A general theory of self-similarity I

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 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, the self-similarity of X is described by a system of simultaneous equations. Here I formalize this idea and the notion of a ‘universal solution’ of such a system. I determine exactly when a system has a universal solution and, when one does exist, construct it. A sequel [Lei2] contains further results and examples, and an introductory article [Lei3] gives an overview.