## 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algebraic operations have been replaced by gluing.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.