A general theory of self-similarity

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)2935-3017
Number of pages83
JournalAdvances in Mathematics
Issue number4
Publication statusPublished - 1 Mar 2011


  • Recursion
  • Self-similarity
  • Final coalgebra
  • Real interval
  • Barycentric subdivision
  • Fractal
  • Categorification
  • Colimit
  • Bimodule
  • Profunctor
  • Flat functor


Dive into the research topics of 'A general theory of self-similarity'. Together they form a unique fingerprint.

Cite this