Jamie Vicary bet me a bottle of wine that the

fairy field would be found in the next few years. Actually, he was willing to bet on the next

two years, but I let him off the hook on that one. (Unfortunately, my total stakes on this question include a little wine and around ten dollars, so even I can afford to lose.) At

QPL,

Vicary spoke about his characterisation of the complex numbers using natural structures in dagger monoidal categories with superposition (see this

paper).

Superposition says that given two morphisms $f,g: A \rightarrow B$, there exists a morphism called $f+g$ and addition is commutative and associative. The crucial notion is that of a $\dagger$ limit for a diagram $D$, defined to be a

limit $L$ such that the arrows $f_{S}: L \rightarrow D(S)$ satisfy

$\sum f_{s} \circ f_{s}^{\dagger} = 1_{L}$,

where the sum is over a set of source objects in $D$. This is a normalisation condition for superpositions. When all objects in a discrete diagram $D$ are sources, this reduces to the categorical

biproduct $\oplus$. Given a category with a zero object and all finite biproducts (such as the category of Hilbert spaces) it turns out that there is a unique superposition rule.

One of the things that Vicary shows is that, for a category with tensor unit $I$ and all finite dagger limits, the semiring of scalars $I \rightarrow I$ has a natural embedding into a

characteristic zero field. This relies on the decomposition of any non-zero ordinal $p: I \rightarrow I$ into a diagonal arrow

$I \oplus I \oplus \cdots \oplus I \rightarrow I$

and its adjoint codiagonal. So, if we

want to work with finite fields of characteristic $p$, we can now identify exactly which pieces of complex number structure break down. For instance, there might be a zero map constructed from a finite diagonal and codiagonal on the unit object.