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Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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Cited by 18 (7 self)
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
Free crossed resolutions of groups and presentations of modules of identities among relations
, 2008
"... ..."
Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using ..."
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Cited by 10 (7 self)
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Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.
On the Twisted Cobar Construction
 Math. Proc. Cambridge Philos. Soc
, 1997
"... this paper is the extension of this result to the case of twisted coefficients given by ..."
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Cited by 10 (4 self)
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this paper is the extension of this result to the case of twisted coefficients given by
Crossed complexes, and free crossed resolutions for amalgamated sums and HNNextensions of groups
 Georgian Math. J
, 1999
"... Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the p ..."
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Cited by 7 (6 self)
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Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating nonabelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNNextensions of groups, and so obtain computations of higher homotopical syzygies in these cases. 1
Free crossed resolutions from simplicial resolutions with given CW Basis
, 1998
"... In this paper, we examine the relationship between a CW basis for a free simplicial group and methods of freely generating the corresponding crossed complex. Attention is concentrated on the case of resolutions, thus comparing free simplicial resolutions with crossed resolutions of a group. A. M. ..."
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Cited by 3 (3 self)
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In this paper, we examine the relationship between a CW basis for a free simplicial group and methods of freely generating the corresponding crossed complex. Attention is concentrated on the case of resolutions, thus comparing free simplicial resolutions with crossed resolutions of a group. A. M. S. Classication: 18D35, 18G30, 18G50, 18G55, 20F05, 57M05. Introduction When J.H.C. Whitehead wrote his famous papers on \Combinatorial Homotopy", [25], it would seem that his aim was to produce a combinatorial, and thus potentially constructive and computational, approach to homotopy theory, analogous to the combinatorial group theory developed earlier by Reidemeister and others. In those papers, he introduced CWcomplexes and also the algebraic `gadgets' he called homotopy systems, and which are now more often called free crossed complexes, [5], or totally free crossed chain complexes, [3]. Another algebraic model for a (connected) homotopy type is a simplicial group and again, there, one...
NORMALISATION FOR THE FUNDAMENTAL CROSSED COMPLEX OF A SIMPLICIAL SET
, 2007
"... Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. Th ..."
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Cited by 2 (2 self)
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Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.
Crossed complexes, free crossed resolutions and graph products of groups’, (submitted
"... The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating nonabelian extensions. We show ..."
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Cited by 1 (1 self)
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The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating nonabelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions of graph products of groups, and so obtain computations of higher homotopical syzygies in this case. 1
Covering morphisms of crossed complexes and of cubical omegagroupoids are closed under tensor product
, 2010
"... The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with t ..."
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The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical ωgroupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.
Scategories, Sgroupoids, Segal categories and quasicategories
, 2008
"... The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in ..."
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The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, [26], or any equivalent text if one can be found! What do the notes set out to do? “Aims and Objectives! ” or should it be “Learning Outcomes”? • To revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today’s resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; • To introduce Segal categories and various other tools used by the NiceToulouse group of abstract homotopy theorists and link them into some of the older ideas; • To introduce Joyal’s quasicategories, (previously called weak Kan complexes but I agree with André that his nomenclature is better so will adopt it) and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and myself; • To ask lots of questions of myself and of the reader. The notes include some material from the ‘Cubo ’ article, [35], which was itself based on notes for a course at the Corso estivo Categorie e Topologia in 1991, but the overlap has been kept as small as is feasible as the purpose and the audience of the two sets of notes are different and the abstract homotopy theory has ‘moved on’, in part, to try the new methods out on those same ‘old ’ problems and to attack new ones as well. As usual when you try to specify ‘learning outcomes ’ you end up asking who has done the learning, the audience? Perhaps. The lecturer, most certainly! 1