Kolmogorov Complexity of Categories

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.Comment: 16 page

A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points

Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses the semantic paradoxes, and how they arise as diagonal arguments and fixed point theorems in logic, computability theory, complexity theory and formal language theory

The Syntax of Coherence

This article tackles categorical coherence within a two-dimensional generalization of Lawvere's functorial semantics. 2-theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded, many coherence results become simple statements about the quasi-Yoneda lemma and 2-theory-morphisms. Given two 2-theories and a 2-theory-morphism between them, we explore the induced relationship between the corresponding 2-categories of algebras. The strength of the induced quasi-adjoints are classified by the strength of the 2-theory-morphism. These quasi-adjoints reflect the extent to which one structure can be replaced by another. A two-dimensional analogue of the Kronecker product is defined and constructed. This operation allows one to generate new coherence laws from old ones.Comment: 44 pages, LaTeX; XY-Pic (with 2-cells). Corrected typos and small change