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24
From Concurrency to Algebraic Topology
, 2000
"... This paper is a survey of the new notions and results scattered in [13], [11] and [12]. Starting from a formalization of higher dimensional automata (HDA) by strict globular !categories, the construction of a diagram of simplicial sets over the threeobject small category gl ! + is exposed. Some of ..."
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Cited by 25 (9 self)
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This paper is a survey of the new notions and results scattered in [13], [11] and [12]. Starting from a formalization of higher dimensional automata (HDA) by strict globular !categories, the construction of a diagram of simplicial sets over the threeobject small category gl ! + is exposed. Some of the properties discovered so far on the corresponding simplicial homology theories are explained, in particular their links with geometric problems coming from concurrency theory in computer science.
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
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Cited by 20 (0 self)
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In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
Postnikov extensions for ring spectra
, 2006
"... We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum. ..."
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Cited by 13 (3 self)
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We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum.
Algebraic invariants for homotopy types
 Math. Proc. Cambridge Philos. Soc
, 1999
"... Abstract. We define a sequence of purely algebraic invariants – namely, classes in the Quillen cohomology of the Πalgebra π∗X – for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Πalgebra can be r ..."
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Abstract. We define a sequence of purely algebraic invariants – namely, classes in the Quillen cohomology of the Πalgebra π∗X – for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Πalgebra can be realized as the homotopy Πalgebra of a space. 1.
Realizing Commutative Ring Spectra as E∞ Ring Spectra
, 1999
"... We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are AndréQuillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for comput ..."
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Cited by 9 (2 self)
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We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are AndréQuillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for computing the homotopy type of mapping spaces between E∞ ring spectrum. The obstruction theory arises out of techniques of Dwyer, Kan, and Stover, and the main application here is to prove an analog of a theorem of Haynes Miller and the second author: the LubinTate spectra En are E∞ and the space of E∞ selfmaps has weakly contractible components.
Investigating The Algebraic Structure of Dihomotopy Types
, 2001
"... This presentation is the sequel of a paper published in GETCO'00 proceedings where a research program to construct an appropriate algebraic setting for the study of deformations of higher dimensional automata was sketched. This paper focuses precisely on detailing some of its aspects. The main ..."
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Cited by 8 (1 self)
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This presentation is the sequel of a paper published in GETCO'00 proceedings where a research program to construct an appropriate algebraic setting for the study of deformations of higher dimensional automata was sketched. This paper focuses precisely on detailing some of its aspects. The main idea is that the category of homotopy types can be embedded in a new category of dihomotopy types, the embedding being realized by the Globe functor. In this latter category, isomorphism classes of objects are exactly higher dimensional automata up to deformations leaving invariant their computer scientific properties as presence or not of deadlocks (or everything similar or related). Some hints to study the algebraic structure of dihomotopy types are given, in particular a rule to decide whether a statement/notion concerning dihomotopy types is or not the lifting of another statement/notion concerning homotopy types. This rule does not enable to guess what is the lifting of a given notion/statement, it only enables to make the verification, once the lifting has been found.
Stratified fibre bundles
, 2003
"... A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the ..."
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Cited by 4 (2 self)
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A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem. AMS SC: 55R55 (Fiberings with singularities); 55R65 (Generalizations of fiber spaces and bundles); 55R70 (Fiberwise topology); 55R10 (Fibre bundles); 18F15 (Abstract manifolds and fibre bundles); 54H15 (Transformation groups and semigroups); 57S05 (Topological properties of groups of homeomorphisms or diffeomorphisms).
COMPARING HOMOTOPY CATEGORIES
, 2006
"... Abstract. Given a suitable functor T: C → D between model categories, we define a long exact sequence relating the homotopy groups of any X ∈ C with those of TX, and use this to describe an obstruction theory for lifting an object G ∈ D to C. Examples include finding spaces with given homology or ho ..."
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Abstract. Given a suitable functor T: C → D between model categories, we define a long exact sequence relating the homotopy groups of any X ∈ C with those of TX, and use this to describe an obstruction theory for lifting an object G ∈ D to C. Examples include finding spaces with given homology or homotopy groups. A number of fundamental problems in algebraic topology can be described as measuring the extent to which a given functor T: C → D between model categories induces an equivalence of homotopy categories: more specifically, which objects (or maps) from D are in the image of T, and in how many different ways. For example:
Simplicial Structured Ring Spectra
, 1999
"... We examine the foundations of simplicial algebras in spectra over a simplicial operad. We are led to simplicial operads and simplicial algebras over simplicial operads because certain operads which are notoriously hard to work with { mainly the E1 operad { can be simplicially resolved by simpler pie ..."
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We examine the foundations of simplicial algebras in spectra over a simplicial operad. We are led to simplicial operads and simplicial algebras over simplicial operads because certain operads which are notoriously hard to work with { mainly the E1 operad { can be simplicially resolved by simpler pieces. Our main goals are to a build spectral sequence for computing spaces of maps between structured ring spectra, and to develop a DwyerKanStover style obstruction theory for deciding when a spectrum actually can be a structured ring spectrum. In this paper we work out some of the foundations of the homotopy theory of simplicial ring spectra over a simplicial operad. This is not a gratuitous act of generalization. Simplicial objects in any category are a standard mechanism for building the resolutions necessary for computations; this is how simplicial spectra arise. The simplicial operads arise as an answer to an immediate practical problem. If T is an E1 operad over the linear isometrie...