Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Thursday, January 04, 2007

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.


Blogger L. Riofrio said...

The notion of higher dimensional equations shows that humans still have a lot to learn. That will keep mathematicians employed for a long time!

January 04, 2007 10:08 PM  
Blogger Kea said...

Hi Louise! Yes, it makes one realise how little we really know.

January 04, 2007 10:11 PM  
Blogger CarlBrannen said...

Can you add some examples of 2 categories in physics? Perhaps from GR?

January 05, 2007 12:04 AM  
Blogger Kea said...

Hi Carl. Yes, there should be a simple GR example, where the surface pieces are directly interpretable as 2-arrows. Probably no one has really tried to work it out from this perspective. I know of some more sophisticated examples, but one needs to be a GR specialist to really appreciate them, because unfortunately, it's not as obvious as it sounds to make such a connection. That's why Twistor Theory is so important, because it provides a step-by-step path from GR to better geometric (categorical) concepts.

This is a great research direction for relativists. One potentially interesting example is the work of Harris - see eg. Topology of the future chronological boundary: universality for spacelike boundaries, Class. Quant. Grav. 17 (2000) 551-603. The guy doesn't care at all about categories, but there is a nice example there: 1-categories of 'causal sets' forming a 2-category of categories. That is, causality introduces arrows into sets, making causal sets themselves into 1-categories.

January 05, 2007 11:18 AM  

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