Glasnik Matematicki, Vol. 51, No. 2 (2016), 255-306.

THE SHAPES IN A CONCRETE CATEGORY

Nikica Uglešić

Abstract.   We show under what conditions, and how, one can obtain a shape theory (various shape theories) in a concrete category. The technique is, roughly speaking, reduced to the quotients by congruences providing the objects of lower cardinalities. The application yields the new (coarser) classifications in every concrete category which admits sufficiently many non-trivial quotients. Thus, the ordering, (ultra)pseudometric, uniform and topological structures, as well as many algebraic and mixed (multi-) structures, give rise to interesting results.

2010 Mathematics Subject Classification.   03E99, 06A99, 16D99, 16S99, 20B07, 46A99, 54B15, 54B30, 54C56, 54E99, 55P55.

Key words and phrases.   (pointed) set, partially ordered set, (ultra)pseudometric space, topological space, monoid, group, ring, module, vector space, equivalence relation, congruence, (infinite) cardinal, concrete category, quotient object, dimension, expansion, pro-category, shape category.

DOI: 10.3336/gm.51.2.01

References:

1. K. Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223-254.
   MathSciNet    

2. K. Borsuk, Theory of shape, Matematisk Inst. Aarhus Univ., Aarhus, 1971.
   MathSciNet    

3. K. Borsuk, Theory of shape, Polish Scientific Publishers, Warszaw, 1975.
   MathSciNet    

4. J.-M. Cordier and T. Porter, Shape theory. Categorical methods of approximation, Ellis Horwood Ltd., Chichester, 1989. (Dover edition, 2008.)
   MathSciNet    

5. J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
   MathSciNet    

6. J. R. Durbin, Modern algebra. An introduction, John Wiley \& Sons, Inc., New York, 1992.
   MathSciNet    

7. J. Dydak and J. Segal, Shape theory. An introduction, Springer-Verlag, Berlin, 1978.
   MathSciNet    

8. D. A. Edwards and H. M. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Springer-Verlag, Berlin, 1976.
   MathSciNet    

9. S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, 1952.
   MathSciNet    

10. H. Herlich and G. E. Strecker, Category theory. An introduction, Allyn and Bacon Inc., Boston, 1973.
    MathSciNet    

11. N. Koceić Bilan and N. Uglešić, The coarse shape, Glas. Mat. Ser. III. 42(62) (2007), 145-187.
    MathSciNet     CrossRef

12. S. Mac Lane, Homology, Springer-Verlag, Berlin, 1995.
    MathSciNet    

13. S. Mardešić and J. Segal, Shape theory. The inverse system approach, Noth-Holland Publishing co., Amsterdam-New York, 1982.
    MathSciNet    

14. I. Moerdijk, Prodiscrete groups and Galois toposes, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), 219-234.
    MathSciNet     CrossRef

15. W. Tholen, Pro-categories and multiadjoint functors, Canad. J. Math. 36 (1984), 144-155.
    MathSciNet     CrossRef

16. N. Uglešić, On ultrametrics and equivalence relations-duality, Int. Math. Forum 5 (2010), 1037-1048.
    MathSciNet    

17. N. Uglešić and V. Matijević, On expansions and pro-pro-categories, Glas. Mat. Ser. III. 45(65) (2010), 173-217.
    MathSciNet     CrossRef

18. N. Uglešić and B. Červar, The concept of a weak shape type, Int. J. Pure Appl. Math. 39 (2007), 363-428.
    MathSciNet