M Theory Lesson 247

A simple way to express a cardinal number $n$ as a dimension is to write $n$ as the $n$ dimensional identity matrix. In this case, the natural way to express prime factorizations is in terms of embedded identity matrices. For example, since $6 = 2 \times 3$ there are two ways to write down a circulant permutation such that in the first case $C^3 = 1$ and in the second case $C^2 = 1$. In other words, the factorization of the number 6 results in roots of unity of order less than 6. In general, one embeds identities into the basic Fourier circulant $(234...n1)$. Since matrix elements are noncommutative, one would not dream of using classical geometry to study categorified arithmetic.