A general theory of self-similarity

Leinster, T. (2011) A general theory of self-similarity. Advances in Mathematics, 226(4), pp. 2935-3017. (doi: 10.1016/j.aim.2010.10.009)

[img] Text
arxiv.html

3kB

Publisher's URL: http://dx.doi.org/10.1016/j.aim.2010.10.009

Abstract

<p>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.</p> <p>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.</p>

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Leinster, Dr Tom
Authors: Leinster, T.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Advances in Mathematics
Journal Abbr.:Adv. Math.
ISSN:0001-8708
ISSN (Online):1090-2082
Published Online:05 November 2010

University Staff: Request a correction | Enlighten Editors: Update this record

Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
373241Higher-dimensional algebra and self-similarityThomas LeinsterThe Nuffield Foundation (NUFFIELD)NAL/00852/GM&S - MATHEMATICS
424771Self similarity - recursively definable objects in topology, analysis category theory and algebraThomas LeinsterEngineering & Physical Sciences Research Council (EPSRC)EP/D073537/1M&S - MATHEMATICS