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48
Combinatorial descriptions of the homotopy groups of certain spaces
 Math. Proc. Camb. Philos. Soc
"... Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3sphere are combinatorially given. 1. ..."
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Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3sphere are combinatorially given. 1.
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Simplicial cohomology with coefficients in symmetric categorical groups
, 2004
"... In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)}n≥0, which allows us to give a representation theorem for our cohomol ..."
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In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)}n≥0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−,n) is right adjoint to the functor ℘n, where℘n(X•) is defined as the fundamental groupoid of the nloop complex � n (X•). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with πi(Y) = 0foralli = n, n + 1andn ≥ 3; and also we obtain a classification theorem for those spaces: [−,Y] ∼ = H n (−,℘n(Y)).
Pasting Schemes for the Monoidal Biclosed Structure on ωCat
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on ωcategories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on ωgroupoids. Immediate consequences ..."
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Cited by 18 (0 self)
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on ωcategories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on ωgroupoids. Immediate consequences are a general and uniform definition of higher dimensional lax natural transformations, and a nice and transparent description of the corresponding internal homs. Further consequences will be in the development of a theory for weak ncategories, since both tensor products and lax structures are crucial in this.
Applications of Peiffer pairings in the Moore complex of a simplicial group
, 1998
"... Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, dn NGn ; of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2crossed modules and quadratic modules are discussed ..."
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Cited by 16 (8 self)
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Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, dn NGn ; of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2crossed modules and quadratic modules are discussed. A. M. S. Classication: 18G30, 55U10, 55P10. Introduction Simplicial groups occupy a place somewhere between homological group theory, homotopy theory, algebraic Ktheory and algebraic geometry. In each sector they have played a signicant part in developments over quite a lengthy period of time and there is an extensive literature on their homotopy theory. In homotopy theory itself, they model all connected homotopy types and allow analysis of features of such homotopy types by a combination of group theoretic methods and tools from combinatorial homotopy theory. Simplicial groups have a natural structure of Kan complexes and so are potentially models for weak innity categories. They d...
Lie Groups and pCompact Groups
, 1998
"... A pcompact group is the homotopical ghost of a compact Lie group; it is the residue that remains after the geometry and algebra have been stripped away. This paper sketches the theory of pcompact groups, with the intention of illustrating the fact that many classical structural properties of compa ..."
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Cited by 13 (1 self)
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A pcompact group is the homotopical ghost of a compact Lie group; it is the residue that remains after the geometry and algebra have been stripped away. This paper sketches the theory of pcompact groups, with the intention of illustrating the fact that many classical structural properties of compact Lie groups depend only on homotopy theoretic considerations.
A COHOMOLOGICAL DESCRIPTION OF CONNECTIONS AND CURVATURE OVER POSETS
"... Abstract. What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theorie ..."
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Abstract. What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general nonAbelian, of a poset with values in a group G. Interpreting a 1–cocycle as a principal bundle, a connection turns out to be a 1–cochain associated in a suitable way with this 1–cocycle; the curvature of a connection turns out to be its 2–coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into G. We discuss holonomy and prove an analogue of the AmbroseSinger theorem. 1.
The functor A min on plocal spaces
"... Abstract. In a previous paper, the authors gave the finest functorial decomposition of the loop suspension of a ptorsion suspension. The purpose of this paper is to generalize this theorem to arbitrary plocal path connected spaces. 1. ..."
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Abstract. In a previous paper, the authors gave the finest functorial decomposition of the loop suspension of a ptorsion suspension. The purpose of this paper is to generalize this theorem to arbitrary plocal path connected spaces. 1.
Effective generalized SeifertVan Kampen: how to calculate ΩX
, 1997
"... 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 ..."
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Cited by 7 (4 self)
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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
Secondary homotopy groups
 Preprint of the MaxPlanckInstitut für Mathematik MPIM200636, http://arxiv.org/abs/math.AT/0604029
, 2005
"... Abstract. Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n − 1)connected (n + 1)types for n ≥ 0. ..."
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Abstract. Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n − 1)connected (n + 1)types for n ≥ 0.