Kolmogorov Complexity of Categories

Abstract

Kolmogorov complexity theory is used to tell what the algorithmic informational content of a string is. It is defined as the length of the shortest program that describes the string. We present a programming language that can be used to describe categories, functors, and natural transformations. With this in hand, we define the informational content of these categorical structures as the shortest program that describes such structures. Some basic consequences of our definition are presented including the fact that equivalent categories have equal Kolmogorov complexity. We also prove different theorems about what can and cannot be described by our programming language.

A while back, I showed some of these ideas to Samson Abramsky and he was, as always, full of encouragement and great ideas. I am very grateful to him for all his help over the years. I would like to acknowledge the help and advice of Michael Barr, Marta Bunge, James Cox, Joey Hirsh, Florian Lengyel, Dustin Mulcahey, Philip Rothmaler, and Louis Thral. I want to thank Shayna Leah Hershfeld for many enlightening conversations about polymorphism and type theory. Support for this project was provided by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.