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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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Ashley N.

Instytut Matematyczny Polskiej Akademii Nauk

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

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1 Ashley N.

1 1988

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