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Effective generalized SeifertVan Kampen: how to calculate ΩX, preprint available at qalg/9710011
"... A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including highe ..."
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A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including higher order coherences as first investigated in [33]. A delooping machine is a specification of the extra homotopical structure carried by the loop space ΩX of a connected basepointed topological space X, exactly the structure allowing recovery of X by a “classifying space ” construction. The first level of structure is that the component set π0(ΩX) has a structure of group π1(X, x). Classically the SeifertVan Kampen theorem states that a pushout diagram of connected spaces gives rise to a pushout diagram of groups π1. The loop space construction ΩX with its delooping structure being the higherorder “topologized ” generalization of π1, an obvious question is whether a similar SeifertVan Kampen statement holds for ΩX. The aim of this paper is to describe the operation underlying pushout of spaces with loop space structure, answering the above question by giving a SeifertVan Kampen statement for delooping machinery. We work with Segal’s machine [28] [36]. Our SeifertVan
Resolutions of spaces by cubes of fibrations
 Hagen Germany T. Porter School of Mathematics University of Wales Bangor Bangor
, 1986
"... J.L. Loday has used wcubes of fibrations, where n is a nonnegative integer, in his study of spaces with finitely many nontrivial homotopy groups [4]. His main result is the construction of an algebraic category equivalent to the weak homotopy category of pathconnected spaces Z with TI^Z = 0 for ..."
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J.L. Loday has used wcubes of fibrations, where n is a nonnegative integer, in his study of spaces with finitely many nontrivial homotopy groups [4]. His main result is the construction of an algebraic category equivalent to the weak homotopy category of pathconnected spaces Z with TI^Z = 0 for /> w+1 [4, 1.7]. One step in
A Simplicial Description Of The Homotopy Category Of Simplicial Groupoids
, 2000
"... . In this paper we use Quillen's model structure given by DwyerKan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characteri ..."
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. In this paper we use Quillen's model structure given by DwyerKan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicial identities and give a simplicial description of the homotopy category of simplicial groupoids. Finally, we show loop and suspension functors in the pointed case. 1. Introduction 1.1. Summary. A wellknown and quite powerful context in which an abstract homotopy theory can be developed is supplied by a category with a closed model structure in the sense of Quillen [16]. The category Simp(Gp) of simplicial groups is a remarkable example of what a closed model category is, and the homotopy theory in Simp(Gp) developed by Kan [12] occurs as the homotopy theory associated to this closed model structure. According to the t...
DÉCALAGE AND KAN’S SIMPLICIAL LOOP GROUP FUNCTOR
"... Abstract. Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios an ..."
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Abstract. Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak homotopy equivalence for any bisimplicial set. In this paper we will give a new, elementary proof of this result. As an application, we will revisit Kan’s simplicial loop group functor G. We will give a simple formula for this functor, which is based on a factorization, due to Duskin, of Eilenberg and Mac Lane’s classifying complex functor W. We will give a new, short, proof of Kan’s result that the unit map for the adjunction G ⊣ W is a weak homotopy equivalence for reduced simplicial sets. 1.
DoldKan Type Theorem for ΓGroups
, 1998
"... Introduction \Gammaspaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on KanThurston theorem we show that any \Gammaspace is stably weak equivalent to a discrete \Gammagroup. By a wellknown theorem of DoldKan th ..."
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Introduction \Gammaspaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on KanThurston theorem we show that any \Gammaspace is stably weak equivalent to a discrete \Gammagroup. By a wellknown theorem of DoldKan the Moore normalization establishes the equivalence between the category of simplicial abelian groups and the category of chain complexes (see [DP]). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gammagroups. This construction is based on the notion of crosseffects of functors [BP], which is a generalizatin of the classical definition of Eilenberg and Mac Lane [EM] to the nonabelian setup. Finally a DoldKan type theorem for the category of \Gammagroups is proved. In abelian case our theorem claims that the category of abelian \Gammagroups is equivalent to the category of functors Ab\Omega , where\Om
THE DECOMPOSITION OF THE LOOP SPACE OF THE MOD 2 MOORE SPACE
"... Abstract. In 1979 Cohen, Moore, and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod p Moore space for primes p> 2 and used the results to find the best possible exponent for the homotopy groups of the spheres and for the Moore spaces ..."
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Abstract. In 1979 Cohen, Moore, and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod p Moore space for primes p> 2 and used the results to find the best possible exponent for the homotopy groups of the spheres and for the Moore spaces at such primes. The corresponding problems for p = 2 are still open. In this paper we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod 2 Moore space. The algebraic problems involved in determining detailed information about this factor are formidable, related to deep unsolved problems in the modular representation theory of the symmetric groups. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod 2 Moore space or to an improvement in the known bounds for the exponent of the 2torsion in the homotopy groups of spheres. 1.
COMBINATORIAL DESCRIPTION OF THE HOMOTOPY GROUPS OF WEDGE OF SPHERES
"... Abstract. In this paper, we give a combinatorial description of the homotopy groups of a wedge of spheres. This result generalizes that of J. Wu on the homotopy groups of a wedge of 2spheres. In particular, the higher homotopy groups of spheres are given as the centers of certain combinatorially de ..."
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Abstract. In this paper, we give a combinatorial description of the homotopy groups of a wedge of spheres. This result generalizes that of J. Wu on the homotopy groups of a wedge of 2spheres. In particular, the higher homotopy groups of spheres are given as the centers of certain combinatorially described groups with special generators and relations. 1.
The Linearisation Map in Algebraic KTheory
, 2005
"... Let X be a pointed connected simplicial set with loop group G. The linearisation map in Ktheory as defined by Waldhausen uses Gequivariant spaces. This paper gives an alternative description using presheaves of sets and abelian groups on the simplex category of X. In other words, the linearisation ..."
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Let X be a pointed connected simplicial set with loop group G. The linearisation map in Ktheory as defined by Waldhausen uses Gequivariant spaces. This paper gives an alternative description using presheaves of sets and abelian groups on the simplex category of X. In other words, the linearisation map is defined in terms of X only, avoiding the use of the less geometric loop group. The paper also includes a comparison of categorical finiteness with the more geometric notion of finite CW objects in cofibrantly generated model categories. The application to the linearisation map employs a model structure on the category of abelian group objects of retractive spaces over X.
Duality For Simple omegaCategories And Disks
"... A. Joyal [J] has introduced the category D of the socalled finite disks, and used it to define the concept of #category, a notion of weak #category. We introduce the notion of an #graph being composable (meaning roughly that 'it has a unique composite'), and call an #category simple if it is fr ..."
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A. Joyal [J] has introduced the category D of the socalled finite disks, and used it to define the concept of #category, a notion of weak #category. We introduce the notion of an #graph being composable (meaning roughly that 'it has a unique composite'), and call an #category simple if it is freely generated by a composable #graph. The category S of simple #categories is a full subcategory of the category, with strict #functors as morphisms, of all #categories. The category S is a key ingredient in another concept of weak #category, called protocategory [MM1], [MZ]. We prove that D and S are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories. In [MZ], this result is one of the tools used to show that the concept of #category and that of protocategory are equivalent in a suitable sense. We also prove that composable #graphs coincide with the #graphs of the form T # considered by M.Batanin [B], which were characterized by R. Street (as announced in [S]) and called `globular cardinals'. Batanin's construction, using globular cardinals, of the free #category on a globular set plays an important role in our paper. We give a selfcontained presentation of Batanin's construction that suits our purposes.
DOLDKAN TYPE THEOREMS FOR nTYPES OF SIMPLICITIAL COMMUTATIVE ALGEBRAS
, 1998
"... A functor from simplicial algebras to crossed ncubes is shown to be an embedding on a reflexive subcategory of the category of simplicial algebras that contains representatives for all n types. ..."
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A functor from simplicial algebras to crossed ncubes is shown to be an embedding on a reflexive subcategory of the category of simplicial algebras that contains representatives for all n types.