## Simon Burton

I am a theoretical physicist at University College London working on the mathematics of quantum computing. This is my third career, having also worked in the areas of machine learning and finance (algorithmic trading).

## Arrow Theory

The term "arrow theory" is a colloquial term for the mathematical study of category theory. This theory is about the arrows, or processes, or representations between objects.

More than this: there are no objects, there are only processes, or representations. The meaning is in the arrows between the objects, not the objects themselves. The nouns only make sense through the verbs that connect them.

Quoting John Baez:

To understand this, the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both herds and try to match each sheep in one herd with a sheep in the other. But one day, along came a shepherd who invented decategorification. She realized one could take each herd and "count" it, setting up an isomorphism between it and some set of "numbers", which were nonsense words like "one, two, three,..." specially designed for this purpose. By comparing the resulting numbers, she could show that two herds were isomorphic without explicitly establishing an isomorphism! In short, by decategorifying the category of finite sets, the set of natural numbers was invented.

According to this parable, decategorification started out as a stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome by means of categorification. While the historical reality is far more complicated, categorification really has led to tremendous progress in mathematics during the 20th century.

Here is an excellent article about category theory from Quanta magazine.

## Articles

"The real significance of quantum computers" November 19, 2019.

## Research Notes

2-vector spaces

Lie algebras and homotopy theory Partial transcript of a lecture by Jacob Lurie.

On the concept of entropy of a finite probabilistic scheme English translation of the 1956 article by D. K. Faddeev.