General self-similarity: an overview
📜 Abstract
Consider a self-similar space X. A typical situation is that X looks like several copies of itself glued to several copies of another space Y , and that Y looks like several copies of itself glued to several copies of X—or the same kind of thing with more than two spaces. Thus, we have a system of simultaneous equations in which the right-hand sides (the gluing instructions) are ‘higher-dimensional formulas’. This idea is developed in detail in [Lei1] and [Lei2]. The present informal seminar notes explain the theory in outline.
✨ Summary
The paper presents an intrinsic, category-theoretic framework for describing spaces through recursive gluing equations. A self-similarity system consists of a small category indexing the component spaces and a two-sided module encoding how copies of those spaces are glued together. This data induces an endofunctor, and the intended spaces are characterized as a terminal coalgebra, or universal solution, of that endofunctor.
The framework generalizes Freyd’s characterization of the unit interval as two copies of itself glued end to end. Leinster introduces nondegenerate functors to prevent the construction from collapsing to trivial solutions, extends the theory from sets to topological spaces, and states existence and recognition results for universal solutions. The paper highlights examples including intervals, products of self-similar spaces, simplices, Cantor spaces, Sierpiński gaskets, iterated-function-system attractors, and Julia sets. Its principal topological conclusion is that, under the broad definition used, a space is self-similar exactly when it is compact and metrizable; with discrete indexing categories, the corresponding class is the totally disconnected compact metrizable spaces.
The document is explicitly an overview of two technical preprints, and the underlying theory was later consolidated and substantially developed in Leinster’s 2011 A general theory of self-similarity, which superseded those preprints. (arxiv.org) Subsequent work in coalgebra and theoretical computer science cites the overview and its companion papers as background for categorical treatments of recursive program schemes and coalgebraic constructions. (www8.cs.fau.de) Leinster’s later materials on coalgebraic topology also use the work as a reference for universal properties of recursively defined spaces. (webhomes.maths.ed.ac.uk) The available evidence indicates research influence in category theory, coalgebra, and mathematical exposition; no specific industrial deployment or product use of this overview was identified.