#### Glasnik Matematicki, Vol. 38, No.2 (2003), 377-393.

### APPROXIMATE RESOLUTIONS AND THE FRACTAL
CATEGORY

### Takahisa Miyata and Tadashi Watanabe

Division of Mathematics and Informatics, Faculty of Human Development,
Kobe University, Nada-Ku, 3-11 Tsurukabuto, Kobe, 657-8501, Japan

*e-mail:* `tmiyata@kobe-u.ac.jp`
Department of Mathematics and Information Sciences, Faculty of
Education, Yamaguchi University, Yamaguchi-City, 753-8513, Japan

*e-mail:* `tadashi@po.yb.cc.yamaguchi-u.ac.jp`

**Abstract.** This paper concerns the theory of approximate
resolutions and its
application to fractal geometry. In this paper, we first
characterize 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 defined for approximate resolutions by the
authors, is invariant in this category. To define the morphisms of
the category, we introduce an equivalence relation on approximate
maps and define the morphisms as the equivalence classes.

**2000 Mathematics Subject Classification.**
54C56, 28A80.

**Key words and phrases.** Approximate resolution, surjective
map, box-counting dimension, category.

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