Results 1 
9 of
9
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 ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
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...
Higher Dimensional Peiffer Elements In Simplicial Commutative Algebras
, 1997
"... Let E be a simplicial commutative algebra such that En is generated by degenerate elements. It is shown that in this case the n th term of the Moore complex of E is generated by images of certain pairings from lower dimensions. This is then used to give a description of the boundaries in dimension n ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Let E be a simplicial commutative algebra such that En is generated by degenerate elements. It is shown that in this case the n th term of the Moore complex of E is generated by images of certain pairings from lower dimensions. This is then used to give a description of the boundaries in dimension n  1 for n = 2, 3, and 4.
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. ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
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...
Freeness Conditions for 2Crossed Modules of Commutative Algebras
, 1998
"... In this paper we give a construction of free 2crossed modules. By the use of a `stepbystep' method based on the work of Andr'e, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. This involves the introduction of ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
In this paper we give a construction of free 2crossed modules. By the use of a `stepbystep' method based on the work of Andr'e, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. This involves the introduction of the notion of a free 2crossed module of algebras. Keywords: Free 2crossed modules, free simplicial algebras. A M S Classification: 18D35 18G30 18G50 18G55. Introduction Andr'e [?] uses simplicial methods to investigate homological properties of commutative algebras. Other techniques that can give related results include those using the Koszul complex. Any simplicial algebra yields a crossed module derived from the Moore complex [?] and any finitely generated free crossed module C ! R of commutative algebras was shown in [?] to have C ¸ = R n =d( V 2 R n ); i.e. the 2 nd Koszul complex term modulo the 2boundaries. Higher dimensional analogues of crossed modules of commutative a...
DOLDKAN TYPE THEOREMS FOR nTYPES OF SIMPLICITIAL COMMUTATIVE ALGEBRAS
, 1998
"... A functor from simplicial algebras to crossed ncubes is shown to be an embedding on a reflexive subcategory of the category of simplicial algebras that contains representatives for all n types. ..."
Abstract
 Add to MetaCart
A functor from simplicial algebras to crossed ncubes is shown to be an embedding on a reflexive subcategory of the category of simplicial algebras that contains representatives for all n types.
Homotopical Aspects of Commutative Algebras
, 2006
"... This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications. 1 ..."
Abstract
 Add to MetaCart
This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications. 1
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 ..."
Abstract
 Add to MetaCart
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
THREE CROSSED MODULES
, 812
"... We introduce the notion of 3crossed module, which extends the notions of 1crossed module (Whitehead) and 2crossed module (Conduché). ..."
Abstract
 Add to MetaCart
We introduce the notion of 3crossed module, which extends the notions of 1crossed module (Whitehead) and 2crossed module (Conduché).