#### Glasnik Matematicki, Vol. 40, No.2 (2005), 347-384.

### A SUBSHAPE SPECTRUM FOR COMPACTA

### Nikica Uglešić and Branko Červar

University of Split, Department of Mathematics,
21 000 Split, Teslina 12/III, Croatia

*e-mail:* `uglesic@pmfst.hr`

*e-mail:* `brankoch@pmfst.hr`

**Abstract.** A sequence of categories and functors between them are
constructed. They form a *subshape spectrum* for compacta in
the following sense: Each of these categories classifies compact
ANR's just as the homotopy category does; the classification of
compacta by the "finest" of these categories coincides with the
shape type classification; moreover, the finest category contains
a subcategory which is isomorphic to the shape category; there
exists a functor of the shape category to each of these
categories, as well as of a "finer" category to a
"coarser" one; the functors commute
according to the indices.

Further, a few applications of the "subshape spectrum theory" are demonstrated.
It is shown that the *S**-equivalence (a uniformization of the Mardešić
*S*-equivalence) and the *q**-equivalence (a uniformization
of the Borsuk quasi-equivalence) admit the category
characterizations within the subshape spectrum, and that the
*q**-equivalence implies (but is not equivalent to) the
*S**-equivalence.

**2000 Mathematics Subject Classification.**
55P55, 18A32.

**Key words and phrases.** Compactum, ANR, inverse sequence, limit, shape type,
quasi-equivalence, S-equivalence.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.40.2.15

**References:**

- K. Borsuk,
Theory of Shape, Monografie Matematyczne 59,
Polish Scientific Publishers, Warszawa, 1975.

- K. Borsuk,
*Some quantitative properties of shapes*,
Fund. Math. **93** (1976), 197-212.

- K. R. Goodearl and T. B. Rushing,
*Direct limit groups and the Keesling-Mardešić shape fibration*,
Pacific J. Math. **86** (1980), 471-476.

- H. Herlich and G. E. Strecker,
Category Theory, Allyn and Bacon Inc., Boston, 1973.

- J. Keesling and S. Mardesic},
*A shape fibration with fibers of different shape*,
Pacific J. Math. **84** (1979), 319-331.

- S. Mardesic,
*Comparing fibres in a shape fibration*,
Glasnik Mat. **13(33)** (1978), 317-333.

- S. Mardesic and N. Uglesic,
*A category whose isomorphisms induce an equivalence relation coarser
than shape*, Top. Appl. (to appear).

- S. Mardesic and J. Segal,
Shape Theory, North Holland, Amsterdam, 1982.

- N. Uglesic,
*A note on the Borsuk quasi-equivalence*, submitted.

*Glasnik Matematicki* Home Page