Back to Basics

There is an elementary idea in Category Theory that should be more emphasised, namely the notion of a higher dimensional equation. If one looks at any equation in a random physics paper, there's a good chance it will be linear, meaning it can be written in a 1-dimensional line. Even if it's a matrix equation, such as $AB = BA$, it will be a linear equation. Now matrices are really arrows in a category Mat whose objects are ordinals $n$. An $n \times m$ matrix is then an arrow from $n$ to $m$. Thus a simple linear matrix equation of the form $AB = CD$ is actually a square diagram of sides $A, B, C$ and $D$. In general, linear equations are then 1-category diagrams.

In a higher category, with at least some non-identity $n$-arrows for $n \geq 2$, we still interpret diagrams as "equations". As an exercise, draw a square composed of four smaller squares, each containing a 2-arrow. Now label objects, 1-arrows and 2-arrows, and write out all the 1-dimensional paths as linear equations. Clearly, insisting on writing out equations this way would get tiresome very quickly.

If we came across a physical or mathematical structure indexed by three ordinal indices, as opposed to the two indices of matrices or numbers, we would naturally guess that equations in these objects would be 2-dimensional. These objects might be triangle 2-arrows in a 2-category whose objects are again the ordinals, but this could only be useful if the 2-arrows satisfied the right pasting conditions to form equations.