The n-Category Café

• 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 $I$ has to be simple and a monoidal separator. This means that $I$ has exactly two subobjects, and that $f,g \colon H \otimes K \to L$ are the same as soon as $f \circ (h \otimes k)$ and $g \circ (h \otimes k)$ are the same for all $h \colon I \to H$ and $k \colon I \to K$.

• 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 $f$ satisfying $f^\dagger \circ f = \mathrm{id}$ - 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 $z \colon I \to I$ equals $z^\dagger$, 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 $z \colon I \to I$ 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 $w+z=0$ implies $w=z=0$; 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 $p$ satisfying $p^\dagger \circ p = p$. They are ordered by $p \leq q$ if and only if $q \circ p = p$. 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 $f^\perp = \mathrm{ker}(f^\dagger)$, and Axioms 3, 4, and 6 makes it a complete lattice. So projections must be a complete lattice too, and clearly $p^\perp = \mathrm{id}-p$ make those into a complete ortholattice.

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

Combining Axioms 3 and 6 make you think about looking at the object $I^A$ which consists of $A$ many copies of the tensor unit $I$ for any set $A$. You think about this as a sort of standard object like the standard Hilbert space $\ell^2(A)$ of dimension $|A|$. And indeed, you find an orthonormal basis of cardinality $|A|$ for $\mathrm{hom}(I,I^A)$ with some fiddling. Now Solèr’s theorem tells you the scalars must be $\mathbb{R}$ or $\mathbb{C}$, and it quickly follows that $\mathrm{hom}(I,H)$ must be a Hilbert space for any object $H$. Now you’re on a roll. It’s all coming back to you: $\mathrm{hom}(I,-)$ 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.