The n-Category Café

How are ${\mathrm{Hilb}}_{ℝ}$, ${\mathrm{Hilb}}_{ℂ}$ and ${\mathrm{Hilb}}_{ℍ}$ related?

Of course the complex HIlbert space ${ℂ}^n$ has an underlying real Hilbert space ${ℝ}^{2n}$, and the quaternionic Hilbert space ${ℍ}^n$ has an underlying complex Hilbert space ${ℂ}^{2n}$. But there’s a slightly more sophisticated way to say what’s going on.

Let’s start with the chain of inclusions $$ℝ\hookrightarrow ℂ\hookrightarrow ℍ.$$ Thanks to the first inclusion, any complex vector space has an underlying real vector space. In other words: if we have a complex vector space, and we deliberately forget how to do scalar multiplication of vectors by complex numbers, and only remember how to multiply them by real numbers, it becomes a real vector space! Similarly, any quaternionic vector space becomes a complex one, thanks to the second inclusion.

The same is true for Hilbert spaces. To make the underlying real vector space of a complex Hilbert space into a real Hilbert space, we take the real part of the original complex inner product, defined by $$\mathrm{Re}(a+bi)=a.$$ Everyone knows that; a bit less familiar is how the underlying complex vector space of a quaternionic Hilbert space becomes a complex Hilbert space. Here we need to take the complex part of the original quaternionic inner product, defined by $$\mathrm{Co}(a+bi+cj+dk)=a+bi.$$ One can check that these constructions give functors $${\mathrm{Hilb}}_{ℍ}\to {\mathrm{Hilb}}_{ℂ}\to {\mathrm{Hilb}}_{ℝ}.$$

A bit more formally, we have a commutative triangle of homomorphisms:

$$\begin{array}{rlr}ℂ& \stackrel{\beta }{\to }& ℍ\\ \alpha \uparrow & \nearrow \gamma \\ ℝ\end{array}$$

There is only one choice of the homomorphisms $\alpha$ and $\gamma$. There are many choices of $\beta$, since we can map $i\in ℂ$ to any square root of $-1$ in the quaternions. However, all the various choices of $\gamma$ are the same up to symmetries of the quaternions. That is, given two homomorphisms $\beta ,\beta \prime :ℂ\to ℍ$, we can always find an automorphism $\theta :ℍ\to ℍ$ such that $$\beta \prime =\theta \circ \beta .$$ So, nothing important depends on the choice of $\beta$. Let us make a choice — say the standard one, with $\beta (i)=i$ — and use that.

Our commutative triangle of homomorphisms gives a commutative triangle of functors:

$$\begin{array}{rlr}{\mathrm{Hilb}}_{ℂ}& \stackrel{{\beta }^{*}}{\leftarrow }& {\mathrm{Hilb}}_{ℍ}\\ {\alpha }^{*}\downarrow & \swarrow {\gamma }^{*}\\ {\mathrm{Hilb}}_{ℝ}\end{array}$$

Now, recall that a functor $F:C\to D$ is faithful if given two morphisms $f,f\prime :c\to c\prime$ in $C$, $F(f)=F(f\prime )$ implies that $f=f\prime$. When $F:C\to D$ is faithful, we say that $C$ is faithfully embedded in $D$, and we can think of objects of $C$ as objects of $D$ equipped with extra structure.

It is easy to see that the functors ${\alpha }^{*},{\beta }^{*}$ and ${\gamma }^{*}$ are all faithful. This lets us describe Hilbert spaces for a larger normed division algebra as Hilbert spaces for a smaller one — but equipped with extra structure. None of this particularly new or difficult: the key ideas are all in Adams’ Lectures on Lie Groups.

First we consider the extra structure possessed by the underlying real Hilbert space of a complex Hilbert space:

Theorem: The functor $${\alpha }^{*}:{\mathrm{Hilb}}_{ℂ}\to {\mathrm{Hilb}}_{ℝ}$$ is faithful, and ${\mathrm{Hilb}}_{ℂ}$ is equivalent to the category where:

• an object is a real Hilbert space $H$ equipped with a unitary operator $J:H\to H$ with $J^2=-1$.
• a morphism $T:H\to H\prime$ is a bounded real-linear operator preserving this exta structure: $TJ=J\prime T$.

This extra structure $J$ is often called a complex structure.

Next we consider the extra structure possessed by the underlying complex Hilbert space of a quaternionic Hilbert space. For this we need to generalize the concept of an antiunitary operator. First, given $𝕂$-vector spaces $V$ and $V\prime$, we define an antilinear operator $T:V\to V\prime$ to be a map with $$T(vx+wy)=T(v)x^{*}+T(w)y^{*}$$ for all $v,w\in V$ and $x,y\in 𝕂$. Then, given $𝕂$-Hilbert spaces $H$ and $H\prime$, we define an antiunitary operator $T:H\to H\prime$ to be an invertible antilinear operator with $$⟨Tv,Tw⟩=⟨w,v⟩$$ for all $v,w\in H$.

Theorem: The functor $${\beta }^{*}:{\mathrm{Hilb}}_{ℍ}\to {\mathrm{Hilb}}_{ℂ}$$ is faithful, and ${\mathrm{Hilb}}_{ℍ}$ is equivalent to the category where:

• an object is a complex Hilbert space $H$ equipped with an antiunitary operator $J:H\to H$ with $J^2=-1$;
• a morphism $T:H\to H\prime$ is a bounded complex-linear operator preserving this extra structure: $TJ=J\prime T$.

This extra structure $J$ is often called a quaternionic structure. We have seen it already in our study of the Three-Fold Way.

Finally, we consider the extra structure possessed by the underlying real Hilbert space of a quaternionic Hilbert space. This can be understood by composing the previous two theorems:

Theorem: The functor $${\gamma }^{*}:{\mathrm{Hilb}}_{ℍ}\to {\mathrm{Hilb}}_{ℝ}$$ is faithful, and ${\mathrm{Hilb}}_{ℍ}$ is equivalent to the category where:

• an object is a real Hilbert space $H$ equipped with two unitary operators $J,K:H\to H$ with $J^2=K^2=-1$ and $JK=-KJ$;
• a morphism $T:H\to H\prime$ is a bounded complex-linear operator preserving this extra structure: $TJ=J\prime T$ and $TK=K\prime T$.

This extra structure could also be called a quaternionic structure, as long as we remember that a quaternionic structure on a real Hilbert space is different than one on a complex Hilbert space! Of course if we define $I=JK$, then $I,J,$ and $K$ obey the usual quaternion relations.

The functors discussed so far all have adjoints, which are in fact both left and right adjoints:

$$\begin{array}{rlr}{\mathrm{Hilb}}_{ℂ}& \stackrel{{\beta }_{*}}{\to }& {\mathrm{Hilb}}_{ℍ}\\ {\alpha }_{*}\uparrow & \nearrow {\gamma }_{*}\\ {\mathrm{Hilb}}_{ℝ}\end{array}$$

These adjoints can easily be defined using the theory of bimodules. As vector spaces, we have:

$$\begin{array}{ccl}{\alpha }_{*}(V)& =& V\otimes {}_{ℝ}{ℂ}_{ℂ}\\ {\beta }_{*}(V)& =& V\otimes {}_{ℂ}{ℍ}_{ℍ}\\ {\gamma }_{*}(V)& =& V\otimes {}_{ℝ}{ℍ}_{ℍ}\\ \end{array}$$

In the first line here, $V$ is a real vector space, or in other words, a right $ℝ$-module, while ${}_{ℝ}{ℂ}_{ℂ}$ denotes $ℂ$ regarded as a $ℝ$-$ℂ$-bimodule. Tensoring these, we obtain a right $ℂ$-module, which is the desired complex vector space. The other lines work the same way. It is then easy to make all these vector spaces into Hilbert spaces. And I can’t resist mentioning that our previous functors can be described in a similar way, just by turning the bimodules around:

$$\begin{array}{ccl}{\alpha }^{*}(V)& =& V\otimes {}_{ℂ}{ℂ}_{ℝ}\\ {\beta }^{*}(V)& =& V\otimes {}_{ℍ}{ℍ}_{ℂ}\\ {\gamma }^{*}(V)& =& V\otimes {}_{ℍ}{ℍ}_{ℝ}\end{array}$$

But instead of digressing into this subject (called Morita theory), all I want to do now is mention that the functors ${\alpha }_{*},{\beta }_{*}$ and ${\gamma }_{*}$ are also faithful. This lets us describe Hilbert spaces for a smaller normed division algebra in terms of Hilbert spaces for a bigger one!

We begin with the functor ${\alpha }_{*}$, which is called complexification:

Theorem: The functor $${\alpha }_{*}:{\mathrm{Hilb}}_{ℝ}\to {\mathrm{Hilb}}_{ℂ}$$ is faithful, and ${\mathrm{Hilb}}_{ℝ}$ is equivalent to the category where:

• an object is a complex Hilbert space $H$ equipped with a antiunitary operator $J:H\to H$ with $J^2=1$;
• a morphism $T:H\to H\prime$ is a bounded complex-linear operator preserving this exta structure: $TJ=J\prime T$.

The extra structure $J$ here is often called a real structure. We have seen it already in our study of the Three-Fold Way.

Next let’s look at the functor from complex to quaternionic Hilbert spaces. It has no name, as far as I know:

Theorem: The functor $${\beta }_{*}:{\mathrm{Hilb}}_{ℂ}\to {\mathrm{Hilb}}_{ℍ}$$ is faithful, and ${\mathrm{Hilb}}_{ℂ}$ is equivalent to the category where:

• an object is a quaternionic Hilbert space $H$ equipped with a unitary operator $J$ with $J^2=-1$;
• a morphism $T:H\to H\prime$ is a bounded complex-linear operator preserving this extra structure: $TJ=J\prime T$.

This result is less well-known than the previous ones, so let me sketch a proof:

Proof: Suppose $H$ is a quaternionic Hilbert space equipped with a unitary operator $J$ with $J^2=-1$. Then $J$ makes $H$ into a right module over the complex numbers, and this action of $ℂ$ commutes with the action of $ℍ$, so $H$ becomes a right module over the tensor product of $ℂ$ and $ℍ$, considered as algebras over $ℝ$. But this is isomorphic to the algebra of $2\times 2$ complex matrices. The matrix $$\left(\begin{array}{rl}1& 0\\ 0& 0\end{array}\right)$$ projects $H$ down to a complex Hilbert subspace $H_{ℂ}$ whose complex dimension matches the quaternionic dimension of $H$. By applying arbitrary $2\times 2$ complex matrices to guys in this subspace we get back everything in $H$, so ${\beta }_{*}{H}_{ℂ}$ is naturally isomorphic to $H$.   █

Composing ${\alpha }_{*}$ and ${\beta }_{*}$, we obtain the functor from real to quaternionic Hilbert spaces. I’ve seen this called quaternification — or occasionally ‘quaternization’, but that means something else in chemistry!

Theorem: The functor $${\gamma }_{*}:{\mathrm{Hilb}}_{ℝ}\to {\mathrm{Hilb}}_{ℍ}$$ is faithful, and ${\mathrm{Hilb}}_{ℝ}$ is equivalent to the category where:

• an object is a quaternionic Hilbert space $H$ equipped with two unitary operators $J,K$ with $J^2=K^2=-1$ and $JK=-KJ$.
• a morphism $T:H\to H\prime$ is a bounded complex-linear operator preserving this extra structure: $TJ=J\prime T$ and $TK=K\prime T$.

Again, let me sketch a proof:

Proof: The operators $J,K$ and $I=JK$ make $H$ into a left $ℍ$-module. Since this action of $ℍ$ commutes with the existing right $ℍ$-module structure, $H$ becomes a module over the tensor product of $ℍ$ and ${ℍ}^{\mathrm{op}}\cong ℍ$, considered as algebras over $ℝ$. But this tensor product is isomorphic to the algebra of $4\times 4$ real matrices! So, the matrix $$\left(\begin{array}{rlrl}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$ projects $H$ down to a real Hilbert subspace $H_{ℝ}$ whose real dimension matches the quaternionic dimension of $H$. By applying arbitrary $4\times 4$ real matrices to guys in this subspace we get back everything in $H$, so ${\gamma }_{*}{H}_{ℝ}$ is naturally isomorphic to $H$.   █

Finally, it is worth noting that some of the six functors we have described have additional nice properties:

• The categories ${\mathrm{Hilb}}_{ℝ}$ and ${\mathrm{Hilb}}_{ℂ}$ are symmetric monoidal categories, meaning roughly that they have well-behaved tensor products. The complexification functor ${\alpha }^{*}:{\mathrm{Hilb}}_{ℝ}\to {\mathrm{Hilb}}_{ℂ}$ is a symmetric monoidal functor, meaning roughly that it preserves tensor products.
• The categories ${\mathrm{Hilb}}_{ℝ},{\mathrm{Hilb}}_{ℂ}$ and ${\mathrm{Hilb}}_{ℍ}$ are dagger-categories, meaning roughly that any morphism $T:H\to H\prime$ has a Hilbert space adjoint $T^{†}:H\prime \to H$ such that $$⟨Tv,w⟩=⟨v,T^{†}w⟩$$ for all $v\in H$, $w\in H\prime$. All six functors preserve this dagger operation.
• For ${\mathrm{Hilb}}_{ℝ}$ and ${\mathrm{Hilb}}_{ℂ}$, the dagger structure interacts nicely with the tensor product, making these categories into dagger-compact categories, and the functor ${\alpha }^{*}$ is compatible with this as well.

For precise definitions of the terms here, click on the links. I’ve spent a lot of time trying explain how these concepts unify physics with topology and other subjects:

• John Baez and Aaron Lauda, A prehistory of $n$-categorical physics, to appear in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson, Cambridge U. Press.
• John Baez and Mike Stay, M. Stay, Physics, topology, logic and computation: a Rosetta Stone, in New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics 813, Springer, Berlin, 2000, pp. 95–174.

For more on dagger-categories see also these papers:

• Samson Abramsky, Abstract scalars, loops, and free traced and strongly compact closed categories, in Proceedings of CALCO 2005, Lecture Notes in Computer Science 3629, Springer, Berlin, 2005, 1–31.
• Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols.
• Peter Selinger, Dagger compact closed categories and completely positive maps, Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), Elsevier, 2007, pp. 139–163.

and this wonderful book:

• Bob Coecke, editor, New Stuctures for Physics, Lecture Notes in Physics 813, Springer, Berlin, 2000, pp. 95–174.

The three-fold way is best appreciated with the help of these category-theoretic ideas. A more $n$-categorical treatment of symplectic and orthogonal structures can be found in my old paper on 2-Hilbert spaces:

• John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125–189.

If I’m feeling exceptionally energetic I may say more about the $n$-categorical aspects, Morita theory, and so on. More likely, I’ll wrap up the story next time by saying how the Three-Fold Way solves some of the problems of real and quaternionic quantum theory.