Cool Cats II
Now ordinary categories are not as interesting as toposes, so we would like to extend the comparison of sets and vector spaces to generalised axioms for higher dimensional toposes. But then we had better consider multicategories as well. This is what logos theory aims to do: utilise physically important structures such as MUBs to determine a recursive series of axioms for $n$-dimensional topos like structures. Since ordinary weak $n$ categories should appear as certain algebras, this is clearly a very difficult problem. Unfortunately, working on gravity is like throwing oneself constantly against the brick wall of major problems in mathematics, so a poor physicist just has to accept this fate.
The dimension of the matrix operators that we study is the dimension $n$ of the MUB problem, which states that there are always $n + 1$ MUBs for prime power dimension. This $n$ also represents the dimensionality of the logos structure. Hence sets, or vector spaces, are two dimensional beasts, partly because the subobject classifier is specified by two elements, but more importantly because the categorical structure has two levels of arrow and the classifying square is planar.
Hence the M theorist's obsession with three object groupoids and triangles and cubes and other trinities. Most 20th century mathematics is set theoretic, meaning that it only deals with one dimensional logoses. For example, a group is usually treated as a set with extra structure (multiplication, unit, associativity). But as a category a group only has one object. It is often more natural to use groupoids, such as the groupoid of directed loops in a space, with concatenation for composition. All the loops from a given point to itself form a group anyway, so in a two point space there would be two groups sitting inside the loop groupoid.