A general theory of self-similarity

TL;DR

This paper develops a general theory of universal characterizations of spaces, extending classical self-similarity concepts like the interval [0,1], and provides explicit conditions and constructions for such universal solutions.

Contribution

It introduces a categorified framework for understanding self-similarity through universal solutions to systems of space-equations, generalizing classical topological properties.

Findings

Characterization of the interval [0,1] as a universal self-similar space

Explicit conditions for the existence of universal solutions

Construction methods for universal spaces satisfying self-similarity equations

Abstract

A little-known and highly economical characterization of the real interval [0, 1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar spaces have similar universal properties; for example, the topological simplices Delta^n may be defined as the universal family of spaces admitting barycentric subdivision. We develop a general theory of such universal characterizations. This can also be regarded as a categorification of the theory of simultaneous linear equations. We study systems of equations in which the variables represent spaces and each space is equated to a gluing-together of the others. One seeks the universal family of spaces satisfying the equations. We answer all the basic questions about such systems, giving an explicit condition equivalent to the existence of a universal solution, and an explicit construction of it whenever it does exist.