A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of is an isomorphism if Y is movable. Recall that is the full subcategory of consisting of inverse sequences in , the homotopy category of pointed connected CW complexes.
@article{bwmeta1.element.bwnjournal-article-fmv160i3p269bwm,
author = {Jerzy Dydak and Francisco Ruiz del Portal},
title = {Bimorphisms in pro-homotopy and proper homotopy},
journal = {Fundamenta Mathematicae},
volume = {159},
year = {1999},
pages = {269-286},
zbl = {0936.55004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv160i3p269bwm}
}
Dydak, Jerzy; Ruiz del Portal, Francisco. Bimorphisms in pro-homotopy and proper homotopy. Fundamenta Mathematicae, Tome 159 (1999) pp. 269-286. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv160i3p269bwm/
[00000] [B] H. J. Baues, Foundations of proper homotopy theory, Draft manuscript, Max-Planck-Institut für Math., 1992.
[00001] [Br] E. M. Brown, Proper homotopy theory in simplicial complexes, in: Topology Conference (Virginia Polytechnic Institute and State University), R. F. Dickmann Jr. and P. Fletcher (eds.), Lecture Notes in Math. 375, Springer, Berlin, 1974, 41-46.
[00002] [C-G] C. Casacuberta and S. Ghorbal, On homotopy epimorphisms of connective covers, preprint, 1997.
[00003] [D1] J. Dydak, The Whitehead and the Smale theorems in shape theory, Dissertationes Math. 156 (1979). | Zbl 0405.55010
[00004] [D2] J. Dydak, Epimorphism and monomorphism in homotopy, Proc. Amer. Math. Soc. 116 (1992), 1171-1173. | Zbl 0768.55007
[00005] [D-S1] J. Dydak and J. Segal, Shape Theory: An Introduction, Lecture Notes in Math. 688, Springer, Berlin, 1978. | Zbl 0401.54028
[00006] [D-S2] J. Dydak and J. Segal, Strong shape theory, Dissertationes Math. 192 (1981).
[00007] [Dy-R] E. Dyer and J. Roitberg, Homotopy-epimorphism, homotopy-monomorphism and homotopy-equivalences, Topology Appl. 46 (1992), 119-124. | Zbl 0760.55005
[00008] [Ed-H] D. A. Edwards and H. M. Hastings, Čech and Steenrod Homotopy Theories with Applications to Geometric Topology, Lecture Notes in Math. 542, Springer, Berlin, 1976. | Zbl 0334.55001
[00009] [En1] R. Engelking, General Topology, Heldermann, Berlin, 1989.
[00010] [F-T-W] F. T. Farrell, L. R. Taylor and J. B. Wagoner, The Whitehead theorem in the proper category, Compositio Math. 27 (1973), 1-23. | Zbl 0285.55011
[00011] [G] S. Ghorbal, Epimorphisms and monomorphisms in homotopy theory, PhD Thesis, Université Catholique de Louvain, 1996 (in French).
[00012] [H-R] P. Hilton and J. Roitberg, Relative epimorphisms and monomorphisms in homotopy theory, Compositio Math. 61 (1987), 353-367. | Zbl 0624.55005
[00013] [H-W] L. Hong and S. Wenhuai, Homotopy epimorphisms in homotopy pushbacks, Topology Appl. 59 (1994), 159-162. | Zbl 0830.55007
[00014] [M-S] S. Mardešić and J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.
[00015] [Mat] M. Mather, Homotopy monomorphisms and homotopy pushouts, Topology Appl. 81 (1997), 159-162. | Zbl 0892.55004
[00016] [Mo-P] M. A. Morón and F. R. Ruiz del Portal, On weak shape equivalences, ibid. 92 (1999), 225-236.
[00017] [Mu] G. Mukherjee, Equivariant homotopy epimorphisms, homotopy monomorphisms and homotopy equivalences, Bull. Belg. Math. Soc. 2 (1995), 447-461. | Zbl 0868.55003
[00018] [P] T. Porter, Proper homotopy theory, in: Handbook of Algebraic Topology, Elsevier Science, 1995, 127-167. | Zbl 1004.55004
[00019] [S] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
[00020] [Sp] S. Spież, A majorant for the family of all movable shapes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 615-620. | Zbl 0275.55026