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Pseudo simplicial groups and crossed modules

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In this paper, we define the notion of pseudo 2-crossed module and give a relation between the pseudo 2-crossed modules and pseudo simplicial groups with Moore complex of length 2. | Turk J Math 34 (2010) , 475 – 487. ¨ ITAK ˙ c TUB doi:10.3906/mat-0807-20 Pseudo simplicial groups and crossed modules ˙ Ak¸ca and S. Pak I. Abstract In this paper, we define the notion of pseudo 2-crossed module and give a relation between the pseudo 2-crossed modules and pseudo simplicial groups with Moore complex of length 2. Key Words: Crossed modules, Pseudo simplicial groups, Moore complex. 1. Introduction Simplicial groups occupy a place somewhere between homological group theory, homotopy theory, algebraic K-theory and algebraic geometry. In each sector they have played a significant part in developments over quite a lengthy period of time and there is an extensive literature on their homotopy theory. Crossed modules were introduced by Whitehead in [15] with a view to capturing the relationship between π1 and π2 of a space. Homotopy systems (which would now be called free crossed complexes [5] or totally free crossed chain complexes [3], [4]) were introduced, again by Whitehead, to incorporate the action of π1 on the higher relative homotopy groups of a CW -complex. They consist of a crossed module at the base and a chain complex of modules over π1 further up. Conduch´e [6] defined the notion of 2-crossed module, as a model of connected 3-types and showed how to obtain a 2-crossed module from a simplicial group. Inasaridze(c.f. [8],[9]) constructed homotopy groups of pseudosimplicial groups and nonabelian derived functors with values in the category of groups. In this paper we analysis the low dimensional parts of the Moore complex of a pseudosimplicial group. We prove that the category of crossed modules is equivalent to the category of pseudosimplicial groups with Moore complex of length 1. We extend this result to 2-dimension by defining pseudo 2-crossed modules and give the relation between the category of pseudo 2-crossed modules and the category of pseudosimplicial groups with Moore complex of length 2. The above theorems, in some sense, are well .