Glasnik Matematicki, Vol. 43, No.2 (2008), 451-479.

PULL-BACKS AND FIBRATIONS IN APPROXIMATE PRO-CATEGORIES

Takahisa Miyata

Abstract.   In this paper we introduce the category Apro-ANR called the approximate pro-category of ANR's, whose objects are all systems of ANR's and whose morphisms are obtained as equivalence classes of system maps for some equivalence relation. We show that any 2-sink X →^f Z ←^g Y in Apro-ANR admits a weak pull-back and it admits a pull-back if they are systems of compact ANR's. Moreover, it admits a pull-back if they are objects of pro-ANRU. Here ANRU is the full subcategory of the category Unif of uniform spaces and uniform maps, whose objects are uniform absolute neighborhood retracts (ANRU's) in the sense of Isbell. We define the approximate homotopy lifting property (AHLP) for morphisms in Apro-ANR and show that the category Apro-ANR with fibration = morphism with the AHLP with respect to paracompact spaces, and weak equivalence = morphism inducing an isomorphisms in pro-H(ANR) satisfies composition and factorization axioms and part of pull-back axiom for fibration category in the sense of Baues. Finally, we show that the limit of the pull-back of any 2-sink X →^f Z ←^g Y in Apro-ANR consisting of systems of compact ANR's is a pull-back in the category Top of topological spaces and continuous maps, and conversely every pull-back in the full subcategory CH of Top whose objects are compact Hausdorff spaces admits an expansion which is a pull-back in Apro-ANR.

2000 Mathematics Subject Classification.   54C56, 54C55, 55U35.

Key words and phrases.   Approximate pro-category, pull-back, approximate homotopy lifting property.

DOI: 10.3336/gm.43.2.15

References:

1. M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics 100, Springer-Verlag, Berlin-New York, 1969.
   MathSciNet

2. H. J. Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, Cambridge, 1989.
   MathSciNet

3. Q. Haxhibeqiri, Shape fibration for topological spaces, Glas. Mat. Ser. III 17(37) (1982), 381-401.
   MathSciNet

4. Q. Haxhibeqiri, The exact sequence of a shape fibration, Glas. Mat. Ser. III 18(38) (1983), 157-177.
   MathSciNet

5. J. R. Isbell, Uniform spaces, Mathematical Surveys 12, American Mathematical Society, Providence, R.I., 1964.
   MathSciNet

6. M. Jani, Induced shape fibrations and fiber shape equivalence, Rocky Mountain J. Math. 12 (1982), 305-332.
   MathSciNet

7. S. Mardesic, Approximate polyhedra, resolutions of maps and shape fibrations, Fund. Math. 114 (1981), 53-78.
   MathSciNet

8. S. Mardesic and T. B. Rushing, Shape fibrations. I, General Topology Appl. 9 (1978), 193-215.
   MathSciNet     CrossRef

9. S. Mardesic and J. Segal, Shape Theory. The inverse system approach, North-Holland Mathematical Library 26, North-Holland Publishing Co., Amsterdam-New York, 1982.
   MathSciNet

10. S. Mardesic and T. Watanabe, Approximate resolutions of spaces and mappings, Glas. Mat. Ser. III 24(44) (1989), 587-637.
    MathSciNet

11. C. R. F. Maunder, Algebraic Topology, Van Nostrand Reinhold Company, 1970.

12. T. Miyata, Uniform shape theory, Glas. Mat. Ser. III 29(49) (1994), 123-168.
    MathSciNet

13. T. Miyata, Fibrations in the category of absolute neighborhood retracts, Bull. Pol. Acad. Sci. Math. 55 (2007), 145-154.
    MathSciNet

14. T. Miyata, Strong pro-fibrations and ANR objects in pro-categories, mimeographic note, Department of Mathematics and Informatics, Kobe University.

15. T. Miyata, On the pro-categories of abelian categories, mimeographic note, Department of Mathematics and Informatics, Kobe University.

16. T. Miyata and T. Watanabe, Approximate resolutions of uniform spaces, Topology Appl. 113 (2001), 211-241.
    MathSciNet     CrossRef

17. T. Watanabe, Approximative shape. I. Basic notions, Tsukuba J. Math. 11 (1987), 17-59.
    MathSciNet