M Theory Lesson 103

Changing topics somewhat: in a classical topos a natural number object enables Peano's axioms for arithmetic to hold in the internal logic. This object $N$ is equipped with a successor function $s: N \rightarrow N$ and a zero object $0: 1 \rightarrow N$. But quantum mechanical logic doesn't belong in a classical topos. In the category Vect of vector spaces, for example, the terminal object $1$ might be replaced by the number field $K$ and the subobject classifier by the qubit object $K \oplus K$.

So what happens to arithmetic? Now Vect is actually a higher dimensional structure, being a symmetric monoidal category. Thus we expect to do a lot more decategorifying before we can count, and this process could take us through a topos like Set in which arithmetic makes sense. For example, how do we count the dimension of a (finite dimensional) vector space? We usually take a basis set and then use arithmetic in Set to count its elements. The logical analogue of dimension in Vect would really be an arrow $K \rightarrow K^{\infty}$ which picks a one dimensional space from $K^{\infty}$. Thus numbers really look like quantum states, and the so-called collapse of the wavefunction would be a simple categorical transition of numbers between different logics.