26 research outputs found
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