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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.
NORMALISATION FOR THE FUNDAMENTAL CROSSED COMPLEX OF A SIMPLICIAL SET
, 2007
"... Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. Th ..."
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Cited by 2 (2 self)
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Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.
THE LOOP GROUP AND THE COBAR CONSTRUCTION
, 903
"... Abstract. We prove that for any 1reduced simplicial set X, Adams ’ cobar construction ΩCX on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan loop group GX. In order to prove this result, we extend the definition of the cobar constr ..."
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Cited by 2 (1 self)
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Abstract. We prove that for any 1reduced simplicial set X, Adams ’ cobar construction ΩCX on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan loop group GX. In order to prove this result, we extend the definition of the cobar construction and actually obtain the existence of such a strong deformation retract for all 0reduced simplicial sets.
unknown title
, 2006
"... Abstract. We give a small functorial algebraic model for the 2stage Postnikov section of the Ktheory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors. ..."
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Abstract. We give a small functorial algebraic model for the 2stage Postnikov section of the Ktheory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.
unknown title
, 2006
"... Abstract. We give a small functorial algebraic model for the 2stage Postnikov section of the Ktheory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors. ..."
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Abstract. We give a small functorial algebraic model for the 2stage Postnikov section of the Ktheory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.
unknown title
, 2007
"... Abstract. We give a small functorial algebraic model for the 2stage Postnikov section of the Ktheory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors. ..."
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Abstract. We give a small functorial algebraic model for the 2stage Postnikov section of the Ktheory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.
unknown title
, 2008
"... Possible connections between whiskered categories and groupoids, many object Lie algebras, automorphism structures and localtoglobal questions ..."
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Possible connections between whiskered categories and groupoids, many object Lie algebras, automorphism structures and localtoglobal questions