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Homotopical excision, and Hurewicz theorems, for ncubes of spaces
 Proc. London Math. Soc
, 1987
"... The fact that the relative homotopy groups do not satisfy excision makes the computation of absolute homotopy groups difficult in comparison with homology groups. The failure of excision is measured by triad homotopy groups πn(X; A, B), with n � 3 (for n = 2, this gives a based set), which fit into ..."
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Cited by 18 (9 self)
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The fact that the relative homotopy groups do not satisfy excision makes the computation of absolute homotopy groups difficult in comparison with homology groups. The failure of excision is measured by triad homotopy groups πn(X; A, B), with n � 3 (for n = 2, this gives a based set), which fit into an exact sequence.
Pasting Schemes for the Monoidal Biclosed Structure on
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a gen ..."
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. 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. Contents 1 Introduction 3 2 Cubes and cubical sets 5 2.1 Cubes combinatorially : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 A model category for cubes : : : : : : : : : : : : : : : : : : : : : 6 2.3 Generating the model category for cubes : : : : : : : : : : : : : : 7 2.4 Cubical sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.5 Duality : : : : : : : : : : : : : ...
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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Cited by 14 (0 self)
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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 ...
Group objects and internal categories
, 2002
"... ABSTRACT. Algebraic structures such as monoids, groups, and categories can be formulated within a category using commutative diagrams. In many common categories these reduce to familiar cases. In particular, group objects in Grp are abelian groups, while internal categories in Grp are equivalent bot ..."
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ABSTRACT. Algebraic structures such as monoids, groups, and categories can be formulated within a category using commutative diagrams. In many common categories these reduce to familiar cases. In particular, group objects in Grp are abelian groups, while internal categories in Grp are equivalent both to group objects in Cat and to crossed modules of groups. In this exposition we give an elementary introduction to some of the key concepts in this area. This expository essay was written in the winter of 19992000, early in the course of my PhD research, and has since been updated with supplementary references. I hope you will find it useful. I am indebted to my supervisor, Professor Tim Porter, for his help in preparing this article. 1. GROUPS WITHIN A CATEGORY Let C be a category with finite products. For this it is necessary and sufficient that C have pairwise products (i.e. for any 2 objects C, D ∈ Ob(C), there is a product C × D) and a terminal object, which we shall denote by 1. Examples of suitable categories include Set, Grp, Top and Ab. Let G be an object of C. Then G × G is also an object of C. Suppose we can find a morphism m: G × G → G such that the diagram G × G × G idG×m
Mikhailov: A colimit of classifying spaces
"... We recall a grouptheoretic description of the first nonvanishing homotopy group of a certain (n+1)ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu’s grouptheoretic description of the ho ..."
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Cited by 7 (3 self)
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We recall a grouptheoretic description of the first nonvanishing homotopy group of a certain (n+1)ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu’s grouptheoretic description of the homotopy groups of a 2sphere. 1
THREE CROSSED MODULES
, 812
"... We introduce the notion of 3crossed module, which extends the notions of 1crossed module (Whitehead) and 2crossed module (Conduché). ..."
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We introduce the notion of 3crossed module, which extends the notions of 1crossed module (Whitehead) and 2crossed module (Conduché).
AND
, 1985
"... A generalised tensor product G 0 H of groups G, H has been introduced by R. Brown and J.L. Loday in [3,4]. It arises in applications in homotopy theory of a generalised Van Kampen theorem. The reason why G 0 H does not necessarily reduce to GUh Oz Huh, the usual tensor product over Z of the abelian ..."
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A generalised tensor product G 0 H of groups G, H has been introduced by R. Brown and J.L. Loday in [3,4]. It arises in applications in homotopy theory of a generalised Van Kampen theorem. The reason why G 0 H does not necessarily reduce to GUh Oz Huh, the usual tensor product over Z of the abelianisations, is that it is assumed that G acts on H (on the left) and H acts on G (on the left), and these actions are taken into account in the definition of the tensor product. A group G acts on itself by conjugation (“g = hgh ‘) and so the tensor square GO G is always defined. Further, the commutator map G x G + G induces a homomorphism of groups K: G 0 G + G, sending g @ h to [g, h] =ghg‘hl. We write J,(G) for Ker IC; its topological interest is the formula [3,4] qSK(G, l)=J,(G).
(PhD supervisor) Acknowledgements
"... This thesis is my own work, except where stated in the text, and has not been submitted for any other degree in this or any other institution. ..."
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This thesis is my own work, except where stated in the text, and has not been submitted for any other degree in this or any other institution.