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Simplicial T-complexes and crossed complexes: a non-abelian version of a theorem of Dold and Kan

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Rozprawy Matematyczne tom/nr w serii: 265 wydano: 1988
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Warianty tytułu
Abstrakty
EN

CONTENTS
Preface (by Ronald Brown)..................................................................5
Introduction..........................................................................................7
Preliminaries........................................................................................9
1. T-complexes and crossed complexes.............................................10
1.1. A groupoid structure for a T-complex..........................................12
1.2. The isomorphism theorem for a T-complex.................................16
1.3. Certain abelian groups associated to a T-complex......................20
1.4. The homomorphism δ..................................................................22
1.5. The groupoid action....................................................................24
1.6. The interchange law between the isomorphisms h and φ............25
1.7. The crossed complex associated to a T-complex........................27
1.8. Some technical results................................................................31
1.9. The isomorphism theorem for T-complexes.................................33
1.10. The T-complex addition lemma..................................................35
1.11. A functor from crossed complexes to T-complexes....................36
1.12. The equivalence of categories..................................................40
2. Special filtered Kan complexes.......................................................41
2.1. Introduction.................................................................................41
2.2. Definitions and examples............................................................42
2.3. The crossed complex associated to the T-complex ϱ(X).............49
3. Simplicial groups............................................................................49
3.1. Group T-complexes.....................................................................50
3.2. Special simplicial groups over a groupoid...................................53
3.3. A filtration of a simplicial group....................................................54
4. Miscellaneous................................................................................54
References........................................................................................58
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 265
Liczba stron
58
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCLXV
Daty
wydano
1988
Twórcy
autor
  • Department of Pure Mathematics, University College of North Wales, Bangor, Gwynedd L57 2UW, United Kingdom
Bibliografia
  • M.K. Dakin, 1977, Kan complexes and multiple groupoid structures, Ph. D. thesis, University of Wales.
  • R. Brown, 1984, Some non-abelian methods in homotopy theory and homological algebra, Categorical Topology: Proc. Conf. Toledo, Ohio, 1983 (ed. H. L. Bentley et al.), Heldermann Verlag. Berlin (1984), 108-146.
  • R. Brown and P.J. Higgins, 1981, Colimit theorems for relative homotopy groups, J. Pure Appl. Alg. 22, 11-41.
  • P. Glenn, 1982, Realisation of cohomology classes in arbitrary exact categories, J. Pure Appl. Alg. 25. 33-105.
  • D.W. Jones, 1988, Poly-T-complexes, University of Wales Ph. D. Thesis (1984), published as A general theory of polyhedral sets and the corresponding T-complexes, Diss. Math. 266.
  • G.N. Tie, 1987, T-groupoids, W, and a Dold-Kan theorem for crossed complexes, Ph. D. Thesis, State University of New York at Buffalo.
  • [1] A. L. Blakers, Some relations between homology and homotopy groups, Ann. of Math. 49 (1948), 428-461.
  • [2] R. Brown, Elements of Modern Topology, McGraw-Hill, New York 1968.
  • [3] R. Brown, Fibrations of groupoids, J. Algebra 15 (1970), 103-132.
  • [4] R. Brown and P. J. Higgins, Sur les complexes croisés d'homotopie de quelques espaces filtrées, C.R. Acad. Sci. Paris Sér. A (1978), 91-93.
  • [5] R. Brown and P. J. Higgins, Sur les complexes croisés, ω-groupoides, T-complexes et ∞-groupoides, C.R. Acad. Sci. Paris Sér. A (1977), 997-999.
  • [6] R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978), 193 -212.
  • [7] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Topologie Géom. Différentielle 17 (1976), 343-362.
  • [8] M. K. Dakin, Kan complexes and multiple groupoid structures, Ph. D. thesis, University of Wales, 1977.
  • [9] S. T. Hu, Homotopy Theory, Academic Press, New York 1959.
  • [10] J. P. May, Simplicial Objects in Algebraic Topology, Van Nostrand, Princeton, 1967.
  • [11] J. H. C. Whitehead, Combinatorial homotopy 11, Bull. Amer. Math. Soc. 55 (1949), 453-496.
Języki publikacji
EN
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Identyfikator YADDA
bwmeta1.element.zamlynska-8d0c18b5-0388-4275-a854-70409f42fffb
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ISBN
83-01-08026-4
ISSN
0012-3862
Kolekcja
DML-PL
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