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Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional localtoglobal problems
 in Michiel Hazewinkel (ed.), Handbook of Algebra, volume 6, Elsevier
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Free crossed resolutions from simplicial resolutions with given CW Basis
, 1998
"... In this paper, we examine the relationship between a CW basis for a free simplicial group and methods of freely generating the corresponding crossed complex. Attention is concentrated on the case of resolutions, thus comparing free simplicial resolutions with crossed resolutions of a group. A. M. ..."
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Cited by 4 (3 self)
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In this paper, we examine the relationship between a CW basis for a free simplicial group and methods of freely generating the corresponding crossed complex. Attention is concentrated on the case of resolutions, thus comparing free simplicial resolutions with crossed resolutions of a group. A. M. S. Classication: 18D35, 18G30, 18G50, 18G55, 20F05, 57M05. Introduction When J.H.C. Whitehead wrote his famous papers on \Combinatorial Homotopy", [25], it would seem that his aim was to produce a combinatorial, and thus potentially constructive and computational, approach to homotopy theory, analogous to the combinatorial group theory developed earlier by Reidemeister and others. In those papers, he introduced CWcomplexes and also the algebraic `gadgets' he called homotopy systems, and which are now more often called free crossed complexes, [5], or totally free crossed chain complexes, [3]. Another algebraic model for a (connected) homotopy type is a simplicial group and again, there, one...
GROUP COHOMOLOGY WITH COEFFICIENTS IN A CROSSEDMODULE
, 902
"... Abstract. We compare three different ways of defining group cohomology with coefficients in a crossedmodule: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossedmodule is braided and the case where t ..."
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Abstract. We compare three different ways of defining group cohomology with coefficients in a crossedmodule: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossedmodule is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossedmodules and also prove the “long ” exact cohomology sequence associated to a short exact sequence of crossedmodules and weak morphisms. Contents
GROUP COHOMOLOGY WITH COEFFICIENTS IN A CROSSEDMODULE
, 902
"... Abstract. We compare three different ways of defining group cohomology with coefficients in a crossedmodule: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossedmodule is braided and the case where t ..."
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Abstract. We compare three different ways of defining group cohomology with coefficients in a crossedmodule: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossedmodule is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossedmodules and also prove the “long ” exact cohomology sequence associated to a short exact sequence of crossedmodules and weak morphisms. Contents
GROUP COHOMOLOGY WITH COEFFICIENTS IN A
"... Abstract. We compare three different ways of defining group cohomology with coefficients in a crossedmodule: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossedmodule is braided and the case where t ..."
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Abstract. We compare three different ways of defining group cohomology with coefficients in a crossedmodule: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossedmodule is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossedmodules and also prove the “long ” exact cohomology sequence associated to a short exact sequence of crossedmodules and weak morphisms. Contents
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.