Coalgebraic Representation Theory of Fractals

We develop a \emph{representation theory} in which a point of a fractal specified by \emph{metric} means (by a variant of an \emph{iterated function system, IFS}) is represented by a suitable equivalence class of infinite streams of symbols. The framework is categorical: symbolic representatives carry a final coalgebra; an IFS-like metric specification of a fractal is an algebra for the same functor. Relating the two there canonically arises a \emph{representation map}, much like in America and Rutten's use of metric enrichment in denotational semantics. A distinctive feature of our framework is that the canonical representation map is bijective. In the technical development, \emph{gluing} of shapes in a fractal specification is a major challenge. On the metric side we introduce the notion of \emph{injective IFS} to be used in place of conventional IFSs. On the symbolic side we employ Leinster's presheaf framework that uniformly addresses necessary identification of streams---such as $.0111\dotsc=.1000\dotsc$ in the binary expansion of real numbers. Our leading example is the unit interval $\mathbb{I} = [0,1]$.