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32
Tensor products and homotopies for ωgroupoids and crossed complexes
, 2007
"... Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed comp ..."
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Cited by 43 (21 self)
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Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product involves nonabelian constructions related to the commutator calculus and the homotopy addition lemma. This monoidal closed structure is derived from that on the equivalent category of ωgroupoids where the underlying cubical structure gives geometrically natural definitions of tensor products and homotopies.
On finite induced crossed modules, and the homotopy 2type of mapping cones
 THEORY AND APPLICATIONS OF CATEGORIES
, 1995
"... Results on the niteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of ..."
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Cited by 21 (18 self)
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Results on the niteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of some nite crossed modules are given, using crossed complex methods.
A homotopy double groupoid of a Hausdorff space II: A van Kampen Theorem
 THEORY AND APPLICATIONS OF CATEGORIES
, 2005
"... This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2dimensional, localtoglobal ..."
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Cited by 20 (11 self)
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This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2dimensional, localtoglobal problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These results have recently been generalised to all dimensions by Philip Higgins.
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.
Computing Crossed Modules Induced By An Inclusion Of A Normal Subgroup, With Applications To Homotopy 2Types
, 1996
"... We obtain some explicit calculations of crossed Qmodules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2types and second homotopy modules of certain homotopy pushouts of maps of classifying ..."
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Cited by 17 (13 self)
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We obtain some explicit calculations of crossed Qmodules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups.
Theory and Applications of Crossed Complexes
, 1993
"... ... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for t ..."
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Cited by 15 (2 self)
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... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for the chain complex version. Given simplicial sets K 0 ; : : : ; K r , we discuss the rcube of homotopies induced on (K 0 \Theta : : : \Theta K r ) and show these form a coherent system. We introduce a definition of a double crossed complex, and of the associated total (or codiagonal) crossed complex. We introduce a definition of homotopy colimits of diagrams of crossed complexes. We show that the homotopy colimit of crossed complexes can be expressed as the
Homotopy types of strict 3groupoids
, 1988
"... It has been difficult to see precisely the role played by strict ncategories in the nascent theory of ncategories, particularly as related to ntruncated homotopy types of spaces. We propose to show in a fairly general setting that one cannot obtain all 3types by any reasonable realization functo ..."
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Cited by 13 (0 self)
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It has been difficult to see precisely the role played by strict ncategories in the nascent theory of ncategories, particularly as related to ntruncated homotopy types of spaces. We propose to show in a fairly general setting that one cannot obtain all 3types by any reasonable realization functor 1 from strict 3groupoids (i.e. groupoids in the sense of [20]). More precisely we show that one does not obtain the 3type of S 2. The basic reason is that the Whitehead bracket is nonzero. This phenomenon is actually wellknown, but in order to take into account the possibility of an arbitrary reasonable realization functor we have to write the argument in a particular way. We start by recalling the notion of strict ncategory. Then we look at the notion of strict ngroupoid as defined by Kapranov and Voevodsky [20]. We show that their definition is equivalent to a couple of other naturallooking definitions (one of these equivalences was left as an exercise in [20]). At the end of these first sections, we have a picture of strict 3groupoids having only one object and one 1morphism, as being equivalent to abelian monoidal objects (G, +) in the category of groupoids, such that (π0(G), +) is a group. In the case in question, this group will be π2(S 2) = Z. Then comes the main
Covering groups of nonconnected topological groups, and the monodromy groupoid of a topological group
, 1993
"... All spaces are assumed to be locally path connected and semilocally 1connected. Let X be a connected topological group with identity e, and let p: ˜ X → X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for a ..."
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Cited by 13 (10 self)
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All spaces are assumed to be locally path connected and semilocally 1connected. Let X be a connected topological group with identity e, and let p: ˜ X → X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ˜e in ˜ X with p˜e = e there is a unique structure of topological group on ˜ X such that ˜e is the
Free crossed resolutions of groups and presentations of modules of identities among relations
, 2008
"... ..."