M Theory Lesson 277

The close connection between MUBs and finite fields makes one wonder how to properly state categorical axioms for modular arithmetic. As far as I can tell, this issue is far from resolved in the literature. For example, in the topos Set the natural number object contains finite sets as subsets, but the axioms of arithmetic rely on the infinite object.

Recall that one dream for logoses is to understand an ordinal $n$ as an elementary category, independently of larger numbers. Just taking oriented simplices, for instance, doesn't say anything at all about modular arithmetic, basically because one never imagines pieces of space disappearing under addition! How can we hope to understand the complex numbers if we don't even understand finite fields?