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Macquarie University  Department of Mathematics

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Categories in Algebra, Geometry and Mathematical Physics

Conference in honour of Ross Street's sixtieth birthday

July 11-16 2005, Macquarie University, Sydney

Mon, 11 July:  15:45 - 16:25

A general theory of self-similarity
Leinster, Tom (Glasgow University)

I will present the beginnings of a general theory of self-similarity.
In principle this concerns self-similar anythings, but I will focus on
the topological case.

Consider a self-similar object $X$. Typically, $X$ looks like several
copies of itself glued to several copies of another object $Y$, and $Y$
looks like several copies of itself glued to several copies of $X$---or the
same kind of thing with a larger family $X_1, X_2, \ldots$ of objects.
Now observe:
%
\begin{enumerate}
\item This is a system of simultaneous equations in which the usual
algebraic operations have been replaced by gluing. The gluing
instructions can be regarded as `higher-dimensional formulas'.
\item Often such a system has a canonical solution $X_1, X_2, \ldots$
In this case, the $X_i$s --- which may be highly complex fractal
spaces --- are determined by a system of algebraic equations.
\end{enumerate}
%
There is a clean and simple formalism for such systems of
simultaneous equations. An explicit criterion determines whether
a system has a canonical solution; if so, an explicit construction
gives it.

Many well-known self-similar spaces arise as examples. One of the
simplest is the real interval $[0, 1]$; indeed, the starting point of the
theory was Freyd's characterization of $[0, 1]$ as a terminal coalgebra.

Typeset PDF of this abstract.