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28
Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 22 (6 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
Double Loop Spaces, Braided Monoidal Categories and Algebraic 3Type of Space
 Math
, 1997
"... We show that the nerve of a braided monoidal category carries a natural action of a simplicial E2operad and is thus up to group completion a double loop space. Shifting up dimension twice associates to each braided monoidal category a 1reduced lax 3category whose nerve realizes an explicit double ..."
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Cited by 20 (2 self)
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We show that the nerve of a braided monoidal category carries a natural action of a simplicial E2operad and is thus up to group completion a double loop space. Shifting up dimension twice associates to each braided monoidal category a 1reduced lax 3category whose nerve realizes an explicit double delooping whenever all cells are invertible. We deduce that lax 3groupoids are algebraic models for homotopy 3types. Introduction The concept of braiding as a refinement of symmetry is the starting point of a rich interplay between geometry (knot theory) and algebra (representation theory). The underlying structure of a braided monoidal category reveals an interest of its own in that it encompasses two at first sight different geometrical objects : double loop spaces and homotopy 3types. The link to double loop spaces was pointed out by J. Stasheff [38] and made precise by Z. Fiedorowicz [15], who proves that double loop spaces may be characterized (up to group completion) as algebras o...
Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Cited by 19 (2 self)
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
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.
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 : : : : : : : : : : : : : ...
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
Applications of Peiffer pairings in the Moore complex of a simplicial group
, 1998
"... Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, dn NGn ; of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2crossed modules and quadratic modules are discussed ..."
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Cited by 10 (6 self)
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Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, dn NGn ; of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2crossed modules and quadratic modules are discussed. A. M. S. Classication: 18G30, 55U10, 55P10. Introduction Simplicial groups occupy a place somewhere between homological group theory, homotopy theory, algebraic Ktheory and algebraic geometry. In each sector they have played a signicant part in developments over quite a lengthy period of time and there is an extensive literature on their homotopy theory. In homotopy theory itself, they model all connected homotopy types and allow analysis of features of such homotopy types by a combination of group theoretic methods and tools from combinatorial homotopy theory. Simplicial groups have a natural structure of Kan complexes and so are potentially models for weak innity categories. They d...
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
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."