Results 1  10
of
232
Modeling Concurrency with Geometry
"... The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer ..."
Abstract

Cited by 150 (13 self)
 Add to MetaCart
The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer one home over the other? We identify dimension as the culprit: 1dimensional automata are skeletons permitting only interleaving concurrency, whereas true nfold concurrency resides in transitions of dimension n. The truly concurrent automaton dual to a schedule is not a skeletal distributive lattice but a solid one. We introduce true nondeterminism and define it as monoidal homotopy; from this perspective nondeterminism in ordinary automata arises from forking and joining creating nontrivial homotopy. The automaton dual to a poset schedule is simply connected whereas that dual to an event structure schedule need not be, according to monoidal homotopy though not to group homotopy. We conclude with a formal definition of higher dimensional automaton as an ncomplex or ncategory, whose two essential axioms are associativity of concatenation within dimension and an interchange principle between dimensions.
The type of the classifying space for a family of subgroups
 J. Pure Appl. Algebra
"... We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the GCWversion EF(G) and the numerable Gspace version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact su ..."
Abstract

Cited by 108 (31 self)
 Add to MetaCart
We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the GCWversion EF(G) and the numerable Gspace version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra, for the FarrellJones Conjecture about the algebraic Kand Ltheory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.
Lecture notes on motivic cohomology
 of Clay Mathematics Monographs. American Mathematical Society
, 2006
"... From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by ..."
Abstract

Cited by 70 (2 self)
 Add to MetaCart
From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by
THE SPECTRAL SEQUENCE RELATING ALGEBRAIC KTHEORY TO MOTIVIC COHOMOLOGY
"... The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilins ..."
Abstract

Cited by 65 (6 self)
 Add to MetaCart
The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilinson [Be] as a natural analogue of the AtiyahHirzebruch spectral sequence from the singular cohomology to the topological Ktheory of a topological space. The expectation of such a spectral sequence has provided much of the impetus for the development of motivic cohomology (e.g., [B1], [V2]) and should facilitate many computations in algebraic Ktheory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [BL]. Our construction depends crucially upon the main result of [BL], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative Ktheory of coherent sheaves on standard simplices over F (recalled as Theorem 5.5 below). A major step in generalizing the work of Bloch and Lichtenbaum is our reinterpretation of their spectral sequence in terms of the “topological filtration ” on the Ktheory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ωprespectra K( ∆ • ) = K 0 ( ∆ • ) ← − K 1 ( ∆ • ) ← − K 2 ( ∆ • ) ← − · · · Thus, even in the special case in which X equals SpecF, we obtain a much clearer understanding of the BlochLichtenbaum spectral sequence which is essential for purposes of generalization. Following this reinterpretation, we proceed using techniques introduced by V. Voevodsky in his study of motivic cohomology. In order to do this, we provide an equivalent formulation of Ktheory spectra associated to coherent sheaves on X with conditions on their supports K q ( ∆ • × X) which is functorial in X. We then Partially supported by the N.S.F. and the N.S.A.
Connected simple systems and the Conley index of isolated invariant sets
 Trans. Amer. Math. Soc
, 1985
"... Abstract. The object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows. Simple, explicit formulae are derived for the maps which play a central role in the theory. The presentation is selfcontained. ..."
Abstract

Cited by 56 (2 self)
 Add to MetaCart
(Show Context)
Abstract. The object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows. Simple, explicit formulae are derived for the maps which play a central role in the theory. The presentation is selfcontained.
Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
Abstract

Cited by 50 (3 self)
 Add to MetaCart
(Show Context)
A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(Z)
, 2006
"... Let F ⊆ SL2(Z) be a finite subgroup (necessarily isomorphic to one of Z2, Z3, Z4, or Z6), and let F act on the irrational rotational algebra Aθ via the restriction of the canonical action of SL2(Z). Then the crossed product Aθ ⋊α F and the fixed point algebra AF θ are AF algebras. The same is true ..."
Abstract

Cited by 33 (12 self)
 Add to MetaCart
Let F ⊆ SL2(Z) be a finite subgroup (necessarily isomorphic to one of Z2, Z3, Z4, or Z6), and let F act on the irrational rotational algebra Aθ via the restriction of the canonical action of SL2(Z). Then the crossed product Aθ ⋊α F and the fixed point algebra AF θ are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z2 on any simple ddimensional noncommutative torus AΘ. Along the way, we prove a number of general results which should have useful applications in other situations.
Ganea's conjecture on LusternikSchnirelmann category
, 1997
"... A series of complexes Qp indexed by all primes p is constructed with catQp = 2 and catQpSn = 2 for either n 2 or n = 1 and p = 2. This disproves Ganea’s conjecture ..."
Abstract

Cited by 30 (7 self)
 Add to MetaCart
A series of complexes Qp indexed by all primes p is constructed with catQp = 2 and catQpSn = 2 for either n 2 or n = 1 and p = 2. This disproves Ganea’s conjecture
Homotopy type of symplectomorphism groups
 of S 2 ×S 2 , Geom. Topol. 6 (2002), 195–218 (electronic). MR1914568
"... Let M be S 2 × S 2. M carries a family of symplectic forms ωλ, where λ ≥ 0 determines the cohomology class [ωλ]. This paper calculates the homotopy type of the group Gλ of symplectomorphisms of (M, ωλ) when 0 < λ ≤ 1. We show that if λ is in this range Gλ contains two finite dimensional Lie group ..."
Abstract

Cited by 25 (5 self)
 Add to MetaCart
(Show Context)
Let M be S 2 × S 2. M carries a family of symplectic forms ωλ, where λ ≥ 0 determines the cohomology class [ωλ]. This paper calculates the homotopy type of the group Gλ of symplectomorphisms of (M, ωλ) when 0 < λ ≤ 1. We show that if λ is in this range Gλ contains two finite dimensional Lie groups that generate its homotopy. A key step in this work is to calculate the mod 2 homology of Gλ. Although this homology has a finite number of generators with respect to the Pontryagin product, it is unexpected large because it contains a free noncommutive ring with 3 generators. Our arguments involve a study of the space of ωλcompatible almost complex structures on M. 1
Directed Homotopy Theory, I. The Fundamental Category
 TOPOL. GÉOM. DIFFÉR. CATÉG
, 2001
"... ..."
(Show Context)