#### Glasnik Matematicki, Vol. 33, No.2 (1998), 287-297.

### PROPER *n*-SHAPE CATEGORIES

### Katsuro Sakai

Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571,
Japan

*e-mail:* `sakaiktr@sakura.cc.tsukuba.ac.jp`

**Abstract.** In this paper, it is shown that the proper
*n*-shape category of Ball-Sher type is isomorphic to a
subcategory of the proper *n*-shape category defined by proper
*n*-shaping. It is known that the latter is isomorphic
to the shape category defined by the pair
(H_{p}^{n},
H_{p}^{n} Pol),
where
H_{p}^{n}
is the category whose objects are locally compact separable metrizable
spaces and whose morphisms are proper *n*-homotopy classes of
proper maps, and
H_{p}^{n} Pol
is the full subcategory of
H_{p}^{n}
whose objects are spaces having the proper *n*-homotopy type of polyhedra. In the case *n* =
∞,
this shows the relation between the original Ball-Sher's category
and the proper shape category defined by proper shapings. We also
discuss the proper *n*-shape category of space of dimension
≤ *n* + 1.

**1991 Mathematics Subject Classification.**
54C56, 55P55.

**Key words and phrases.** Proper *n*-shape category,
proper *n*-homotopy, proper *n*-fundamental net,
proper *n*-approximative net, proper *n*-shaping, expansion.

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