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A general theory of polyhedral sets and the corresponding T-complexes

Seria
Rozprawy Matematyczne tom/nr w serii: 266 wydano: 1988
Zawartość
Warianty tytułu
Abstrakty
EN
Preface
This paper is essentially David Jones' 1984 University of Wales Ph. D. Thesis, "Poly-T-complexes". It is published concurrently with Asley, 1988.
The main aim is to find a setting for the most general kinds of geometrically defined compositions. Thus it comes under the slogan: "Find an algebraic inverse to subdivision". In the background is the Generalised Van Kampen Theorem, whose proof uses in an essential way general compositions of cubes. An even older background is the idea in topology of cycles in a space as some kind of composition of small pieces.
The technicalities of even this stage of the theory mean that a number of problems are left unresolved. These are summarised in David Jones' final chapter. However the fundamental nature of the basic ideas should make this paper a useful stimulus to further work.
N. Ashley, 1988, Simplicial T-complexes: a non-abelian version of a theorem of Dold and Kan, Diss. Math. 265.
Ronald Brown, Bangor, 1988
EN

CONTENTS
Preface......................................................................................................5
Introduction...............................................................................................6
1. A class of model categories.................................................................10
1.1 Categories with models......................................................................10
1.2. Cone-complexes...............................................................................12
1.3. Three standard constructions..........................................................14
1.4. Marked cone-complexes and the category Poly...............................16
1.5. Consequences of the marked face structure of a polycell................19
1.6. Subcategories of Poly appropriate as model categories..................23
2. Posets and shellability.........................................................................26
2.1. Shelling............................................................................................26
2.2. The class ET of model categories....................................................29
2.3. Face posets of S-polycells...............................................................32
2.4. The equivalence SPoly → SPos.......................................................36
3. Equivalences of categories of T-complexes.........................................40
3.1. T-complexes.....................................................................................40
3.2. The isomorphism ∆TC → ∆₁TC........................................................42
3.3. Structures and collapsing.................................................................45
3.4. Particular collapses in Sd X and VX..................................................47
3.5. The collapse $A(∆^n)$ in Sd $∆^n - p ∆^n$.......................................55
3.6. The functor $e_ℳ$, from ∆₁T-complexes to ℳ T-complexes...........62
3.7. The natural equivalence $r_ℳ ∘ e_ℳ ≃1$.....................................63
3.8. The equivalence of categories.........................................................65
4. Degeneracy structures in ℳ T-complexes...........................................67
4.1. An approach to degeneracy structures in ℳ T-complexes...............68
4.2. Pseudocylinder structures on SC-complexes...................................70
4.3. Rectifiers on pseudocylinders..........................................................74
4.4. Degenerate elements in an ℳ T-complex........................................77
4.5. Functors between categories of T-complexes..................................83
4.6. Suggested proof of Claims 5.4 and 5.10..........................................86
5. Comments and possibilities for further work........................................92
Appendix. S-shellability of cone-complexes.............................................98
Glossary of symbols..............................................................................106
References...........................................................................................109
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 266
Liczba stron
111
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCLXVI
Daty
wydano
1988
Twórcy
  • Department of Pure Mathematics, University College of North Wales, Bangor, Gwynedd L57 2UW, United Kingdom
Bibliografia
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  • [3] N. Ashley, T-complexes and crossed complexes, Ph. D. thesis, University of Wales, 1978 published as Simplicial T-complexes and crossed complexes: a non-abelian version of a theorem of Dold and Kan, Diss. Math. 165 (1988).
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Języki publikacji
EN
Uwagi
Errata
Page, line: 5⁴
For: Asley, 1988
Read: Ashley, 1988
Identyfikator YADDA
bwmeta1.element.zamlynska-5249ebed-3daf-4fca-866d-334e4e45afee
Identyfikatory
ISBN
83-01-08117-1
ISSN
0012-3862
Kolekcja
DML-PL
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