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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 54 (23 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.
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 31 (17 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
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 31 (19 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.
Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional localtoglobal problems
 in Michiel Hazewinkel (ed.), Handbook of Algebra, volume 6, Elsevier
"... ..."
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 19 (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.
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 18 (2 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
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 18 (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
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 16 (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.
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 16 (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.
On Yetter’s invariant and an extension of the DijkgraafWitten invariant to categorical groups
 Theory Appl. Categ
"... We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the ..."
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Cited by 15 (0 self)
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We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined