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

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Jones D. W.

Instytut Matematyczny Polskiej Akademii Nauk

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

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1 Jones D. W.

1 1988

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