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53
SemiAbelian Categories
, 2000
"... The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories ar ..."
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Cited by 37 (3 self)
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The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to "old" exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semiabelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar nonabelian structures. Mathematics Subject Classification: 18E10, 18A30, 18A32. Key words:...
Determination Of A Double Lie Groupoid By Its Core Diagram
 J. Pure Appl. Algebra
, 1992
"... In a double groupoid S, we show that there is a canonical groupoid structure on the set of those squares of S for which the two source edges are identities; we call this the core groupoid of S. The target maps from the core groupoid to the groupoids of horizontal and vertical edges of S are now base ..."
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Cited by 27 (14 self)
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In a double groupoid S, we show that there is a canonical groupoid structure on the set of those squares of S for which the two source edges are identities; we call this the core groupoid of S. The target maps from the core groupoid to the groupoids of horizontal and vertical edges of S are now basepreserving morphisms whose kernels commute, and we call the diagram consisting of the core groupoid and these two morphisms the core diagram of S. If S is a double Lie groupoid, and each groupoid structure on S satisfies a natural double form of local triviality, we show that the core diagram determines S and, conversely, that a locally trivial double Lie groupoid may be constructed from an abstractly given core diagram satisfying some natural additional conditions. In the algebraic case, the corresponding result includes the known equivalences between crossed modules, special double groupoids with special connection (Brown and Spencer), and cat 1 groups (Loday). These cases correspon...
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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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 !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 ...
Computations and homotopical applications of induced crossed modules
 J. Symb. Comp
"... We explain how the computation of induced crossed modules allows the computation of certain homotopy 2types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications. ..."
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Cited by 10 (8 self)
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We explain how the computation of induced crossed modules allows the computation of certain homotopy 2types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications.
Higher Hopf formulae for homology via Galois Theory, preprint math.AT/0701815
, 2007
"... and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case ..."
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Cited by 10 (3 self)
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and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A
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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Cited by 9 (0 self)
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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
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
Semistrict models of connected 3types and Tamsamani’s weak 3groupoids
, 2006
"... Homotopy 3types can be modelled algebraically by Tamsamani’s weak 3groupoids as well as, in the path connected case, by cat 2groups. This paper gives a comparison between the two models in the pathconnected case. This leads to two different semistrict algebraic models of connected 3types usin ..."
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Cited by 7 (3 self)
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Homotopy 3types can be modelled algebraically by Tamsamani’s weak 3groupoids as well as, in the path connected case, by cat 2groups. This paper gives a comparison between the two models in the pathconnected case. This leads to two different semistrict algebraic models of connected 3types using Tamsamani’s model. Both are then related to Gray groupoids.
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