Abstract
This paper concerns the theory of approximate resolu- tions and its application to fractal geometry. In this paper, we rst charac- terize a surjective map f : X ! Y between compact metric spaces in terms of a property on any approximate map f : X ! Y where p : X ! X and q : Y ! Y are any choices of approximate resolutions of X and Y , respectively. Using this characterization, we construct a category whose objects are approximate sequences so that the box-counting dimension, which was dened for approximate resolutions by the authors, is invariant in this category. To dene the morphisms of the category, we introduce an equivalence relation on approximate maps and dene the morphisms as the equivalence classes.
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