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Combinatorial descriptions of the homotopy groups of certain spaces
 Math. Proc. Camb. Philos. Soc
"... Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3sphere are combinatorially given. 1. ..."
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Cited by 28 (21 self)
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Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3sphere are combinatorially given. 1.
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:
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
Joins for (Augmented) Simplicial Sets
, 1998
"... We introduce a notion of join for (augmented) simnplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category \Delta. 1991 Math. Subj. Class.: 18G30 1 Introduction The theory of joins of (geomet ..."
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Cited by 4 (0 self)
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We introduce a notion of join for (augmented) simnplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category \Delta. 1991 Math. Subj. Class.: 18G30 1 Introduction The theory of joins of (geometric) simplicial complexes as given by Brown, [2], or Spanier, [13], reveals the join operation to be a basic geometric construction. It is used in the development of several areas of geometric topology (cf. Hudson, [8]) whilst also being applied to the basic properties of polyhedra relating to homology. The theories of geometric and abstract simplicial complexes run in a largely parallel way and when describing the theory, expositions often choose which aspect  abstract combinatorial or geometric  to emphasise at each step. Historically in algebraic topology geometric simplicial complexes, as tools, were largely replaced by CW complexes whilst the combinatorial abstract complex became pa...
On algebraic models for homotopy 3types
 J. Homotopy Relat. Struct
"... We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2. ..."
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Cited by 3 (0 self)
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We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2.
Homotopical Aspects of Commutative Algebras
, 2006
"... This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications. 1 ..."
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This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications. 1
School of Mathematics,
, 2008
"... We introduce a notion of join for (augmented) simplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial ..."
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We introduce a notion of join for (augmented) simplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial
Email: rgbea@algebra.us.esTo my grandfather Mariano and my grandmother Carmen. Acknowledgements
, 804
"... Luis Narváez Macarro ..."