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The Convenient Setting of Global Analysis
, 1997
"... ichor i Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I Calculus of Smooth Mappings . . . . . . . . . . . . . . . . . . . . 4 1. Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1a. Completeness . . . . . . . . . . . . . . ..."
Abstract

Cited by 199 (48 self)
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ichor i Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I Calculus of Smooth Mappings . . . . . . . . . . . . . . . . . . . . 4 1. Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1a. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1b. Smooth Mappings and the Exponential Law . . . . . . . . . . . . . 17 2. The c 1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . 29 3. Uniform Boundedness Principles and Multilinearity . . . . . . . . . . 47 3a. Some Spaces of Smooth Functions . . . . . . . . . . . . . . . . . 59 Historical remarks on the development of smooth calculus . . . . . . . . . 63 CHAPTER II Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . 68 5. D
A model category for the homotopy theory of concurrency
 Homology, Homotopy and Applications
"... Abstract. We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are Shomotopy equivalent. This result provides an interpretation of the notion ..."
Abstract

Cited by 37 (13 self)
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Abstract. We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are Shomotopy equivalent. This result provides an interpretation of the notion of Shomotopy equivalence in the framework of model
ON DAVISJANUSZKIEWICZ HOMOTOPY TYPES II; COMPLETION AND GLOBALISATION
, 2009
"... For any finite simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CWcomplexes whose integral cohomology rings are isomorphic to the StanleyReisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be hom ..."
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Cited by 19 (6 self)
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For any finite simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CWcomplexes whose integral cohomology rings are isomorphic to the StanleyReisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to the original examples. It is therefore natural to investigate the extent to which the homotopy type of a space X is determined by such a cohomology ring. Having analysed this problem rationally in Part I, we here consider it prime by prime, and utilise Lannes’ T functor and BousfieldKan type obstruction theory to study the pcompletion of X. We find the situation to be more subtle than for rationalisation, and confirm the uniqueness of the completion whenever K is a join of skeleta of simplices. We apply our results to the global problem by appealing to Sullivan’s arithmetic square, and deduce integral uniqueness whenever the StanleyReisner algebra is a complete intersection.
Colimits, StanleyReisner Algebras, and Loop Spaces
, 2003
"... We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: rightangled Artin and Coxeter groups (and their complex analogues, which ..."
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Cited by 4 (3 self)
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We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: rightangled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); StanleyReisner algebras and coalgebras; Davis and Januszkiewicz’s spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend wellknown results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops ΩDJ(K). We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for ΩDJ(K) for an arbitrary complex K, and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.
Categorical aspects of toric topology
"... Abstract. We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces XK, such as momentangle complexes ZK, quasitoric manifolds M, and DavisJanuszkiewicz spaces DJ(K). First we exhibit ever ..."
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Cited by 3 (0 self)
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Abstract. We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces XK, such as momentangle complexes ZK, quasitoric manifolds M, and DavisJanuszkiewicz spaces DJ(K). First we exhibit every XK as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as algebraic models for XK. Such models encode many standard algebraic invariants, and their existence is assured by the Quillen structure. We provide several illustrative calculations, often over the rationals, including proofs that quasitoric manifolds (and various generalisations) are rationally formal; that the rational Pontrjagin ring of the loop space ΩDJ(K) is isomorphic to the quadratic dual of the StanleyReisner algebra Q[K] for flag complexes K; and that DJ(K) is coformal precisely when K is flag. We conclude by describing algebraic models for the loop space ΩDJ(K) for any complex K, which mimic our previous description as a homotopy colimit of topological monoids. 1.
A CONVENIENT CATEGORY FOR DIRECTED HOMOTOPY Dedicated to Walter Tholen
"... at the occasion of his sixtieth birthday ..."