occasional meanderings in physics' brave new world

Wednesday, February 25, 2009

An important characteristic of the topos Set is the existence of a natural number object, namely the object $N$ of counting numbers along with a diagram

$1 \rightarrow N \rightarrow N$

where the second arrow is a successor function, plus one. This diagram is universal in the sense that it is initial in the category of all such diagrams. A general diagram in this category replaces the object $N$ by another set $A$.

In the quantum world, however, $N$ is better described by the dimensions of simple vector spaces. Including the ordinal maps, we can think of $N$ as a whole category, usually called $\Delta$. But $\Delta$ lives in a category of categories, Cat, rather than Set. So instead of maps $u: N \rightarrow A$ characterising the universality of arithmetic, we end up looking at functors $U: \Delta \rightarrow C$, which are basic mathematical gadgets known as cosimplicial objects.

The commuting square in Set that compares the successor function with a map $f: A \rightarrow A$ is replaced by a (weakened) commuting square that compares an increment in dimension to a functor $F: C \rightarrow C$ via the cosimplicial functor $U$. In other words, quantum arithmetic really is about cohomological invariants after all.

And let's not forget that in this higher dimensional operadic world, $1$-ordinals are merely the simplest kind of trees. The category $\Delta$ should really be replaced by a category whose objects are trees.