probably missed something, I hope it’s obvious how much symmetry is implied by being daggered at a primitive level.

Posted by: Corbin on September 23, 2021 8:39 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

I am very much in favour of taking dagger category theory seriously as a study in its own right. It behaves quite differently from ordinary category theory in many ways. In my book with Jamie Vicary, we call this philosophy ‘the way of the dagger’: everything in sight has to cooperate with the dagger. It’s also the title of Martti Karvonen’s thesis, that includes our papers showing how monads and limits really work differently in the presence of a dagger.

There is one sneaky thing in the axioms, though, that seems to connect the dagger and non-dagger behaviour of morphisms: the tensor unit has to be simple in the non-dagger sense. That is, it has to have exactly two subobjects. It is not enough to ask that it has exactly two dagger subobjects. (Well, I don’t know a proof in that case, but I also have no counterexample.)

Posted by: Chris Heunen on September 23, 2021 10:12 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

I’ve recently been investigating dagger-categories in the context of understanding Evil.

Anything to add to this discussion?

Posted by: David Corfield on September 24, 2021 8:23 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

After reading through Karvonen’s The Way of the Dagger in the neighboring comment, I think that we might do well to completely refactor the page on dagger-categories so that the current definition is called “classical”, and then add a new “way of the dagger” definition which quotes and cites Karvonen’s definition. We should directly tackle the philosophy of how DagCat and Cat relate. The upshot is that I think that only the classical definition has the problems with equivalence.

To quote myself from a discussion in #categorytheory on Libera Chat:

The notion of “category” is relative to Cat. Cat-sized 2-categories might have objects which don’t behave like categories; a category is only an analogy at that level, just like how the typical categorical object is not really a set.

The Way of the Dagger is that, further, 2Cat-sized 3-categories might have objects which don’t behave like 2-categories. If we consider DagCat as a 2-dagger-category, then it fundamentally is not a 2-category, but only something which has analogous structure to a 2-category.

Posted by: Corbin on September 28, 2021 1:38 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

I’m pretty sure that dagger categories aren’t evil, because the type of objects in a dagger category is defined as a 1-truncated type in homotopy type theory. If dagger categories were evil, the type of objects would be 0-truncated instead.

As groupoids are an example of a type of dagger category, where the dagger of a morphism is equal to the inverse of the morphism, if dagger categories were evil, then groupoids would have to be evil as well. I don’t think many people would accept that groupoids are evil.

Posted by: Madeleine Birchfield on September 24, 2021 7:38 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

What’s “evil” is the particular definition “a dagger category is a category with an identity-on-objects endofunctor…” since it refers to identity on objects. In particular, you can’t give a correct definition of dagger-category in HoTT by starting from that version. The better definition is the one given in the HoTT Book, and generalized to other categorical structures here, where a dagger is an additional “field” added to the dependent record type defining a category. This enables its “identity-on-objects”-ness to be recorded by type dependency, in the same way that when we write gfg\circ f in an ordinary category the codomain of ff must be exactly equal to the domain of gg.

Posted by: Mike Shulman on September 25, 2021 4:37 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

Is there a way to combine this result with the idea that Finite dimensional Hilbert spaces are complete for dagger compact closed categories (in the logical sense)? They seem tantalizingly similar, but I couldn’t figure out how to put any of it together.

Posted by: Corbin on September 23, 2021 8:46 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

That’s a good question, that I also wondered about. The immediate obstacle is that you’ll first need to adapt the axioms to characterise finite-dimensional Hilbert spaces. As in the very last paragraph of the paper, you can look at the largest compact subcategory within the category given by the axioms. But it’s unsatisfactory to characterise a small thing by really characterising a big thing and saying how the small thing sits inside the big thing. Instead I’d like to characterise the small thing directly. But for that the axiom about directed colimits has to go, because of course finite-dimensional Hilbert spaces don’t have that.

Posted by: Chris Heunen on September 23, 2021 10:16 PM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

But it’s unsatisfactory to characterise a small thing by really characteristing a big thing and saying how the small thing sits inside the big thing.

I think that whether it is satisfactory depends on your perspective! This sounds very much like the Grothendieckian approach (of replacing the study of nice objects by the study of nasty objects in nice categories) … but maybe I am misunderstanding the force of “characterise” here.

Posted by: L Spice on September 30, 2021 4:17 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

If we drop the condition that I is simple, can we characterize dagger monoidal categories of W*-modules over commutative von Neumann algebras in this manner?

Posted by: Dmitri Pavlov on September 24, 2021 12:17 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

I’m hoping the answer is yes, but will have to get back to you. With the methods of this paper you could hopefully reduce from a setting where II is not simple to a sheaf of categories where it is, and hence characterise the category of Hilbert modules over a fixed commutative C*-algebra CC. But probably the axiom that II is simple will have to be replaced by a property that the central idempotents of II are well-behaved enough to form the opens of the spectrum of CC.

Posted by: Chris Heunen on September 24, 2021 9:33 AM | Permalink | Reply to this

Re: Axioms for the Category of Hilbert Spaces (bis)

Chris and Andre, it’s so nice to see this amazing result, congratulations! What surprises me most about it is the fact that works for all Hilbert spaces, not just finite ones. That’s great and a real strength of the result.

Some people here might be interested in the history behind this. Dagger-monoidal categories as an abstract setting for quantum theory were popularized by Samson Abramsky and Bob Coecke in 2004, which I guess covers your Axioms 1 and the “dagger symmetric monoidal” of 2. Dagger-biproducts were introduced by Peter Selinger in his paper “Idempotents in dagger-categories” (2008), which gives your Axiom 3. Peter suggested to me I should think about “dagger-equalizers”, which led to my paper arXiv:0807.2927 showing dagger-equalizers and a simple tensor unit, your Axiom 4 and part of Axiom 2, guarantees that the scalars are a subsemiring of the complex numbers, and has other nice consequences. Chris’s paper arXiv:0811.1448 followed soon after, adding your Axiom 5, and proving a stronger embedding result. We all then got bored or did other things for 13 years (or at least I did.) Then you guys added this nice Axiom 6 which enables you to prove the Holy Grail result you have here.

It’s extremely satisfying to see all this resolved, and I have to do some thinking about Axiom 6.

Posted by: Jamie Vicary on October 1, 2021 12:30 PM | Permalink | Reply to this

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